AN EFFICIENT ALGORITHM FOR THE ELEMENT VALUES OF MID-SERIES AND MID-SHUNT LOW - PASS LC LADDER NETWORKS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY CHIN CHENG LIN 1968 ;._ .‘ h '91" '5' .11. \ w r,‘ ‘10-. "Wls ' +4 13'...“ This is to certify that the thesis entitled An Efficient Algorithm for the Element Values of Mid-Series and Mid—Shunt Low-Pass LC Ladder Networks presented by Chin Cheng Lin has been accepted towards fulfillment of the requirements for ELL degree in __E_:_E_ ‘— / Majmyésor Date November 19, 1968 0—169 egg WI; « nu. um. ABSTRACT AN EFFICIENT ALGORITHM FOR THE ELEMENT VALUES OF MID-SERIES AND MID-SHUNT LOW-PASS LC LADDER NETWORKS by Chin Cheng Lin In this thesis, an algorithm for the computation of element values of mid-shunt or mid-series low-pass ladder networks is given. The algorithm is based on the network interpretation of the Darlington formulas described in a recent article by Amstutz. It is shown that Fujisawa's procedure can readily be included in the algorithm. Thus, starting with an arbitrary attenuation pole sequence, the computer can find a realization with positive element values and a rearrangement of the sequence into a proper one. Compared with conventional methods, this algorithm gives more accurate results for ladder networks with large numbers of sections and it is insensitive to the change of variable used in filter synthesis to overcome the accuracy problem. As an application of the algorithm, the synthesis of inverse Chebyshev filters are considered. Some useful properties and a design procedure for the inverse Chebyshev filters are established. Besides the practical application of the design method, the established properties are quite general and may be used for any low-pass ladder filters. The group delay characteristics of the inverse Chebyshev filters are also studied thoroughly. The last part of this thesis considers a new approach to the synthesis of mid-shunt low-pass LC ladder networks by using state equations. Some common properties of the coefficient matrices of the state equations are established. Interpretation of the relations that exist between functions defined in insertion-loss theory and the Armatrix of a state model are given. The formulation of the problem is such that it suggests some possible synthesis procedures. Although no general method for such synthesis is given, the results obtained from the analysis, and two suggestions for-the solution of the synthesis problem are useful for further study in this area. AN EFFICIENT ALGORITHM FOR THE ELEMENT VALUES OF MID-SERIES AND MID-SHUNT LOW-PASS LC LADDER NETWORKS By Chin Cheng Lin A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1968 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to his major professor and thesis advisor Dr. Yilmaz Tokad for his counsel and guidance throughout the program. He also wishes to thank the other members of his Guidance Committee for their interest in this work: Professors Richard C. Dubes, Harry G. Hedges and in particular Dr. J. S. Frame, who suggested improvements and Dr. Gerald L. Park, who became the thesis advisor during the sabbatical leave of Dr. Tokad. Thanks is also due to Mr. Ronald L. McNally for the offer of his optimization program which is used in carrying out the computation of Example 4.3-1 in Chapter IV. The author especially wishes to express his gratitude to the Division of Engineering Research of Michigan State University for the financial support of this research. ii TABLE OF CONTENTS Page MTERI INTRODUCTION ......OOOOOCOOOOOOOOI......OOOIOOOOOOOOO 1 CHAPTER II REALIZATION OF DRIVING-POINT IMMITTANCE INTO Low-PASS MDDER NENORKS 000............OOOCOOOOOO... 7 2.1 General ......OOOOO...O.......OOOOOOOOOOOOOOOOO.....O 7 2.2 Mid-shunt and Mid-series Ladder Network Synthesis ... 7 2.3 Realization of Yd with a Prescribed Attenuation Pale sequence ......0.000............OOCOOOOOOOIOI... 15 2.4 Realization of Y Starting with an Arbitrary Attenuation Pole Sequence ........................... 23 2.5 Numerical Examples .....OOOOOIOOOOOOOOOOO0.00.0000... 25 2.6 Some Properties of the Algorithms ................... 29 CHAPTER III FILTERS WITH INVERSE CHEBYSHEV ATTENUATION CHARACTERISTICS 0.00.00.00.00.......CCOOOIOCO0.00.... 31 3.1 Introduction ........................................ 31 3.2 Insertion-loss Theory ............................... 31 3.3 Synthesis of Inverse Chebyshev Filters .............. 36 3.4 Design Procedure and Numerical Examples ............. 52 3.5 The Group Delay Characteristics ..................... 59 CHAPTER IV STATE EQUATIONS APPROACH TO THE SYNTHESIS OF LOW-PASS LC LADDER NETWORK sosoosssosoosssooossosooss 67 4.1 Introduction ........................................ 67 4.2 State Equations for Low-pass Mid-shunt Ladders ...... 69 4.3 Synthesis Methods ................................... 77 CHAPTER V CONCLUSIONS ......................................... 85 APPENDIXI 000............OOOOOOOOOOOOOO......OOOQOOOOOOOOOOOOOO 86 BIBLIOGRAPHY 0.00.00.........OOOOOOOOOO.........OOOOOOOOIOOOOOOOOO 87 iii Figure I-l I-2 2.2.1 2.2.2 2.3.1 2.3.2 2.3.3 3.2.1 3.3.1 3.3.2 3.3.3 3.4.1 3.4.2 3.5.1 3.5.2 3.5.3 4.1.1 4.3.1 4.3.2 LIST OF FIGURES Filter Network 0................OOOOOOOOOOOOO.......OOOOOOO Mid-shunt and Mid-series LC Low-pass Networks ............. The First Step in the Fujisawa Procedure .................. Reactance Curve Used in Zero Shifting for the FUJisawa Procedure .0......OOOOOOOOIOOOOO0..........COCOOCO Mid-Shunt Ladder Networks 00.0.0.0.........OOOOOOOOOOOOOOOO Equivalent Networks for the Illustration of the Property Proved in Appendix I ............................. Figures Used in the Proof of the Algorithm ................ Definition of the Insertion Voltage Ratio ................. Specification of Attenuation Characteristics .............. Zeros of the Characteristic Polynomials for Butterworth and Chebyshev Filters IO..OOOOOOOOOO......OOOOOOOOOOO0.0... Zeros of the Characteristic Polynomial for an Inverse ChebYShev Filter 00.0.00.........OOOOOCOOOOOOOOOOOO Graph Used for Determining the Design Parameters for the Inverse Chebyshev Filters ......................... Selection of Design Parameters by Graphical Method ........ Group Delay Characteristics for Inverse Chebyshev Filters (0-105, wa-l.2, and n-7,ll,15,19,23) .............. Group Delay Characteristics for Inverse Chebyshev Filters (n-ll, wa-1.2, and D-105,3XI05,SX105,7XI05)........ Group Delay Characteristics for Inverse Chebyshev Filters (nail, 0-105, and wa-l.l,l.2,l.3,l.4) ............. A Simple Ladder Network and Its Graph ..................... One-section Mid-shunt Ladder Network ...................... TWO‘Section Mid-Shunt Ladder Network osssssosssosossssssoso Networks Used in Proving the Property Given in Appendix 1.. iv Page 11 14 16 19 20 32 37 4O 42 53 55 65 66 66 68 79 80 Table 303-1 30 3-2 3.4-1 4.2-1 LIST OF TABLES Page Some Preperties of the Chebyshev Polynomials .............. 38 The Degrees of the Numerator and the Denominator Polynomials for Terminated Mid-shunt LC Ladder Networks ... 47 Critical Values of Am for Inverse Chebyshev Filters ..... 56 in Submatrices Appearing in the State Equations for Three Cases of Mid-shunt Ladder Networks .................. 72 CHAPTER I INTRODUCTION In the synthesis of filter networks, on the basis of insertion-loss theory, the final step is the realization of the driving-point immittance function as either an impedance function, Zd(s), or an admittance function, Yd(s), which is seen from the input port 1-1' as indicated in Fig. I-l. Because of practical considerations, it is always preferable to realize the filter network in a ladder form. For low-pass filters, the most important forms are the mid-shunt and the mid-series ladders shown in Fig. I-2 in which “a is the resonant frequency of a mid-shunt k branch or a mid-series branch. Since at each wak the corresponding branch becomes open circuited for the mid-shunt ladder and short circuited for the mid-series ladder, no signal can be transmitted at these frequencies. For this reason the wwk's are called the transmission zeros or the attenuation pole frequencies. The establishment of the explicit formulas for the element values of a low-pass LC ladder filter from a given driving-point immittance function and a set of transmission zeros, is one of the outstanding problems in filter synthesis. Even in the simple LC ladder, where all transmission zeros occur at infinity, i.e., when in Fig. I-2 Pk's (or ?k'8) are zeros, such explicit formulas are available only for special kinds of Butterworth and Chebyshev filters [BEN], [N0], [TA], [BE], [OR], [GR], [WEI], [NAl]. For non-simple LC low-pass ladder filters, i.e., when the wk's are finite, Darlington [DA] has given explicit formulas for the element values in terms of three distinct determinants, based on the R1 1 2 '—*’ LC Filter 1?.2 1' 2' -Zd(s) or Yd(s) Fig. I-l L 1 k 2 r k ”wk - lA/kak (a) Mid-shunt LC low-pass ladder network 1 3k 2 Wk T 8 9 l: T“ I la, wak - 1/ dek (b) Mid-series LC low-pass ladder network Fig. I-2 work by Norton [NO]. In practice, however, these formulas are useful only for a few sections [ZU], [SKZ], because as the number of sections increases, one has to cope with higher order determinants which have very small numerical values. In this thesis, an attempt is made to develop new approaches to the solution of this problem which do not suffer from numerical inaccuracies. The first approach is complete and discussed in Chapter II and its application is given in Chapter III by considering a little known class of filters, namely, the Inverse Chebyshev filters. The second approach is based on the state model of low-pass ladder filters. This approach is developed in Chapter IV where the formulation of the problem is discussed thoroughly and two approaches to the synthesis problem are suggested. The contents of the various chapters may be briefly outlined as follows: In Chapter II, an algorithm is given through which all the element values of the non-simple low-pass ladder networks (or their duals) in Fig. 2.3.1 can be determined iteratively from a knowledge of some initial parameters. Two of these parameters are, in fact, the element values in the first or last sections of the filter, the other parameters being used only for the intermediate computations. This algorithm automatical- ly eliminates the difficulties encountered in the application of Darlington's formulas for complicated ladders. Such an algorithm is suggested and used, but not explicitly proved, in a recent article by Amstutz [AM]. The pair of general recursion formulas derived in this chapter are equivalent to those given for special cases in Table-II of Amstutz's paper. Because of the duality, only mid-shunt configurations are considered throughout this thesis. For the realization with positive element values, it is proved that Fujisawa's procedure [FU], [BA], [MB], [ME], [NE], [LE] can be included in the algorithm. A program was written for the computation of element values, based on the above mentioned algorithm in which one can specify the order of the attenuation poles so that the algorithm yields positive element values. If, however, the order of the attenuation poles is not specified, then the program starts with an arbitrary ordering of poles and rearranges the poles according to Fujisawa's criterion so that the algorithm produces positive element values. Examples given at the end of Chapter II demonstrate in detail the scheme of computations involved in the process for different cases. In Chapter III, the amplitude and group delay characteristics of a low-pass ladder filter with inverse Chebyshev attenuation characteristics are studied. One reason the filters with inverse Chebyshev characteristics are studied is that in the work by Beletskiy [BL], [TR], where the linear phase characteristics are studied, the attenuation characteristics turned out to be very similar to those of the inverse Chebyshev attenuation characteristics. Another reason for considering inverse Chebyshev filters is the fact that, in the literature, no details of the properties or design formulas for this class of filters have been published. In Chapter III, after a brief introduction of the insertion-loss theory, the relations that exist among filter functions for Butterworth, Chebyshev and inverse Chebyshev filters are studied. With the aid of these relations, formulas for the realization of inverse Chebyshev filters are derived which utilize the algorithm obtained in Chapter II. A design procedure, starting from the approximation of the specified attenuation characteristics, is presented with examples. In the last section of Chapter III, a general formula for the group delay function, 1(m), for low-pass filters is derived and some useful properties are established. By using this general formula, the group delay characteristics for inverse Chebyshev filters for different values of the design parameters D, ma, and n, defined in Chapter III, are obtained. The last part of this thesis considers a new method of realizing the non-simple low-pass LC ladder network with arbitrary attenuation characteristics by using state equations. In this approach, the natural frequencies and transmission zeros of the network are assumed to be given. An approach similar to this was first given by Marshall [MAl], [MAZ] for simple low-pass ladders. The essence of this approach as included in Chapter IV is as follows: Let P§(t) - RW(t) + Be(t) be the state equations for the network shown in Fig. I-l, where V is the state vector, e(t) is the input voltage function, and P, R and B are coefficient matrices. Entries in P are linear combination of network elements while the entries in R and B are equal to 1's and the terminating resistances. The forms of P, R and B depend on the network configuration. Since the eigenvalues of P-lR are the specified natural frequencies and the forms of P and R are known, the network element values can be obtained from the given natural frequencies and terminating resistances. For simple ladders, P is diagonal and R is a special kind of tridiagonal matrix. Therefore, the element values can be obtained by transforming the companion matrix constructed from the given eigenvalues into the tridiagonal form of P-lR. However, for non-simple ladders, P is tri- diagonal. Therefore, the form of P-lR is complicated and the problem cannot be solved by a simple transformation. In Chapter IV, without loss of generality, only the mid-shunt configuration is used to demonstrate this approach for synthesis. Although no general solution for the synthesis problem is obtained, a careful formulation of the problem is presented which may be used in future research work. Two possible synthesis methods are suggested together with simple examples. CHAPTER II REALIZATION OF DRIVING-POINT IMMITTANCE INTO LOW-PASS LADDER NETWORKS 2.1. General This chapter deals with the realization of driving-point impedance 16 «I. 31*“. "- .. -\ (admittance) in terms of the mid-series (mid-shunt) ladder network defin- J. —---l ed in Chapter I with positive element values, i.e., without mutual reactances. It is well known that only a certain class of driving-point fi— ‘ ——:_. immittances that satisfy the necessary and sufficient conditions found by Fujisawa can be realized in terms of mid-series or mid-shunt ladders without mutual reactances. In this chapter, without loss of generality, only the realization of driving-point immittances in terms of the mid- shunt ladder with non-negative elements is considered, since the mid- series network is the dual of the mid-shunt and their realizations are essentially the same. Two algorithms for the realization of a driving- point admittance are established under the assumption that all the attenuation pole frequencies of the corresponding filter are distinct. 2.2. Mid-shunt and Mid-series Ladder Network Synthesis d or Yd which is to be realizable as a mid-series or mid-shunt ladder network, three conditions stated in Given a driving-point immittance Z the following realizability theorem must be satisfied. Let Zd or Yd be written in the form “1+nl m2 + n2 where m1, n1 and m2, n2 are even and odd parts of the numerator and Zd(s) or Yd(s) - denominator polynomials of 2d or Yd, respectively. Let n1 - sgl, n2 - 832, where g1 and g2 are even polynomials, and M(s) - mlm2 - nln2 The zeros of M(s) are the zeros of transfer functions [VV] which are the transmission zeros of the filter shown in Pig. 1.1.1. The necessary and sufficient conditions for realizability of a given Zd (or Yd) were given by Pujisawa [PU], [MB], [BA], [WEZ] and are stated in detail in the following theorem: Realizability Theorem The Brune function Zd with n1 I 0, can be realized as the driving-point impedance of a resistance terminated mid-series low- pass ladder network if, and only if, the following three conditions are satisfied: (1) 2 has a pole or zero at infinity. d (2) M(s) is a positive constant or M(s) - Kl (.2 + si)(s2 + si)...(s2 + s3) 12 2 2 2 where n - degree of Zd and O < s1 s 82 < ...g an (3) Let m be the degree of g1 in s2 and 2 2) where G > O and the wi's are distinct. Then m a n g1(s) - C(sz + wi)(sz + w ...(s2 + mi) and for each a: there exists at least i wfi's which are not greater than si. Note 1. In the special case when n1 5 0 the above three necessary and sufficient conditions reduce to the necessary and sufficient condition that Note 2. The necessary and sufficient conditions for the realization of a given driving-point admittance function as a mid-shunt ladder network terminated in a resistor are the same except in condition (3) g1 is replaced by g2. Note 3. If in the Realizability Theorem 2 is replaced by Y then the d d term "mid-series" should be replaced by "mid-shunt". The Realizability Theorem is important in low-pass ladder filter synthesis and can be interpreted as follows. Let yll’ y22, y12 and 211’ 222, 212 be the short-circuit and open-circuit parameters, respectively, for the 2-port LC network, then from Case A (low-pass ladder networks belong to this case) of Darlington's synthesis [BA] we have: z . £1.21 . 3.2—(11.211. 211 1 +(1/y22) (2 2 1) d m2 + n2 n2 1 + (m2/n2) 1 + z22 where the short-circuit admittance function y22 - Y corresponds to Zs the reactive network looking from the output port of the filter with the input port short-circuited. The conditions of the Realizability Theorem simply say that for Z to be realizable as a mid-series ladder network d with all positive element values, at the i-th transmission zero of the mid-series ladder network, Y must have at least i poles on the imaginary Zs axis with magnitudes not greater than that of the transmission zero. For the mid-shunt case 222 should replace y22 in the above state- ment. A similar interpretation can be given when 2 is replaced by Y d d' The necessity of the condition (3) interpreted above becomes evident if one looks at the zero-shifting procedure utilized in ladder network synthesis. There will not be a sufficient number of zeros to be shifted to generate the Specified transmission zeros if condition (3) is not satisfied. 10 In the proof of sufficiency of the Realizability Theorem Fujisawa described a procedure based on a very simple criterion which is applied at each step of the conventional zero-shifting procedure used in the synthesis of a ladder network [VV]. This procedure by several authors is sometimes referred to as "Fujisawa's procedure". A driving-point immittance which satisfies the conditions of the Realizability Theorem, in general, may have more than one realization with all positive element values. This is due to the multiple choices of ordering the transmission zeros used in the realization procedure. Fujisawa's procedure, however, corresponds to one of these choices of ordering and will always give a realization whenever the given driving-point immittance satisfies the realizability conditions. The algorithm to be shown in section 2.4. is based on Fujisawa's procedure which is given together with its network interpretation in the following. Let Yd be the given driving-point admittance function satisfying the realizability conditions, then Y can be realized as a mid-shunt d ladder network without any negative element values by the following procedure: Step 1. (A) If Yd(s) has a zero at s-w, remove a series inductance L from Yd as shown in Fig. 2.2.1-(a). For this case 1/Yd has a pole at infinity with the residue L: l/ld - l/Yd - Ls (B) If Y has a zero s-jw which coincides with one of the d transmission zeros, remove a parallel resonator as shown in 11 Fig. 2.2.1-(b). For this case l/Yd has a pole at s-jwk with the residue Lui/Z: 2 ka 2 s + m 1/'\3'd - 1/Yd - R‘s: rd_.,, r. LC R Y d_.,“ r. LC R G I ‘ C l _____J yd Qd (a) (b) Fig. 2.2.1 Step 2. The next step is to carry out the zero-shifting procedure by removing a shunt capacitance C determined by the following relation: 1 c1 - min.[Yd(jm1)/jw1,..., Yd(jmn)/jwn, sigma/5)] where jw ,..., jw are the transmission zeros. In determining the l n minimum element of the set of numbers, all negative or infinite values are ignored. If it is found that C (jmk)/jwk, then the first section of the 1'Yd mid-shunt ladder to be realized has shunt capacitance C and a mid-shunt 1 arm Ll-Pl (see Fig. 2.3.1 in the next section) with resonance frequency equal to wk. Then the same procedure is applied to the new driving-point admittance obtained from the removal of the section (C1, L P1) until 1’ all the transmission zeros are realized. Darlington [DA] gave a pair of formulas for computing the values of 12 the elements C1 and L1 as follows: C1 - F(zk) (2.2-2) d L1 - -1/[-d-; F(2)]z_zk where F(z) - [Yd(jw)/jw]z.1/w2 and 2k - l/w: In the above discussion realization of the 2-port LC network is accomplished by the consideration of Y But this is not necessary. d. Indeed, one can always realize the same network by using short-circuit admittance functions Y18 and Y28 obtained from Yd. This is preferred since Y18 and Y28 have much simpler forms as compared to Yd. On the other hand, realizing the network from both Y and Y28 yields a check Is on the accuracy of the element values. In order to demonstrate the validity of the procedure for Y18 and Y28, let yll’ y22’ y12 be the short-circuit admittance functions for the LC ladder network. Therefore Y - y - y2 /(1 + y ) d 11 12 22 Note that, u.c_ : 2: 2 Yd yll 2"125'12“1 I ’22) + 3'123'22“1 + ’22) Since the transmission zeros are the zeros of y12, it follows from the last two equations that Yd(jwk) - y11(1wk) - Y18(ka) and Yé(jwk) - Yi8(jwk) 13 Since in Darlington's formulas 61 and L1 are expressed in terms of Yd and Yd’ this justifies the use of Y A similar conclusion is valid for Y 3 instead of Y in the realization. 1 d 23' Before giving a network interpretation to the Fujisawa's procedure, it is necessary to show first that L --l/F'(zk) is always positive for 1 any zk. This property can be proved as follows: Without the transforma- tion of the variable z-l/wz, Darlington's formula for L1 is L - -2/['s31 l s-jw k or €E-- t-s3 Yls(jw)/jw , which implies 2/L1 > 0, and the property, L1 > O. In fact this, in turn, also justifies Fujisawa's procedure: Let the reactance curve of Y18 be as shown in Fig. 2.2-2 in which m's, i-1,2,3, i are transmission zeros and ei-tan-llYls(jw1)/jw1] Fujisawa's procedure is to find C1 from the condition C1 - min.(tan61, tane3, tanem) From Fig. 2.2.2 , one can see that Cl-tan63. Note that tane2 < 0 must be ignored because that corresponds to negative capacitance. If tanel, rather than tane is used for C1, then the difference of the Y and the line ls is not a reactance curve anymore. Therefore, the first 3 with slope tane1 transmission zero to be realized is w3. l4 Y18(jw) i I ”f l V.- M‘GL 2 1 3 in 0 :2 ‘m‘I ( m 4F---- - L for k-l,2,..., n (2.3-4) Lk1 - -1/F'(ck) then it will be proved that C and L1 (i-l,2,..., n) can be generated by i the following algorithm which is written in the form used by Wendroff [WD] where the abbreviations Pji - (cj - c1_1) Bji ' Cj,i-l ' Ci-1,i-l Dji ' P31 + BjiLi-l,i-l are used: For i-l (the first column) For k-1,2,sss, n Ck1 - F(ck). Lk1 - -1/F'(ck) For i-2,3,..., n For j-i, i+l,..., n cji - PjiBji/Dji (2.3-5) 2 ) - 311L1_1’1_1] (2.3-6) 2 _L31 - ”3.1/“P — 2 JilLJ.i-l 18 From this the element values are obtained as follows: i ii’ Li - Lii and r1 3 Ci/Li for 1.1,2’...’ n For example, for a 4-section ladder network with the sequence corresponding to the specified sequence of attenuation pole frequency, the steps involved in the scheme of computation can be put into a clearer form as follows: :1 c2 c3 :4 C1 (Cll’Lll c2 (CZl’L21) (czz'Lzz) C3 (C31'L31) (C32’L32) (633'L33)::: ‘4 (C41’L41) (Caz'Laz) (caa’Las) (C44’L44) In the above array arrows indicate how each pair of elements is obtained from two other pairs of already computed elements by the use of iterative formulas (2.3-5) and (2.3-6). For each pair (C L11) four ii’ parameters computed in the previous step are required, two of which called C1_1’1_1 and L1_1’1_1, are the actual element values and the other two, C and L1 1_1, can be considered as the element values if in the D i,i-l pole sequence the poles Ci-l and C1 are interchanged. Therefore, for an n-section ladder the total number of parameters that should be computed in the first step, by using Darlington's formulas in (2.3-2) is 2n, and the total number of computations needed is n(n+l), among which 2n appear in the leading positions of the array which give the actual element values and the remaining n(n-l) are parameters used for intermediate computations. In order to prove the algorithm the following property is needed 19 whose proof is given in Appendix I at the end of this thesis. Let the one-port networks N1 and N2 in Fig.2.3-2 be obtained from two non-simple LC ladder network terminated by identical resistances. With the identical admittance functions, let N1 and N2 have identical parts indicated by N. Then Y1(s)-Y2(s), if and only if Yr1(s)EYr2(s). Fig. 2.3.2 For the proof of the algorithm, let us assume that Y18 is realized for two distinct attenuation pole sequences as shown in Fig. 2.3-3. Note that both sequences are identical up to the (k-l)-th element, but differ in that the k-th element and the (k+l)-th element are interchanged, whereas the rest of the elements in both sequences may have arbitrary orders. If we let Fr1(c)-Ck+ Lk #r— 1 Cl:- c + ° (2.3-7) and 20 Fig. 23.3 21 N Fr2(C) ' Ck + ~ Ck+l ' C 8 + (2.3-8) then according to the above theorem Fr1(c)EFr2(c). Therefore, for C-Ck+l we have 1 c + - 8 k Lk 1 k c{CH1 Ck+l (Ck+l-Ck)(ek-Ck) (‘k+1"k) + (Bk-Ck)Lk 01' Ck+1 This is Eq.(2.3-5) with different index notations. a v . ..mJ-duum (2.3-9) To prove the iterative formula (2.3-6) the relation F;1(c)EF;2(;) is to be used. Applying Darlington's formulas, given in (2.3-3), to Eq. (2.3-8), we have V - _ N Fr2(‘k+1) l/Lk From the derivative of Fr1 in Eq.(2.3-7), we have 1 . : Frl(‘k+1) {3%k+1Fr1(‘) Lk 1 Lk 1 2 - -[ 2 + -3-]/sz: - '+'5--] (ck- Ck+1) ck+l k k+l k+l I - ' Then from the relation Frl(§k+l) Fr2(ck+l)’ we have ck+lLk Ck+l ‘ Ck 2 2 l ’ chk+l/(Ck+l ’ Ck) 22 Substituting C of equation (2.3-9) into the above equation gives k+l (c - t )2 l k+l k 2 2 37- - [—-—-—;——— - Lkdfk - Cr) mum - ck) + (8k - cpnk] k+l Lk or finally 2 L - [em - ck) + (3, — Ck)Lk] k+l 2 ~ 2 (cm - ck) n, - (Bk - ck) Lk which is identical to that in Eq.(2.3-6) with different index notations. This proves the algorithm. Needless to say, if instead of Y18(s), Y28(s) is used for the computation of element values, the algorithm proved above is also valid. When st is used that means we are realizing the LC ladder network from the output end, therefore, in order to obtain the realization identical to that of Yls the pole sequence in reverse order should be used. The algorithm as given, when applied to compute the element values in the network of Fig. 2.3.1-a can not give the last element in the apposite end, i.e., it misses C when Y is used and C can not be n+1 ls 1 obtained when st is used. But since Yd is given it is obvious that Cn+1 and C1 can always be obtained separately. In actual ladder network synthesis, in order to check the accuracy, it is usual practice to realize both Y18 and Y28 and compare the element values obtained in both computations. The reason is that, because of the subsequent algebraic sum operations in iterative computations the errors will accumulate as the computation proceeds and cause a loss of significant digits. If both Y18 and Y28 are realized then all the element values can always be obtained. 23 For the network of Fig. 2.3.1-b which exists when the degree of Yd is even (see Table 3.3-2), the same algorithm is valid without any change. This network can be considered as the same network in Fig. 2.3.1-a with -0 and C -0, and for this case, since C -0, realization of Y Cn+l n+2 n+2 ls yields all the element values. 2.4. Realization of Yd Starting with an Arbitrary Attenuation _‘—‘-: Pole Sequence As mentioned in section 2.2, not all attenuation pole sequences < wj>;-1's can yield positive element values in the realization of associated driving-point admittance functions Yd, even though Yd satisfies the realizability conditions. In this section, it is shown that Fujisawa's procedure can readily be included in the algorithm described in the preceding section. Thus, starting with a sequence < cj>;_1, -1/m2, corresponding to a pole sequence < w selected arbitrarily J n ‘3 J ’3-1 from the set {mi}:.1, the computer can find a realization with positive element values by rearranging the given sequence into a preper one in the following manner: At any step of the algorithm, kti, i+l,..., n, is Cki’ computed for the remaining poles, c1, c1+1,...,§n, then the non-negative minimum element, Cm -min.{C >0, k-i, i+l,..., n} is determined. Inter- i ki changing (C Lmi) with (C and Cm with :1, also interchanging mi’ ii’ Lii)’ those elements in the same rows computed previously, i.e., interchanging (C Lmt) with (C Lit) for l-l,2,..., i-l a rearranged array is mi ’ il’ obtained. Cmi's always exist whenever Y conditions given in section 2.2. d satisfies the realizability By referring to section 2.2 and from the proof of the algorithm it is easily seen that the above procedure is equivalent to applying 24 Fujisawa's procedure in the conventional zero-shifting synthesis method. As is shown in section 2.2, when Fujisawa's procedure is applied, at any step i, the Lki’ for k-i, i+l,..., n, are always positive. Negative element values, if such exist, will be among the Cki’ k-i, i+l,..., n, and the procedure is equivalent to the Operations stated in the preceding paragraph. The algorithm, including Fujisawa's procedure, is as follows: For column 1 (1-1) For k-l,2,..., n L_' ck1 - F(ck), Lk1 - -l/F'(Ck) Find the minimum non-negative element, le; interchange le and Cll’ Lml For i-2,3,..., n and L11, also Cm and c1. _ For j-i, i+l,..., n C B /D 11 ' P3111 11 2 2 2 L ' Dji/[(Pji/Lj,i-l) ’ BjiLi-l,i-l] ji it..— Find the minimum non-negative element Cmi' For k-l,..., i Interchange ka and Cik; Lmk and Lik . Interchange Cm and :1. From this procedure the element values and the attenuation pole sequence < C1>:-1 are obtained as follows: ci-cii, Li-Lii, riaci/Lii’ i-1,2,..., n and n n ‘ ‘i’i-l'< Liri>i-l 25 It should be emphasized, therefore, in both algorithms, that with or without Fujisawa's procedure, the expression for Y18 or Y28 is needed only for the computation of Zn parameters, the use of Darlington's formulas in (2.3-3). can be generated by the algorithm using these 2n parameters. 2-5: Numerical Egggples Ckl and Lkl’ k-l,2,..., n, by All other element values Both examples given in this section are constructed from tables obtained by Saal and Ulbrich [80]. The degree of the driving-point admittance, Yd(s), for the first example is even which corresponds to the ladder of the type shown in Fig. 2.3.1-b. according to the pole sequence given in the table. example, the degree of Y d The filter is realized For the second is odd and Yd is realized into a ladder of the type shown in Fig. 2.3.1-a both with and without application of the Fujisawa criterion. Exagple 2.5.1 For the element values of the filter with specification C-08-20c, 9-85', the coefficients of the numerator and denominator polynomials of Yd(s) are found as follows: (sk) k OHNw§UIO|Nm Numerator 2.6362218751+000 2.3857211540+OOO 9.1942771683+000 6.9928022992+OOO l.1488488046+001 6.794736526l+000 5.9301643998+000 2.1875999999+000 1.0000000000+000 From these calculations Y Is and Y 2 Denominator 0.0000000000+000 2.3863732874+000 2.1596138348+OOO 6.9946429436+000 5.2927668720+000 6.7963234209+000 4.1329735998+000 2.1880000000+000 l.OOOOOOOOOO+OOO s can be obtained. Applying the algorithm to Y C C 13 1:! Cs reverse order we have: C c11 11 11 ls 7.070395-001 5.0010000000-001 -8.6405977862-001 ij -l.5500865651-001 1.1046980324+OOO 6.8050000000-001 1.6914529924-002 1.3506378822-001 1.6263605667+OOO 1.0390000000+000 5.8284368790+001 6.9999695143+OOO 0.0000000000+OOO From the above array, element values with 4-digit C1-5.001 C2-0.5856 C3-O.3525 L1-0.6805 L2-0.1267 L3-O.2758 F1-1.039 T2-7.781 P3-3.428 26 9.454424-001 5.8560000000-001 7.5304195949-001 1.4460739189+OOO 1.2670000000-001 3.3406518193-001 l.49052834l6+000 7.7BlOOOOOOO+OOO 2.8301135561+OOO 0.0000000000+OOO can be written as follows: the following array is obtained: 9.858527-001 0.000000+OOO 3.5250000000-001 9.6746222531-001 7.4940000000-001 2.7580000000-001 1.3772334767+OOO 1.1050000000+OOO 3.4280000000+OOO 0.0000000000+OOO 0.0000000000+OOO accuracy, except C4-0.7494 L4-1.105 P4-0.0000 Applying the same algorithm to Y28, using the pole sequence in 0.000000+OOO 0.0000000000+OOO 6.0373407151+000 l.l305337200+000 -l.4l37388487+000 1.1050000000+000 3.5665232712—003 1.3061351187-002 1.6240692289-001 0.0000000000+OOO 2.6508796610+002 7.5478615182+001 4.3535059184+000 9.454424-001 7.4940000000-001 4.9865486101-001 l.l688927606+000 2.7580000000-001 6.8434818073-002 8.4093963802-001 3.4280000000+OOO 1.4405718138+001 8.4077318756-001 9.858527-001 7.070395-001 3.5250000000-001 8.1501834776-001 5.8560000000-001 1.2670000000-001 7.9028547008-001 6.8050000000-001 7.7810000000+OOO 8.9466341818—001 1.039000OOOO+OOO 27 Here C r 1, i=1,2,3,4 are the element values when labeled from 11’ L11’ 1 the output end except C Both realizations give identical element values. 1. Exapple 2.5.2 In this example, the filter is with specifications C-09-20, 6-85‘. The coefficients of Yd(s) are: (sk) k Numerator Denominator 9 3.6604045718+000 0.0000000000+000 8 2.6837214038+000 2.6836561893+000 7 l.413l69098l+001 1.9239848286+000 6 9.0626489556+000 9.0623464387+000 5 2.0345308683+001 5.3996189587+000 4 1.1069134057+001 1.1068719167+001 3 l.2934901904+001 5.0303220756+000 2 5.6902015400+000 5.6900239700+OOO 1 3.0609000000+000 1.5547000000+000 O 1.0000000000+000 l.0000000000+000 Applying the algorithm to Y we obtain: C 4.756319-001 9.6880000000-001 C -2.5387082592-001 ij -8.5808331031-001 3.2801247137-001 9.4110000000-001 L 6.7614671644-002 ij 1.1586329922-002 3.2830884751-001 F11 5.0540000000-001 From Y we have: 23 C 8.716365-001 3.2810000000-001 -8.593409ll74-001 ij -2.5407984338-001 9.6881200799-001 C Is 9.722537-001 9.2840000000-001 7.8248187472-001 1.2256282291+OOO 1.9090000000-001 7.3174915806-002 4.6983947323-001 5.0930000000+OOO 9.912000-001 5.2530000000-001 6.4717406874-001 1.3666191216+OOO 9.912000-001 3.1030000000-001 6.8160375866-001 5.2530000000-001 9.4400000000-002 4.1114083662-001 3.2830000000-001 l.0500000000+001 2.6550000000+OOO 9.722537-001 3.1030000000-001 9.9451958949-001 9.2840000000-001 8.716365-001 4.756319-001 3.2830000000—001 1.1559163379-002 6.7540355836-002 9.4123135087-001 1;] F11 2.6SSOOOOOOO+OOO As in Example 1 the diagonal elements 28 9.4400000000-002 2.1212508419-001 1.9090000000-001 1.0825726775+OOO l.0643853284+000 9.4110000000—001 l.OSOOOOOOOO+OOl 5.093000OOOO+OOO 5.0540000000-001 the realized element values can be written from of the above arrays as follows: -Ti-Wfii.‘ c1-0.9638 c -o.9234 c3-o.3103 “4'0'5253 05-0.3281 L1-0.94ll L -o.1909 13-0.0944 L4-0.3283 fl rl-o.5054 r -5.093 r3-1o.50 r4-2.655 For the same filter, applying the algorithm with Fujisawa's procedure to Yls C 8.716365-001 3.2801247137-001 -8.5808331031-001 -2.5387082592-001 9.6880000000—001 cij 3.2830884751-001 Lij 1.1586329922-002 6.7614671644-002 9.4110000000-001 F11 2.6549284511+000 For st c 4.756319-001 9.6881200799-001 -2.5407984338-001 -8.593409ll74-001 3.2810000000-001 C11 we have: gives: 9.912000-001 9.722537-001 4.756319-001 5.2554323391-001 6.4750404909-001 1.3670045022+OOO 3.1048753397-001 9.9457681501-001 9.2804475276-001 9.4326307134-002 2.1199179497-001 1.0826306257+OOO 1.9083349449-001 1.0645244895+000 9.4123135087-001 l.0508203174+001 5.0947749116+OOO 5.0532947034-001 9.722537-001 9.912000-001 8.716365-001 9.2804475276-001 7.8209332515-001 1.2254063320+OOO 3.1048753397-001 6.8199785079-001 5.2554323391-001 29 9.4123135087-001 Lij 6.7540355836-002 1.9083349449-001 1.1559163379-002 7.3168006653-002 9.4326307134-002 3.2830000000-001 4.6964132660-001 4.1104882472-001 3.2830884751-001 F11 5.0532947034-001 5.0947749ll6+000 1.0508203174+001 2.6549284511+000 From the above arrays, element values with 4-digit accuracy, are obtained as follows: C ="0.3280 C -O.5255 C -O.3105 C -O.9280 C -O.9688 l 2 3 4 5 L1-0.3283 L2-0.0943 L3-0.1908 L4-O.9412 F1-2.655 F2-10.51 P3-5.095 F4-0.5053 Note that in terms of ci's in the order §l< §2< c3< :4, the original attenuation pole sequence is now changed into in accordance with Fujisawa's procedure. Incidentally, in this particular example the pole sequence determined by Fujisawa's procedure is the reverse of the original one and the element values come out to be close to those found when the filter is reversed. What is observed above of course does not happen in general. 2.6. Some Ppgperties of the Algorithms Although a precise error analysis is not considered, the simplicity of the algorithm requires fewer algebraic sum operations than with the conventional zero-shifting procedure. Since the algebraic sum operations are the main cause of accumulation error it is expected that by using the algorithms more accurate results can be obtained. In the examples given in this chapter driving-point admittances are constructed from the 4-digit table given by Saal and Ulbrich [SU]. Thus errors are introduced during the computation of coefficients of polynomials for the driving- 30 point admittance which involves many algebraic operations. Nevertheless, the experience gained through several computations showed that the ll-digit single precision computation is sufficient for 4-digit accuracy element values even for the most complicated 5-section filter given in the table. One interesting property of the recurrence formulas used in the algorithms is that they remain identical for the case where a transforma- tion of variable is applied in improving the accuracy problem [52], [BI]. Only the first step of the algorithms should be modified by using the transformed admittance function and different values of transmission zeros according to the transformations. Indeed, the transformations o2-82/(l+s2) and z2-1+(l/s)2 followed by other transformations ;--l/¢2 and c-zz, respectively, yield the same continued fraction expansion for F(t) as shown in equation (2.3-2) except that the term L1/(C1’C) is now changed into -L1/[(l-ci)-t]. However, the relation ti-tj-(l-c )-(1-c1) shows 1 clearly that the recurrence formulas have not been changed. Very useful piece of information that is obtainable through the computations relates to the negative signs appearing in the array of C13. Negative C implies that when a pole sequence with :1 as its first il element is used, it will yield a negative element value. Similarly the presence of any negative element in the other columns of the C array, 11 say C < O, k< i, indicates that when a pole sequence which contains :1, ik ""Ck-l as the one indicated by the array, except that Ck is replaced by C1. is used, a negative C will be produced. Thus by inspecting the k negative signs that appear in the C array some of the pole sequences 11 that will produce negative element values can be detected. CHAPTER III FILTERS WITH INVERSE CHEBYSHEV ATTENUATION CHARACTERISTICS 3.1. Introduction In this chapter a design method is considered for filters, which uses the algorithm developed in Chapter II as the final step and with inverse Chebyshev attenuation charateristics. These filters are called the inverse Chebyshev filters and their group delay characteristics are also studied. The inverse Chebyshev filters, which may be considered as the limiting cases of the more general elliptic filters [MB], are relatively simpler to study than the elliptic filters, since they are somewhat related to the Chebyshev filters and therefore one can obtain the explicit formulas for the zeros of the polynomials required in the course of synthesis. Nevertheless, the filter network, which is of the type of Fig.2.3.l-a, has the same network configuration as that of the elliptic filter and accordingly involves the same problem of ladder network realization. 3.2. Insertion-Loss Theopz In this section a brief introduction to Darlington's [DA], [TO] insertion-loss theory is given. With reference to the quantities indicated in Fig. 3.2.1, the following definitions and notations are introduced: Insertion voltage ratio-V20(s)/V2(s) Reflection function £(s)-[Zd(s)-R1]/[Zd(s)+R1] (3.2-l) 31 32 1 1 1.2 + 4- 2(8) V20(3) R2 2+4} 1',2' Fig. 3.2.1 2 2 2 2 0 '4R1R2/(R1+R2) or a -4r/(l+r) a where r'R1/R2' It is evident that Os 02$ 1. '“- Transmission function T(s)-a[V2(s)/V20(s)] (3.2-2) or T(s)-a/(Insertion voltage ratio) ....) _.__) .(zexze) V2"’ V2“" s-jw v2 v2 * s-jw where A(m) is defined as the Insertion-loss function. Among the quantities defined above, the following important rela- tion can be derived either from the power relation or simply from the network equations [TO]: Z(s)2(-s) + T(s)T(-s) 1 Using the sub-star notation as in Eq. (3.2-3) this becomes 22* + TT* 1 (3.2-4) Eq. (3.2-4) is the basic relation in the theory of insertion-loss used in the design of filter networks with the specified attenuation characteristics and the specified ratio of terminating resistances. In the synthesis of low-pass filters one approach is to consider the function TT* in the following form: 33 2 a TT I"““"" t l + f2(w) (3.2-5) where f(w) is called the filter function. f(w) in general is a real rational function. There are other forms of TT*, each of which is convenient for a certain case; however, none of them has significant advantages over the others as far as the synthesis is concerned. In Eq. (3.2-5), f(w) is obtained from an approximation so that the function TT* will yield an attenuation function A(w) which meets the specified attenuation constraints. From Eq's. (3.2-2), (3.2-3) and (3.2-5) A(w) can be put into an explicit form: A(w) - %ln[l + f2(m)] (3.2-6) The rational character of f(w) follows from two factors: (1) The filter to be realized is a lumped LC filter with a finite number of elements, hence the driving-point impedance Z is a rational Brune function. (2) d From the theory of approximation of a function by using rational functions, it is known that a rational function can always be found to meet the requirement. Since f(w) is a rational function, one can write F F T1} '( Q HT L (3.2-7) and .13.. J}— £Z*-1- TT* -(Q)( Q )* (3.2-8) where P, Q, H are real polynomials satisfying the relation (3.2-4), i.e., Since the zeros of F, the transmission zeros, are also the zeros of z12 34 of the LC filter network, F must be either even or odd. Therefore FF*-;F2, and the polynomial Q must be a strictly Hurwitz polynomial. For lowbpass filters, since no transmission zero is allowed at zero frequency, F must be an even polynomial. This is a condition which must be satisfied when f(w) is determined for a desired lowbpass filter. In order that the filter be realizable as a mid-shunt or mid-series ladder network, addi- tional restrictions for F and Q are required. These are discussed later. With the notations introduced in the above discussion, a procedure for designing a filter is as follows: .§£gg_l Find a filter function f(m) that can yield a desired attenuation function A(w) which meets specified attenuation characteristic. f(m) also has to satisfy the conditions required for F and Q discussed above. ‘§232_; (a) Find FFluqu2 and QQ* from f(w) determined in Step 1. (b) Find the zeros of F which are the transmission zeros. (c) Find the zeros of QQ. which are distributed with quadrantal symmetry on the s-plane. Using the zeros of QQ* in the left half plane, form the strictly Hurwitz polynomial Q. (d) From Eq. (3.2-9): HH*-QQ*-FF*, find the zeros of HH* and choose half of them to form a real H. For the choice of H the only restriction is that H must have real coefficients. Each different choice of H will give a filter that yields identical attenuation and group delay characteristics but the element values are, in general, different. It can be shown that if the degree of HH* is 2n, then the number of (n/Z-l) (n-l)/2 choices which exist for H are 2 for n-even and 2 for n-odd. Step 3 From H and Q obtained in Step 2 obtain: n. ”man-film i 35 H 2 ai— Q Substituting this relation into Eq. (3.2-l) and solving for 2d, we have 1 + 2 Z a R'-——‘—- or 1 - 2 Z 3 R ———-—— d 1 1 + 2 (3.2-11) corresponding to the choice of positive or negative sign for 2, respectively. The filter networks for these Zd's are dual to each other. Step 4 Realize the filter network from the driving-point impedance Zd’ or driving-point admittance Y -l/Zd, obtained in Step 3 together d with the transmission zeros, i.e., the attenuation poles computed in Step 2. The subject studied in Chapter II is a special case of the final step(Step 4) where the mid-shunt or mid-series filter network configura- tions are considered. The filter design based on the insertion-loss theory was establish- ed by Darlington [DA] and Piloty [P1] in 1939. However it did not become practical until the late 1950's when the large scale electronic computers became available. The reason for this delay is mainly because the computations involved in Step 2 and Step 4 are too complicated and require too many significant digits even for a filter with only 3 sections. Since the late 1950's several papers have appeared which give both formulas and tables for element values for polynomial filters, i.e., f(m) is a 36 polynomial, and for elliptic filter where f(w) is a special kind of rational function. For two special polynomial filters, namely the Butterworth and Chebyshev filters, explicit formulas for the element values were obtained [BE], [TA], and tables of element values for normalized filters now exist [WEZ]. The network configuration for polynomial filters is a simple-ladder and the computation of element values can be obtained easily by using Routh arrays [T0]. For elliptic filters, formulas for less than 4-section filters are available [DA], [TT], [GRO], [SK], and tables of element values for normalized filters were prepared by several authors [80], [SK]. In the rest of this chapter the inverse Chebyshev filter is considered. Thus far in the literature no detail of the properties or design formulas for the inverse Chebyshev filter has been published. 3.3. Synthesis of Inverse Chebyshev Filters In this section the relations that exist among Butterworth, Chebyshev and Inverse Chebyshev filter characteristics are studied. Then by utilizing these relations, formulas for the realization of inverse Chebyshev filters are derived by making use of the algorithm obtained in Chapter II. The filter functions for n-th order Butterworth, Chebyshev, and inverse Chebyshev filters are defined as follows: Butterworth: wn Chebyshev: Cn(w) Inverse Chebyshev: l/Cn(l/w) where Cn(w) is the Chebyshev polynomial whose properties are listed in 37 Table 3.3-1. The above filter functions have unit value at wnl. The specification of a required attenuation characteristic is usually given as shown in Fig. 3.3.1. A Amin . _______________ Neper E; or /, Decibel 5; ¢ ¢ / % A... ' Z 0 i ///////////////A d, ,, w m ra sec l..— Pass Band ——.[ :——-——Block Band Fig. 3.3.1 It is convenient to introduce a positive constant e such that the attenuation at m-l will equal %1n(l+cz) nepers or lOlog(l+c2) db which is equal to or less than Amax’ the maximum allowable attenuation in the pass-band. Therefore at w-ma the attenuation will be-;£n[l+ezf2(m‘)] 2 nepers which is equal to or greater than Amin’ the minimum allowable attenuation over the stop-band. The filter functions after the necessary modification to incorporate the above properties are as follows: Butterworth: awn (3.3-l) Chebyshev: eCn(m) (3.3-2) Inverse Chebyshev: :Cn(ma)/Cn(wa/w) (3.3-3) For these filter functions, the function QQ*-l+f2 defined in 38 Eq. (3.2-S) for the Butterworth filter, is l+€2s2“, s-jw. The zeros of 2 2 1+: 3 n are distributed on a circle of radius p-(l/E)1/n and are separated by equal distances. Fig. 3.3.2-(a) shows the zeros of the 3rd order SOME USEFUL PROPERTIES OF THE CHEBYSHEV POLYNOMIALS cos( n cos-1x) for Ix] < l Cn(x) - _1 (3.3-1) cosh( n.cosh x) for le ; l or * 1 2 n 2 -n Cn(x) - T [(x -'/ x -1) + (x -,/ x -l) ] - —;— [(x + 23-1)“ + (x + #4)”) (3.3-2) Note that x +J x2 - 1 - l/(x -J x2 - l). Co(x) I l Cn+1(x) - 2an(x) - Cn_1(x) (3.3-3) 2 C2n(x) - ZCn(x) - l (3.3 4) n-l Cn-even - even polynomial with constant term - (-l) cn-odd - odd polynomial Coefficient of leading term - 2n-l n-Zk Coefficient of x , k - O, l, 2, ... is given by (_1)k2n-2k-l[ 2 n;k) _ ] Butterworth filter with e-O.S. For the Chebyshev filter, the zeros of QQ*-l+c2C§(s) lie on an ellipse as shown in Fig. 3.3.2-(b) with semiaxes 39 a and b given as follows [GU]: l/n _ K-l/n 23 - K (3.3—4) 2b - K1," + K-lln (3.3-5) K -J l/e2+l + 1/5 (3.3-6) where The zeros of QQ* are a - a cose + jb sine k k (3.3-7) k where 2k-l 6k - 2n u when n is even 2k-2 6k - 2n I when n is odd k-1,2,3,...,2n The ek's are the same for the corresponding Butterworth case. In fact for both cases, with the same a, the polynomial QQ* for the Chebyshev filter can be transformed from the QQ* of the Butterworth filter: 2 2 2n 1 + c c:(s) - T[1 + s w 1 where T represents the transformation s=%(w+l/w) whose mapping is conformal [AL]. As indicated in Fig. 3.3.2 the circles (p - constant) and radial lines (6 - constant) in wbplane are mapped by T onto the s-plane as ellipses and hyperbolas, respectively. All have the same foci ; j and the zeros of l + €2C§(s) are located at the intersections of these curves. By using this transformation, formulas for the zeros of QQ* for the Chebyshev filter can be obtained from the formulas for the zeros of QQ* for the correSponding Butterworth filter. Note that the inverse of T is w - s t/ s2+l and therefore for the filters to have the same a, it is necessary that the constant 1/5 for the Butterworth filter be replaced by a. 1.2: a , i 40 the constant 1/e :-/1/e2+l. Depending upon the positive or negative sign, the constant has two different values which are reciprocal to each other. For both values the same a and b given by Eq. (3.3-4) and Eq. (3.3-5) are obtained. Im.lk - 3 - 0.5 - 61—)U“ - 1.25992 120’ w-plane s - %-( w - %;) (w-sf‘/32+l) l-+'!'--+/--1+ 1 Ian A t e 62 s-plane n - 3 e - 0.5 a - 0.500 b - 1.118 r 3 1.000 "<2 ; ,f : ' —> ‘v I... A. (b) .1. .. I‘LL Fig. 3.3.2 41 The relation between the inverse Chebyshev filter and the Chebyshev filter is as shown in Eq. (3.3-2) and Eq. (3.3-3),from which it is evident that the zeros of 00* for the Chebyshev filter are wa/sk, where s are the corresponding zeros for the Chebyshev filter. The equation k of the ellipse, on which the zeros of QQ* for the Chebyshev filter are located, can be written in polar coordinate form as follows: g r cos 2 )2 g r sin 9 )2 + . 1 (303-9) 2 2 a b where b2 - a2 - 1 (3.3-10) Therefore, for the inverse Chebyshev filter, the equation for the curve on which the zeros of 00* are located can be found by replacing r in Eq. (3.3-9) by wa/r and ¢ by - $3 2 r2 - cos + sin 2 a2 b2 Now using the relation in Eq. (3.3-10), the above equation can be written as 2 1 r - aZbZ [ a2 + coszo ] (3.3-11) The curves corresponding to Eq. (3.3-ll) with D - e2C:(wa) - 104, mg - 1.3 for n-7 and n-l3, respectively, are given in Fig. 3.3.3 where the dots on the curves indicate the zeros of 00*. In the above discussion it is shown that the Butterworth, the Chebyshev, and the inverse Chebyshev filter characteristics are related and hence the explicit formula for the zeros of 00* for the inverse E s: if ll I[Jill]lllllllljlllllllllliilllllllllllllllllllIlllllllllll|IJJlllllllllllllllllllllllllllllllllll Fig. 42 3.3.3 43 Chebyshev filters can be established from the corresponding formula for the Chebyshev filters. Assume that Step 1 in filter synthesis is completed; namely, the determination of the constants n, D and ma which will produce a filter satisfying the desired attenuation characteristics. The synthesis of the inverse Chebyshev filter can then continued as follows. From the filter function defined in Eq. (3.3-3) and the relations among the polynomials 22*, HH* and QQ* we have 2 o 1 + D/C:(wa/m) 22* - 1 - 11* - 1 - w-s/j l - 02 D l 2 l + —6-Cn(wa/w) 2 1 + Cn(walm) w-s/j where D - £2C2(w X n a Upon multiplying both the numerator and the denominator in the above expression by the factor 82“, and observe that the limiting values of 22* at s - 0 and s - a remain the same. Therefore, considering the relation 22* - HH*/QQ*, the polynomials HH* and 00* can be defined as 2n 1-022 33* . 3 [ 1 + D Cn(wa/w) lw=s/j (3.3-12) 2 l 2 QQ* . 3 n [ 1 +-—5-Cn(wa/w) lw-s/j (3.3-13) The zeros of HH* and 00* are located on curves similar to that shown in Fig. 3.3.3 and the curve on which the zeros of HH* are located is enclosed within the curve for 00*. In the special case for the filter with equal 44 terminations (u - 1), HH* reduces to 82“. The explicit formulas for the zeros of 00* mentioned earlier in this section can be derived from equations (3.3-4) through (3.3-10) in the following manner. Replace :2 in Eq. (3.3-6) by 1/D. Then the (ma/sk)'s become the zeros of QQ* for the inverse Chebyshev filter. Let s1 - rif¢i be the zeros of QQ* for the inverse Chebyshev filter, then 2 2 2 2 2 1/r1 - a cos 61 + b sin 61 - azcosze1 + (a2 + 1)sin26i £1, a2 + sinze1 i.e., '1 I 1//a2 + sin2 9 (3.3-14) and an'1 [ (a/b)tan6 (3.3-15) '9 p. I 11 where 2a,- Kl/n - K-1/n (3.3-l6) 1/n -l/n N O‘ I K K -J D + 1 +5 (3.3-17) i - l 61 n w + K for i - 1,2,3,...,2n, with n odd. It will be shown later that this filter is not realizable when n is even. The formulas for the zeros of HH*, when a f l, are the same as those for QQ* except that D is replaced in Eq. (3.3-l7) by D/(l - o2). This is due to the fact that HH* and QQ*, as shown in Eq's. (3.3-12) and (3.3-l3), are identical in form. From the zeros of HH* and 00*, Q and H are obtained by the method 45 described in Step 2 of the synthesis procedure in the preceding section. As mentioned earlier, Q is strictly Hurwitz and is formed from the left (n-l)/2 different choices half plane zeros of QQ*. However, there are 2 for H to be determined from HH* for odd n. It is interesting to note that for n odd, if H is formed by selecting the alternating zeros of HH*, then it can be shown that n V - 11(3) - 63—) [ 1 + 3[(1 - 02)/D cnuma/s) ] (3.3-18) ‘TR—imm J « I Thus H(s) can be obtained without computing the zeros of HH*. For this case, however, the attenuation pole frequencies wk are the zeros of n Cn(jwa/8) and therefore H(wk) - wk. After Q and B have been obtained, the driving-point impedance can be formed from Eq. (3.2-10) or Eq. (3.2-11) : 2d“) . R1 .LLL Q - H (3.3-19a) or . .9421. 2d“) R1 Q + a (3.3-20a) or, alternatively, the driving-point admittance is . ..9...:JL Yd") G1 Q +11 (3.3-19b) or + 11 Yd“) ' G1 Q - n (3.3-20b) where G1 - 1/R1. The filter with desired attenuation characteristics can now be obtained by realizing the above driving-point function Z or Y d d with transmission zeros (attenuation poles), m which are the zeros of k, T - F/Q. For the inverse Chebyshev filter, these zeros of F are the 46 positive roots of the equation Cn(wa/w) I cos[ n cos-1(wa/w) ] These roots are m “k ' cos(2k - 1 I ) 2n (3.3-21) for k I l,2,...,(n-l)/2, with n odd. However, in order that the given Zd or Yd be realizable as a mid-series or mid-shunt ladder network, the degree of Q + H written as 6(Q + H) should be exactly one more than that of Q - H. i.e., 6( Q + H ) -6( Q - H ) + 1 (3.3-22) This necessary condition can also be obtained from the condition (1) of the Realizability Theorem given in Chapter II and from the fact that Zd and Yd are Brune which require that the difference of the degrees of their numerator and denominator polynomials is at most one. Eq. (3.3-22) implies that the leading coefficients of Q and H are identical, further- more the identity HH* + FF* 5 QQ* implies that 6(QQ* - HH*) - 6(FF*) - 6F2 and hence we have 6F + 1 - 60 - 6H (3.3-23) Consider the inverse Chebyshev filter characteristics: FF* (s/j)2nC:(jwa/s) a2/D 09* (ex/5)“! 1 + cficjwa/swn 1 TT I If n is even, Cn contains a constant term. Therefore, the filter is not 47 r' ——————————————— n I L1 TABLE 3.3-2 w... F 2d ~:c1 T 1 c c c: ‘ v v v L _______________ __1 GR I Degree of numerator polynomial 211,222,212 6D I Degree of denominator polynomial n a 2 6N/6D for Elements which 2 z or z z or Z or Y 0 Y have zero values d 11 1o 22 20 y11 ls y22 ’ 2s All non-zero Some or all Pi's 2n 2n 2n 2n+1 2n+l 2n-l C1 or Cn+1 or both C F 2n 2n 2n-2 2n ’ _ — fl _ 1 1 2n-l 2n-l 2n-l 2n-l c , r 2.11:1. 23;; Zn 29;; n+1 n 2n 2n-1 Zn-l 2n-l ‘EBESEL R2211 + '2' y + R Iyl (1) zd'R+ ’Yd'zl'ilxz 2 222 d + 2"22 (2) n a 2 not necessary for all cases. (3) The results are obtained by induction under the assumptions that 0 < R < I and C1, Li’ T1 are non-negative. realizable in terms of a ladder network, since in this case 6F I 6Q, and this does not satisfy the condition given in Eq. (3.3-23). In Table 3.3-2 the degree relations of the numerator and the denominator polynomials of Z for the mid-shunt ladder network are given. When n is odd, from d Table 3.3-2 and from the fact that Eq. (3.3-21) yields all finite and 48 non-zero mk's, it can be concluded that, if the filter is realizable, it is either a full mid-shunt ladder (all elements are non-zero) realized from Eq. (3.3-20b), or a full mid-series ladder realized from Eq. (3.3-19a). Under the assumptions that Q and H yield a realizable Yd (or Zd), two methods of realization of Yd by using the algorithm established in Chapter II can be given. These are discussed in the following: Method 1 In this method, the initial 2n parameters, CR1 and Lk1 (kIl,2,...,n) needed as required by the algorithm for the computations of remaining parameters are obtained directly from polynomials Q and H. The initial 2n parameters are computed by Darlington's formulas as C I Yd(B)/s kl stmk (3.3-24) and -2 Lkl ' 3 . s [ Yd(s)/s ] stmk (3.3-25) for k - l,2,...,(n-1)/2. Since the normalized driving-point admittance (with respect to Cl) is Yd - (Q + H)/(Q - H) (3.3—26) we have Llhl.l_2.i§_'.9_:£9£.l_l_9_t_§ , ds s s (Q _ H)2 s2 Q - H (3.3-27) Both functions are expressed in terms of H, H', Q and Q'. Therefore,the parameters Ck1 and Lk1 can be computed in terms of H, H', Q and Q', for stwk. However,another way to compute the parameters is as follows. ~Let 49 n n H I n (s — si) and Q I n (s - 8i) iIl iIl Then by using the relations I n g n ‘3" 2 .1 .... ‘3‘" 27*:— 1-1 1 1.1 1 Eq. (3.3-27) can be written as Yd, 211 n 1 1 2 7-4—7 zls-s'-s-s l-Yd/s (3.3—29) (Q ‘ H) 8 1'1 1 1 Therefore, Ck1 and Lk1 can be computed from H, Q and the summation that appears in the right hand side of Eq. (3.3-29). In this method the computation of the coefficients in the numerator and the denominator polynomials are avoided and the computation of Yd(sk) u _ v _ and [Yd/slsIsk involves only the factors (sk 31) or (sk s1). The Ckl’ Lk1 obtained by this method will therefore be much more parameters accurate. In the design of filters, especially for filters with large number of sections, it is convenient to compute the element values from both the input and output ends as an accuracy check. This method, with slight modification, can also yield two sets of C corresponding to those kl generated from the values of Yls and Y28 at stwk, respectively. Indeed, if we let Qka) + H(wk) I E11‘ + jOlk and 50 then (Elk/02k)/wk ' Y18(jwk)/jwk and (3.3-30) (Ezk/Ozk)/“k ’ st(j“k)/3“k I are the Ck1 s obtained from‘Y18 23 obtainable from Yis and Y28’ since there are no simple formulas for the t o . values of Y18 and Y28 at s jwk, the L in the case of the parameters C - I [and Y when s jwk. Note that for Lk1 s kl's cannot be computed as easily as kl' However, since the Lkl's are the same from both computations, except for the ordering, we may use the same Lkl's obtained from Eq. (3.3-25) for the out-put side and they are now ordered consistently with the pole sequence seen from the output end. Method 2 In this method the coefficients of the numerator polynomial (Q + H) and those of the denominator polynomial (Q - H) of Yd(s) are obtained by using Newton's formulas [CH]. The algorithm established in Chapter II then is utilized to realize the Y and Y as discussed in Chapter II. ls 23 Let n (n-l)/2 _ Q(s)-H(s-S)-(8-s) II (s-S)(s-s) 1-1 1 O j'l j j n n-1 n-2 I s + pls + p28 + ... + pn-ls + pn (3.3-31) n (n-l)/2 _ H(s) I H (s - s') I (s - s') H (s - s')(s - 3' 1-1 1 ° 3-1 3 3 . 8n+pisn-1 +pésn-2 + +pt'1_18 + p; (3.3-32) then Q + H I an n-l Q H o + (p1 : pi>s + ... + (Pu-1 : p;_1)s + (pn : pg) (3.3-33) 51 Let n 11 5k - z a: and GL - x sik (3.3-34) 1-1 1-1 then from Newton's formulas we have 1 v-_.__ Pk 1 Pk k [ (6k I 5k-11’1 + "° + 6zpk-z + 51Pk-1) I v o v t v v _ 1» (6k + 5k_1p1 + + 52pk_2 + 51pk_1) ] (3.3 35) . . l.‘ I for k l,2,...,n. If s1 rilil’ s1 tiltif then by combining the complex conjugate pairs, Eq's. (3.3-34) become (n-l)/2 6k I so + 2 151 rjcos(k¢j) (3.3-36) k (n-1)/2 k a; - s; + 2 351 r5 cos(k¢i) (3.3-37) where r and ¢ J 1 ly, and so and s; are real. Note that so I -a and a; I ia' are obtained are computed by formulas (3.3-14) and (3.3-15) respective- by Eq. (3.3-l6) for Q and H respectively and the sign of 3; depends on the choice of H. The coefficients in the numerator (Q + H) and the denominator (Q - H) of Yd are found first by computing 6 and 6i (kI1,2,...,n) and k then carrying out the iterative computations given in Eq. (3.3-3S). The advantage of the second method lies in the fact that the coefficients of Y18 and Y28 are also determined once the coefficients for Yd are computed. Therefore the realization from both the input and the output ends of the filter can now be accomplished by applying the algorithm to Y18 and Y28. 52 3.4. Design Procedure and Numerical Examples In this section a design procedure for the inverse Chebyshev filter is outlined. The design procedure also includes the approximation step. This step enables one to select the design parameters D,e:and n so that when the filter is designed based on these parameters, it will meet the required specifications i.e., the attenuation characteristics. As indicated in the preceding section, when the filter function is ecn(wa)/Cn(wa/w) then for a given set of parameters, Amin’ Amax and mh(see Fig. 3.3.1), the design parameters D,6 and n are determined according to the following relations: 1 2Ln(l + D) ; Amin nepers (3.4-1) and l1n(l + :2) < A ne era (3 4-2) 2 ‘ max p ' or if db(decibel) unit is used, then 10 log(l + D) ) Amin db (3.4-la) 2 10 log(l + c ) 5 Amax db (3.4-2a) 2 2 D I e Cn(wa) (3.4-3) First the mdnimum value of D and the maximum value of :2 are determined from Eq's. (3.4-1) and (3.4-2), respectively, and then they are substituted into Eq. (3.4—3). Since ma is known, this relation yields n. In general the parameter n so determined is not an integer. Therefore for the value of n, the minimum integer which is greater than the comput- ed n must be taken. For the determination of n, the formula in Eq.(3.3-2) of Table 3.3-1 may be used: 53 3.4.1 Fig. 54 can.) - £1 ((1) +,/ m2 - 1)n + (w +,/ .12 - 1f“ 1 (3.4-4) When n is large, the second term in the right hand side of the above equation is small compared with the first term and it may be ignored to simplify the computation. Note that this omission will increase the value of n slightly. The computations needed in determining the values of the parameters D, :2 and n may be done graphically. One always has the freedom of choosing D and 62 for a fixed n as long as the inequalities (3.4-1) and (3.4-2) are satisfied. Therefore a graphical method is convenient to the determination of a most suitable set of parameters without the repeated computations. Two families of curves plotted in Fig. 3.4.1 can be used for finding these parameters. The first set of curves (in the upper part of the graph) are computed from Eq's. (3.4-la) and (3.4-3) in which each curve corresponds to a different 62 which in turn implies Amax through Eq. (3.4-2a). The second set of curves (in the lower part of the graph) are the plots of the relation, C§(wa) I D/tz, in which each curve corresponds to a different integer n. The following example illustrates how the graph is used in finding the desired parameters. Exapple 3.4-1 Find the design parameters D, e and n for an inverse Chebyshev filter satisfying the following specifications: A I 1 db max min I 55 db and w € 1.5 55 With the given specifications, regions in Fig. 3.4.1 in which the specifications are not satisfied can be determined. In this case the graph will appear as shown in Fig. 3.4.2. From Fig. 3.4.2, a solution for the parameters will exist if W:2 > 1.2 106 and n > 9. Of course the Amin ‘} db A I58.57 6Q::::___ naug!i’—; ___________ mc -*———;;‘ ll " /’”’” I“ WWW / // / / ' , 4 1.0 : i 2 2 ' 0]: IC (w ) [[12:106 Lam" r8x106 “ a _ nI7 1.4Q l. AaW%%%&%02%%awwwwwwaaflwoza7 w b a‘, Fig. 3.4.2 simplest filter is obtained if nI9. The selection of nI9 imposes an upper bound on D/ez, 8x106, and a lower bound on :2, 0.038. Therefore, 1.2x1o6 < -—D-i- < 8x106 5 and 2 0.038 ( E < 0.26 If we select D/e2I4XlO6 and 52-0.25 then with nI9 we have, m I 1.46 a A A 56 I 60 db n x I 0.961 db (at wIl) 6 6 D I c X4Xl0 I 10 It is evident from the above example that a set of design parameters always exist for any given set of specifications: meal and O, the computer arranged the pole sequence according to Fujisawa's procedure 35“C3a C1. 42. C4 > and gave the element values as 1‘ listed below: 01 - 0.011771 02 - 1.0420 c3 - 1.3817 c4 - 1.0804 1.1 - 0.54464 L2 - 1.1726 L3 - 1.2585 L4 - 0.76581 ‘1 - 0.35589 r2 - 0.38801 r3 - 0.27958 ‘4 - 0.071660 L1‘1 ' ‘3 L2‘2 ' ‘1 L3‘3 ' ‘2 L4‘4 ' ‘4 c5 - 0.22564 Since the element values are all positive, the filter can be realized in a ladder form. In the above example, although Newton's formula involves a large number of algebraic sum operations for nI9, in the 4-section filter computed with single precisions arithmetic (11 digits) both element values realized from Y and Y check up to at least 6 digits. The 1s 28 numerical values given in Example 3.4-2 are rounded to 5 digits. 3.5. The Group Delay Characteristics In section 3.2 the following definitions are given: 60 ...LLL 02 Q Q (3.5—1) (v2 v2) . v v 2 20* 2 where V20(s) I [E(s)R2/(R1 + R2)], 0 I 4R1R2/(R1 + R2)2, and V2(s) “NI :3 I- is the output voltage function and T(s) is the transmission function of the filter network as indicated in Fig. 3.2.1. For LC filters, F(s) is an even or odd polynomial and Q is a strictly Hurwitz polynomial. Since the filter network is a linear system, Q is the characteristic polynomial of the system. In other words, Q characterizes the transient response of the network. The zeros of Q are the natural frequencies which are complex valued and are located in the left half plane. A more explicit form for Q, in terms of the configuration and the element values of the filter have been established and will be given in the next chapter. Two functions, the phase function B(m) and the group delay function T(w), which are important in the study of transient response characteristic of transmission network, are defined as follows: 8(0)) I arg T(jw) I arg-E— Q 8_jw (3.5-2) . _ .22121 T(“) dw ' (3.5-3) From Eq. (3.5-1) and Eq. (3.5-2) the phase function B(m) can be expressed in terms of polynomial Q(jw). Indeed, since argT(jm) is an odd function of m, we have, B(w) I arg T(jw) I -arg T(-jw) then for stm 23(w) I arg T - arg T* 61 I arg[ ] (3.5-4) Since F is an even or odd polynomial, we have F/F*I; l and 9* 28(w) I arg'-6-'s_jw for F I even 9. 23(0)) I Iarg T 8-3” for F I odd We note that for stw e3 arg(T/T*) ..I. . .41. T* T* Substituting Eq. (3.5-4) into the last expression we have, _;3_ _ j23(w) 2* e or T F F 28 ) I 2 -———’- 2 "" /'—-' 3 (w n 1* nt( Q ) ( Q )* 1 Therefore Q 1 t B“) i “—23- in T 8ij (3.5-5) where the positive or negative sign will be used according as F is even and odd, respectively. From Eq. (3.5-5) 1(w) is obtained as follows: _ _ 1 ngwz . - 1 d Qg-jwz ‘(“) i 23 dw + 23 dwl ‘“ 0(30) ] .;L_._é__. SlkzbEL 1 2 d: ”n 004)] 62 Since (Q*)' I -(Q')*, we further have .. ...1... .93. .0; 7(0)) 1' 2 [ (Q ) + (Q )* ls-jw (3.5-6) 0t 9 r(w) I Even part of -%- stw In what follows we shall consider only the low-pass filter where F is even and hence the upper sign of Eq. (3.5-6) is to be used. It is shown in section 3.3 that for inverse Chebyshev filters only the odd degree case is realizable in terms of low-pass ladders. This implies that the degree of Q is odd and, therefore, the characteristic polynomial Q has the following form n-l Q-(s-8)H(s-s) (3.5-7) 0 3.1 i where so is on the negative real axis and the s 's are complex and occur 1 in complex conjugate pairs. It can be shown that n-1 n-1 .91.. I 1 and (fig) =I - Z 1 Q 1-0’"1 Q* 1-0‘+'1 and hence so n-l 8 1(8)) I 2 + Z 4—2 2 s + so iIl s - s1 Since the 81's occur in complex conjugate pairs, letting siIoi + jwi and 3 I o - jm we have i i i 2 2 2 s1 - (a1 + jw1)(w1 - m - o + 2j01w1) 2 2 2 2 2 2 2 2 s - 81 stw («01 - m - 01) + 40101 and 63 s 3 s [ 1 + -——-l-- 1 - - 2Re 1 82 - 82 82 32 w + s i 1 BIjm i -201(m1 + m2 + 02) 2 2 2 2 (m1 - w - a 22) + 401101 from which [sol (8-1)/2 |,.1|(m2 + oi + 82) 1(0)) I fl" 2 2 2 2 2 212 2 (3.5-8) w + so iIl (m + 01 - mi) + 401m1 Eq. (3.5-8) is a general formula, i.e., it is true for any low—pass filter with a characteristic polynomial Q as shown in Eq. (3.5-7). 1(a) is a positive function of w. The expression in Eq. (3.5-8) can also be obtained by applying the result proved by Gilbert [GI]. Indeed, if M N N T I A[ n (s - s1)/ H(s - s1) ] where A is an arbitrary constant, s1I0~1 + jgi’ and 81.01 + jmi then, as Gilbert has shown, 1(a) can be written as M 31 N 01 T(w) I X - Z 1-1 3: + (w - 31)2 1_0 oi + (w - m1)2 (3.5-9) For filter networks, :1 are the zeros of polynomial F all of which lie on the imaginary axis. This implies that the first summation on the right hand side of Eq. (3.5-9) is zero. The second summation, after combining the conjugate pairs, yields the expression in Eq. (3.5-8). It is interesting to note that the m—th derivative of 1(w) is d m! n m m+l 1’00) .-—- 2 [ (-l) + l ]/s am“ 8-0 2 1-1 1 64 i.e., 0 for m I odd m m T(m) I n dw wIO m! X [ 1/8 iIl n+1 1 ] for m I even This result d2 ' L232