SOME EMPROVEMWW OF THE IMAGE-PARAMETER METHOD FOR THE EESIGN 0F LC FELTERS Thesis §ov “no anvoc of DH. D. MICHIGAN STATE UNIVERSETY Yilmaz Tokad 1959 F- 1 "r . ' H! - 5. 5 H E..: bu;.l cs’ This is to certify that the thesis entitled SOME II~IPRUVEEENTS OF THE Ili‘iAGE- PARAIVIETER I‘IETHUD FOR THE DESIGN OF L-C FILTERS presented by YILI‘EAZ 'I‘UKAD has been accepted towards fulfillment of the requirements for DOCTOR 0F pmmsopmdegm in ELECTRICAL ENGINEbBING - .A . 9 ft 2 / 1/1/14 l, k / t; "‘1 ..// v" " t. K ,- Major professor / \ \‘~ Date November 12, 1959 0-169 LIBRARY Michigan Strata University ‘ ‘ .’ u . .. - . ._ 'L . - .. -. . \ r- \ ~ »- V "Fifi" ‘I e ' Tl‘v“"' ("I o “W. ‘.'.J\ --‘_J L11 i~4-\ . r AurJ. - . l -' -.L Lu n ‘x,’ ‘7, w ' i u } “‘r‘rv .j,‘ rhfioAIMVL—l L -J- \, - .r - - -. .- ,-: -*" ' ' ‘ fl - e x . “ac;_;4n Qulur tab/Q 0-». r ”+1|.1hie.e ”He ‘ j I V ‘ I ~ I . _ . 1 fi‘ ' r‘ . ~- 7 r“. r- ' . - v a . . - c v Hie-1.1. u JL L81- J. J. e. _ 11;. a 1m: 1e ,_ . . . a F) \ u - V. .1- 1 fl ‘ ‘ "' ‘. "t a"? ' -\‘~" . - Lab re\1“.-"v.tlL-Lwo -'-.', If». ’s\‘ié-:(.€ A ‘ 1:17 * . "T‘T 3 fl . "V ~ 1 *fl . - u".)v"‘ "“ w t , 1 1 _)r\""“ "0‘ . ,4 ~ '1 \- r‘ 384‘ v" Ml- » -..L L|A¢'{.,.1 ’ \A- “*C.\,b~ rh‘v', #4 .J .,)‘.A- .O-L.4 .LQA \ Apprcved 171BSTRACT Ihe image paremeter method ior iilter design is not yet excelled for its simplicity. Darlington's technique, although qaite straightfor— ward, has the following restriCticns: (l) A special type 0; insertion— loss requirement is Considered in tne blocs-band. (2) The determination 0: element values involves long calculations. In recent years the image parameter method is being increasingly exploited to overcome the above diriiculties. iuttle has shown tr means of image parameter techniques that in tne special case oi two cascaded Zobel sections, one can get Darlington's type 01 insertion—loss charac— teristics. helevitch has pointed out that there is promise oi the two apparently differing techniques coming closer to each other. In this thesis, the image parameter method is studied in detail with the specific purpose of eliminating the drawbacns which are with- holding its Wider application in filter design. rhe results or this study have yielded a derinite improvement over tne existing method. ihe salient reatdres or the method delined in this theSis are: (i) The difficulties of Zobel's decomposition formula are elimi- nated by considering a new rormulation. (ii) A method due to Feldtneller, greatly extended by nelevitch, applies the image parameter method to the design in the pass- hand. This iormulation is now extended to the biocx-oand and a detail study or this is presented. (iii) Exact requirements on the tranSier—loss funct-on in the blue;- band are given. Consequently, the number of intermediate sec- tions in the filter could be minimized. ihis important Cnnse- (iv) (v) (vi) -2” quence or the new rsrmulation is demonstrated Ly means oi an example. Properties oi tne general terminating sectLons are c nsidereu. oeneral iormulas are derived ior the calculation or the ele~ ment values of the terminating seCtions. The rormulas ior tne insertion function valid itr the cut-oi; Irequency are given. rhese general iormulas Cover ielevitcn's formulas as a special case. The eiiect or dissipation on the insertion fanction is deter~ mined by means or the electronic digital computer. A general program is written for this purpose. The new results presented in this tneSis are a consequency or a angle. detailed study UL the image parameter method, viewed iron more tnan one It can be safely concluded that tne study has certainly raised hopes of making use or image parameter method - witn the suggested in— A t rovements - for a wider application in ii ter oesien. &.L o SOILS Il-LPROVEfx-ETFT‘S 03‘ {SEE IE 245113-- PAI’JC ZEN”. MSEJOD FOR THE DESIGN OF L—C FIL”ER3 By Yilmaz Tokad F THESIS Sitsitted to the School of Advanced Graduate Studies of Michigan State University or Agriculture and Applied Science in g l fulfillment of the degree or a a J. .. via. v. A. a f9 Q l the requirements DOCTOR OF PHILOSOPHY Department or Electrical Engineering 1959 Approved ;_2; ." (1- a '1‘ v "-r r! r.‘ w P1774) g Jinx/J" t.'\/.v"iL"_‘J Q'Ll Lb-._L-J The author is indebted to Dr. M. 3. Reed, his thesis adVLSur, his guidance in preparing this thesis. inanhs are also dJC t; Dr. Von Tersch, Dr. E. E. Koenig and Dr. J. 5. Frame for their enc urn ‘ l. ’ A. II. a , r -_ “can; 21 b . 4., a ,— rx- v‘ _'~q-‘ --‘ vr V I" ‘ f1 ‘D‘lt‘ r “1 \ ‘7 Ll...'~nll.,1_.l .14 -.J.-..L. o o o o o o o o o o o o o c o o o c o o o o c o o 0 ...~ CELAP'I‘ER I " II: 1‘1‘KODUCI‘IO:: o o o o o o o o o o o o o I o o o o o o o o o o o 1 II - SURV"? OF I"SEHTIO" LL 3” DESI13 IECFKIQUE . . . . . . . . . . 3 2.1 Insertion Function . . . . . . . . . . . . . . . . . . . 5 2.2 eneral Discussion on Insertion Loss of Symmetrical ' Lossless Filters . . . . . . . . . . . . . . . . . . . . L . Advanta5e of Image Parameter fictncd . . . . . . . . . . ll Formula . . 2 W FULL IJL‘i'i‘IO FOL: IBSEE‘I‘I‘IOL‘ FLTI‘FCI'IOEF OF IMAGE FIR-LESLIE? 2 3 2.4 S.nc Disadvantages cf Zs:el's Decthnositi0 5 5 General Discussion of the L;cation or the Critical Frequencies of the Image Impedance of a'rS . . . . . . . jl 5.o Element Values of TS . . . . . . . . . . . . . . . . . . c2 VI A DESIGN PROCEDURE FOR sn-I ‘RI... . Lou PAS s IILAGE Pi: thiE'l‘rjR FILL’ER 7:. o.l General Review of Image Parameter Filters . . . . . . . VA s.2 The Desi5n Procedure . . . . . . . . . . . . . . . . . . {c (A) Choice and Calculation of Terminating Kali Sections . . . . . . . . . . . . . . . . . . . . . Y; (B) Calculation of Intermediate Secnions . . . . . . . 7) Example . . . . . . . . . . . . . . . . . . . . . . . . ' Some Remarks on Image Parameter hethod . . . . . . . . . a, Tuttle's Problem . . . . . . . . . . . . . . . . . . . . 67 (x 0‘ C \J #‘Lx O"! 011' IT'ISE‘I'Z-‘IOLT LOSS AID .‘ ‘1‘) .‘r-f—m 1 T.v1‘fi|‘ “2.13.13. 1' IL; 11am (3 1’4 ()1 t: ' '7‘ 61 {U 1*! E. (a General . . . . . . . . . . . . . . . CONCLUSIONS . . . . . . . . . . . . . . . . . . . :11. :.LI O}I{1u)}rlr o o o o o o o o o o o o o o o o o o I. INTRODUCTION An image parameter filter is obtained by cascading sections (four- terminal netwarx) with matched image impedances. The Image-transfer function of this filter is the sum of the image-transfer functions of the individual sections. If the filter contains L's and C's only and operates between its image impedances, it will transmit without any loss over a certain band (or bands) of frequency (pass—band) and attenuate for all other frequencies (block—band). In actual practice, we are interested in knowing either the opera~ tion loss* or insertion loss or the filter. In general, neither of these loss functions is the same as the image transfer loss of the fil- ter. Zobel has given an expression for insertion loss 01 the filter in terms or image parameters. When the insertion loss is Speciiied and the filter is to be found, the different factors in Zobel's expression '1 1" must be considered. ihe investigation oi Lobel' s Ioimula shows that it is possible to mane some approximations on the insertion loss of the filter, and consequently the design procedures can be simplified. bode has made further investigations on the image parameter filter theory and has shown how the filter can be constructed by considering only one image impedance and the poles or image transfer loss. his matched cascaded sections generally differ from those of Zobel's, out for the practical case, he arrives at Zobel's composite filter but with more complicated terminating sections. "Operation loss" is defined as the logarithm of the absolute value of the ratio of two voltages or currents. One of these measurements is at the output of the filter when it operates between the terminal resist- ance; the other is measured wnen the source is connected directly toa resistance equal in value to its intexnal 1esistance. ;1e te ‘in '— i'n 1:5 is deIIned.iIIfX eter II. -2» Darlington and others have considered the cut-and~try method in- volved in the image parameter filter design as cumberSome. Darlington‘s method of filter design based on insertion loss is well-known. In this method, a special type of insertion loss function is considered and after finding a characteristic function, ¢, the problem Is reduced to finding the element values of the filter. Darlington's special insertion loss function has been extended by Fromageot and others. But, in the determination of the ¢ - function, some approximations must be made, and after designing the filter, it must be checked as to how good this approximation is. In spite of this generalization, the inherent disadvantage of the method, viz., of deter- mining the element values of the filter, still exists. The above disadvantage of insertion loss filter design (Darlington) compels the designer to loon for more practical methods. Classical image parameter theory does not have this disadvantage of Darlington's method, i.e., after the design parameters are chosen, the element values of the filter can be determined very easily. futtle has shown that two cascaded, matched Zobel sections can produce Darlington's type of in- sertion loss function by proper choice of the design parameters. 'Iuttle's method is based on image parameter theory. Consequently, the calcula- tion of the element values, which is the difficult part of Darlington’s method, is made extremely simple Ior this particular case. Belevitch considers a new formulation for the insertion loss Iunc- tion. oased on this formulation, he discusses the insertion loss func- tion in only the pass—band. In this thesis: -3- (l) Belevitch's formulation is here generalized to include a dis— cussion of the insertion function of the block-band. On the basis of new properties obtained from this generalized formulation, procedures for filter design are described. (2) At the cut-off frequency, the general expression Ior the in- sertion Iunction is given. (3) Low pass filter terminating half sections are considered in general: general expressions are found for determining the element values of the terminating half sections. (4) After designing the filter, the effect of dissipation on inser- tion loss and phase functions are inveStigated by using a digital computer. II. SURVEY OF INSERTIOH-LOSS FILTER DESIGN TECHNIQUPS 2.1 IhSERIION FUNCTIOH The insertion function, Ps’ of a four terminal networx Is defined in terms of two currents, IR and 15 (or two voltages V and Vé) of fig. R 2.1.1 and by the ratio If: V; P = 2n —3 = in ‘n (2.l.l) S a 1’? MTN Fig. 2.l.l Since in general the voltages or cuIrents in eq. 2.1.1 are Cgi lex, then PS is also complex and can be written as P = A + j a (2.1.2) where the insertion loss function Ac is in ncpers and the insertion phase 0 function B is in radians. Therefore, from eq. 2.l.l we also have, S 2411 I ’ 2 V.I 2 ’ I I S _ R _ HI: .1 _ WT R _- r 1‘ 1 Q 8 ~ ’1‘: °‘ VJ OS '— aLQ i”- ‘- arts V: (20+0J) ix R ; [I h The expression for inserti n function in terms of the image Lari.etrrs was first given by Zobel [1). In this thesis, we shall use the sane no- tations as used in reference [2]. In terms of these notations, a .4 ' IQ - P e. = ,. -—~---«~- ~B———~~-~-~‘= l - xflxfle 2P1 e I D I\ (2.1.h) The terms within the parenthesis on the risht hand side of eq. 2.l.h are designated, from left to right, as the: l) Transformer, 2) Input re- flection, 3) Out put reflection, h) Interaction, and 5) Image transfer. The parameters KS and KR appearing in eq. 2.l.h are, respectively, the input and output reflection Coefficients. The logarithms of the factors 1) through 5) correspond to certain functi ns whtse properties are well known [2]. As may be seen from eq. 2.1.h, if 58 and ZR are equal respectively to the image impedances, Z11 and Z12, of the four terminal network, then P ZVZI 11274]: P 2 I , e I 7...“... z e \2.l.5) On the other hand, if only, say, Z0, is equal to the image ihpedance L11, then -j- P a1 éIa r A“ p e S 2 T 1 Est; .1 (2.1.1-) I2 I1 “R For a symmetrical four terminal network, where 311 : 212 : Ll, eqs. 2.1.5 and 2.1.6 yield Ps PI e = e or P = P 2. . S I ( 17) Therefore, for a symmetrical four terminal network, ’f there is a matching at least at cne pair of terminals, the insertion function and the image transfer function are identical. Also, P8 is independent 9f the load im— pedance ZR. The result in eq. 2.1.7 is also obvious from the delinition of P.‘ D For consider a symmetrical four terminal network with 311 2 312 : ZS as in fig. 2.1.lb. The replacement of the sub-network to the left of the terni~ nals 2 - 2’ by its Thévenin equivalent will yield the networh in fig. 2.l.la. The only difference is that such a netwo k will have a different voltage source. In practice, the terminating impedances, Z8 and ZR, of the filter are mostly pure resistances. On the other hand, the image impedances of a filter are real in the pass-band but not constant. Therefore, to provide a matching in the pass—band, the image impedance must be as Constant as possible in this region. This can be done to a certain degree of appr~xi~ nation by increasing the o der of image impedance (e.g., by use of hibel's multiple-derived sections). If perfect matching is possible, then eq. 2.1.7 is valid and the design of the filter is reduced tn the design of an unCorrected image parameter filter. 2.2 sexiest DISCLSSIONS or run IHSERTIOJ Loss OF srnnrriICAL REAerIvs (LC) FILTERS In this section we shall have a general look at the filter synthesis technique on an insertion loss basis. Tle purpose of this thesis will then be clarified. The filter synthesis technique :n an insertion loss basis has two main parts: (A) To find an approximating function, ¢, which is related tw the insertion loss function as [3]. e2 AS = .f\ [1 + ‘¢]2] (2.2.1) such that ¢ must satisfy all the imposed conditions 4n the As-function. (3) Find the element values of a lattice or ladder filter which is to be obtained from the above approximating function, ¢. These two parts are considered in the fallowing. (A) Determination of the ¢-Function: From the general theory of insertion loss filters, the insertion power ratio for a symmetrical filter with equal terminations is oi the fo‘h [5] 2 A, "e = 1 + ¢|2 (2.2.2) where ¢ is an odd rational function of p (= 3w). The poles of ¢ are the poles of the AS-function and the zeros of ¢ are the zeros of the An»func— 0 tion. For example, let w(a312 - (i=2) (0:112 - c122) (1-Cfi%fi).n “-'@:%% (82¢) l¢i=fi where H, p. m- Figure 2.2.2 where k = di/w the modulus of the elliptic sine function and K is the 2 corresponding complete elliptic integral. In a general case, in actual practice, the values of the insertion loss As, is to be less than a given value an’ in the effective pass—band. n the other hand, in the effective bloch-band, AS is not always neces~ sarily greater than a given value a . a If the contour of requirement for AS in the effective block-band is not a horizontal line (flat), it is difficult to find a corresponding d — functian satisfying this given non—flat requirement. In such a case, as .2 is usually done, an equation similar to eq. 2.2.h can be used. In the eerctive block-band, for frequencies of high A”, there is a 0 linear relation between A8 and AI [3]. Therefore, AI can apprwximately be determined. The problem now is reduced to finding a reference filter which will have an image attenuation in the block-band, approximated to I AI just determined. Since the poles of A are known, poles of Am are identical. With D I this information the ¢ - function can be determined [3), [7). -10- In a recent article [5], the deternination of ¢ - function is con- sidered Corresponding to the requirements on As in the effective block— hand as: (1) As will not be less than a given set of different values of'aMi's in the corresponding frequency hands as shown for a low-pass fil- ter in fig. 2.2.3, and (2) in the effective pass hand As SCI. 0‘ A i /. /, f A ,, - , , 4, // // K, I // // ‘/. /./ ."/ 1,. ,1 //, // ’ ' p, , ‘ / , I C /7 /( ;/. .;,/' 7 (11.11 / ”I ' 1/, / I, , a1 W l ' ///'//// /‘ ' I am E //’ . l . __ ‘ 0 fl f2 Fig. 2.2.3 Also, ¢ - functions for different types of filters (i.e., symmetrical, antimetrical, and dissymetrical) which satisfy the imposed restrictions are investigated. This investigation follows exactly the same procedures stated earlier, i.e., of determining the approximate locations of the poles of AS. Since the form of the ¢ - function is similar t, that as in eq. 2.2.h, AS has equal-ripple characteristics in the effective pass- hand. Now, in conclusion, it can be summarized that in filter design on the basis of insertion loss, the important part is that a ¢ — function must be found which satisfies the imposed restrictions on the AS function. (B) Determination of Element leues: -11- After finding 3 ¢ - function, the next problem is to find the element values of the filter, either with lattice or ladder configuration orres- \O ponding to this ¢ — function. Explicit formulas for these elenent values exist far some special cases such as: l) Butterwcrth or Esch‘yscheff he- havior in the effective pass band, 2) Tschehyscheff behavior in both ( effective hands (Darlington's filter) {C1, [5], L,j, [le, L11}, L12}, [13], [361. Although Darlington's method of determining element values includes the general case, [3], [7}, [18], the technique entails long calculatigns and hence is laborious for any practical application. However, the digital computer overcomes this difficulty somewhat, LC], [37]. 2.3 ADVAiTAGE OF IMAGE PARAMETER METHOD The possible use of digital computers with the image parameter meth d is mentioned only in passing. Alth ugh it is believ d that such a digita- investigation holds a promise, it is not the gain idea of this thesis. The key notion of this thesis, is the utilization of the image parame- ter method to overcome the difficulty encountered in the calculation of the element values for the general case, as given by Burlington. Tuttle [1%] and others have emphasized the need for a fuller investigation of the classical image parameter method. Specifically, one of the main ad~antages of the image parameter method is that once design parameters are found, the element values can easily he determined since there exist simple rela ionships hetwecn the element values and the design parameters. however, this method also suffers fron a serious drawwbach, i.e., the cut-and~try method, necessitated because of the fact that the design parameters are closely interrelated. The above -12.. ‘aw-bach is a direct consequence of zohel's decamposition formula on which the existing mage parameter method is based. 2.h SOME DISADVANTAGES OF ZOBEL'S DECOMPOSITION FORMJLA zobel's decomposition formula is considered in section (2.1). This formula is'rewritten here for convenience. p .~ 24:: z z + 2.1 z~ + 2.1 -2P P s s R s 2 H 2 , I I e 2 :7- .73.... “.....- ___._..._.-._..- 1 Jonas. e (2.1+.l) “s a 24:43le 24/413312 “’ The main disadvantages are: (1) (2) In the block band, the interaction term is commonly neglected. It is not at all clear hJW eacn term affects the inser- tion function. Specifically, consider the example of a symmetrical filter with equal terminating resistances. For this case, tne insertion loss function in the pass band is zero at frequencies where the image impedance is equal to the terminating resistance, i.e., when the second and third factors within parantheses in eq. (2.#.l) are equal to unity. But this fact, will not enable us to conclude that these are the only frequencies at which AS is zero. In fact, other frequencies do exist at which A5 is zero in the pass band. However, these other fre- quencies cannot be determined by considering conditions at which the logarithm of the individual factors in eq. 2.h.l vanish. Any determination of these frequencies, for this particular example, can only be accomplished by -13- Considering the combined effect of the interaction and reflection terms [2]. (3) Zobel's decomposition formula does not include the possi- bility of considering tne case of degenerated four termi- nal networks [l5]. (h) At cut-off frequency, the reflection function (factors 2, 3, h in eq. 2.h.l) tend to infinity while AI tends to zero. Consequently, the As - function has an indeterminate form {at cut-off frequency. However, it is well known that the AS — function at the cut-off frequency has a finite value, a fact which does not manifest in Zebel's formulation. On the basis of Feldtxeller's study [lb], Belevitch has expressed the insertion loss function in a different form than Zobel's decomposition [IS], [17]. This new formulation of insertion loss, As, is used in in- vestigating the As in only the pass-band of the filter. In Chapter III of this thesis, Belevitch's formulation is derived for different types of filters, i.e., symmetrical, antimetrical, and dissymmetrical including lossy cases. For the symmetrical filter case, the factors which appear in this new formulation are discussed in general, and new properties of these factors obtained in both pass- and stop-band. Also, a design pro- cedure on the basis of this new investigation is described in Chapter VI. III. A NEW FORMULHTION FOR INSERTION JNCTION OF IMAGE PARAMETER FILTERS The insertion function is here first expressed in a different form than Zobel's decomposition formula. This new form is derivable either by starting from eq. 2.1.h, or from the definition of the insertion function. -11} .. 3.1 ‘l'ZiE TEEW FOPJJULWIOE‘I OF II‘JSERTIOEI FEWCTION Consider a passive general four terminal network, N, as in fig. 3.1. The input-output voltage and current relationships in terms of the image parameters are [2]. L._.. 7' 7 '~ - /.rr .1. 1 . . 7 , 1 Cosn P Z11 oI2 Sinh PI V = ... (3.1.1) ———-Cosh P ‘ ‘ V2311 21.2 I H H U E t’U H fl] HH HR) H H R) l 2 + | \/V\IV D R I R E 2 4.1 —--- (PI) ‘- 4312 l a 0—— —-0 Fig. 3.1.1 Eq. 3.1.1 in symbolic form can be written as V A B V l 2 2 (3.1.2) Il C D 12 In addition, we have from the fig. 3.1.1, v z , q 12 R2 12 (“1.2) = E - 7. J vl R1 11 4 (3 1 J.) From Eqs. 3.1.2, 3.1.3 and 3.1.h, after eliminating V1, I1 and V2, a re- lation for 12 in terms of E is given by E 1 I2 CRP. +11? +D:izi’+B (31°?) -1)- On the ether hand, when the source E is directly connected to the load, (N is taken away in fig. 3.1.1), we have V‘ I 11; I 2 Wm ' (3.1.6) 2 n1 + R2 Therefore, the ratio which determines the insertion function is I2_CR1R2+AH2+DRl~i-B ‘ (31.7) ——'- R + R J' " 2 1 2 Let the parameters, _ Z 1 Z1 T11 _ (3.1.0) L12 22 " .132 be the normalized image impedances with respect to the terminating re- sistances. Substituting the values of A, P, C and D from eq. 3.1.1 into eq. 3.1.7 and from eq. 3.1.6, we have I 4,—— 77 I7 "’ + Z Eg = ePS ~ 31 fig__ E—:~—l:g Sinh P +':l—~——g Cosh P (3 l 9) -- ’17 ’7 7 l O 0 I2 R1 + R2 4‘41 '2 4 I W1 Z2 I . From eq. 2.1.3 and 2.1.10, we have 2.13 R1112 l + z. .4 z. + Z. 2 e =.—~ ~~"§ -;HTW::'Sinh PI-+-E;;::r Cosn PI (3.1.10) 1 I- ..4 Z Y; F ..J F B :: are ~‘01r-:_;.:_:::_g Slim PI 1‘ *‘T:;*"7::—2. COSh PI (3.1.11) 5 ~141ng ”1‘42 ‘fhe formulas (3.1.10) and (3.1.11) fzr insertion loss and inserti»n phase are used throughout this thesis. In the following sections, dis- cussions or these formulas either in general er in some special form are presented. -10- 3.2 SYMMEIRICAL FIDTERS For symmetrical filters, Z11 = 212, let us assume that equal termi- * nating resistances are always used , i.e., 31 = R2 = R (3.2.1) then 2.1 = 2.2 = 2 (3.2.2) Substituting the relations (3.2.1) and (3.2.2) into the eqs. 3.1.10 and 3.1.11, yields the following 2A8 1 + Z2 . 2 e - T 811m PI + Cosh PI (3.2.3) 2 i _ 1 + Z . . . i . as u arg [ 23 - Sinn PI + Cosn P£] (3.2.4) (A) If N is a purely reactive four terminal network, then, in the pass-band, P = ‘ B = 1.1103 0) Therefore, eqs. 3.2.3 and 3.2.A take the forms 2A 2 e 5 z 1 + (L3 Eve—Z) Sin2 13I (3.2.5) 2 1 + Z ' ’ 7 :: B ’ ’ ‘ LS. arctan I: 22 tan 1] (3 2 L) respectively. In the block-band, since 2 is purely imaginary PI:AI+31;I: (1:20,:1,+2,...) o...- h-m- J a 139 _ 441~ must be added to the insertion loss H1 + R2 f} in NIH * Otherwise the term expression. then + Sinh A Sinh PI H I Cosh PI II + Cosh A - I where if the plus (minus) sign is used in one expression it must also be used in the other; therefore, eqs. 3.2.3 and 3.2.4 in this case take the following forms: 2A" ,2 2 e = 1 - (24—) sinn A1 (5.2 7) 1 - Z2 38 = arctan [j-Zir-tan A1] + k n (n = 0, I 1, I 2, ...) (3.2.o) fherefore, in both cases, i.e., in the pass~band and blocx~band, for insertion loss function we have from eqs. 3.2.) and 3.2.7, 2A5 ’1 4 Z2 2 .. 2 . e a 1 — (i«§Z—— Slnh PI (3.2.7) (B) If N is a lossy four ternanal networx, then we cannot distin- guish between pass~ and stop-bands since 2 and P are complex quantities I for allcn. Let, z z r + j x (3.2.10) PI = AI + J BI (3.2.11) then -2 2 11-114 _ 1:: (r3291 .1: o 1 , _. 95 .1 2% _ 2 (r + JX) - 2 l k r2 + xé) + J 2 <} - r2 + xé) = wl + Jw2 (3.2.12) and binh PI = oinh AI Cos BI + J Cosh AI Sin b1 _ _ , . (3.2.1:) Cosh PI = Cosh A Cos B + j Sinh AI Sin 5 I I I -10- Substituting eqs. 3.2.12 and 3.2.13 into eqs. 3.2.3 and 3.2.4, it can oe shown that s 1 J _ ‘ 2 , “.2 . 2 . 2 1 e - (t . 112 + 1) COon AI - (wl . W2 - 1) Cos s: + w Sinh (2A (3'4'14) 1 _ w din (2sI) - 1 I) 2 W olnh AI Cos hI s #1 Cosn AI sin LI + alnh AI bin o1 w Sinh A Cos s - w Cosn A sin n + Sinh A Cos s 1 I , I 2 I I I .-.—— ...— BS 2 arctan (3.2.15) .3. AurinErRICAL FILTERS (Iivsnss IMPEDANCE FILTERS) LA.) For an antimetrical filter, image impedances are the inverse of eacn other with respect to the product R 2 i.e., 1“2’ Z" Z r I 3.3. 11 12 R112 (a 3 1) then Zlue = l (3.j 2) and let 1 = Z =‘~— 3.3. Z l 22 (- J 3) Substituting the eqs. 3.3.1 and 3.3.3 into eqs. 3.1.10 and 3.1.11, we have 2A 1 + 32 2 . e S = Sinh PI + —-§Z—* CQSh PI (303-4) 1 + “2 BS = arctan [Sinh PI + ”“22 Cosh PI ] (3-3-2) (A) If U is a purely reactive four terminal network, then in the pass-band, z is real PI = 3 Bl (AI = O) In this case, eqs. 3.3.h and 3.3.5 can be written as follows 2A. 92 2 e ° = 1 + l—£;=~ Case R (3.3.6) 54 I L . B = arctan 2-—~ tan 3 (3.3.7) 5 2 I 1+2. In the block band therefore, eqs. 3.3.4 and 3.3.5 can be written as e2A5 - l + l 7-93 2 015‘s2 “ ( - 2‘; J . AI 1 + Z . . 1 b ;‘ aI‘CtEI‘I [ . -----. ,— COtIl L) + l. K :- : O + l *- 2 o o o I o I. o k] S ‘ ' J 214 I ( ‘ } _ .’ ___ ) ) ( .) .3 / U) D.) (’1 v I Eqs. 3.3.5 and 3.3.? can be given as one equation. Therefore, for anti— metrical reactive filters, the expression for insertion loss is ~3 2A 2 2 S l ' Z 12 a a e - l + < 22*) 0001]. PI (3.4.10) (B) If . is a lossy antimetrical four terminal networh, then the image transfer 1583 function does not vanish in Some interval, i.e., there is no pass-band. For all values ofch, z and PI are complex quantities. Letting PI : A1 + J 3I and substituting eqs. 3.2.12 and 3.2.13 into eqs. 3.3.3, we have, for a lessy antimetrical four terminal network, 2As 2 2 2 2 2 2 s t + ' + 1 ”i: J — ‘ + “ - “in L e (wl ”2 ) s 1h 11 (”1 mg 1) s I fl - 1 . ‘q (fl_a (a ) hr fi_. ( .I 2 x, 2 (;l..:.—l.l) 1 1 oinh XI + 2 bin 2B1) + («l + a2 ) -20- Cosh A Sin B + w Cosh A Cos h + w Sinh A Sin 3 r arctan l I 2 I I 1 ~ I 1 I “s ” ‘ oinh A Cos D + W Cosh A Cos h — w Sinh A Sin 3 ‘I I 1 I ° I 2 I ,1 (3.3.12) 3.h DISSYMUETRICAL LOSSLESS FILTERS In this section, a general expression for insertion loss and p ase for a general lossless filter is given. In the pass-band: Therefore, eqs. 3.1.10 and 3.1.11 give ,Y 1 7 2 , , 2 7 3 1 ~ h h - 2As » ‘1‘2 (1 ‘ 1Z2) , 2 D ( 1 * 42) 2 o 1 e ~ 2 Z Z Sin oI + £~;—*——-Cos II (3.h.1) (Al + R2) 1 2 1 2 1 r LIL B : arctan U —-r—-tan B (3.h.2) S 1:11 + 2.2 I In the block-band: Zl Z2 are purely reactive PI 2 AI - J h n (r = o, i 1, i 2, ) then in this case . T a s 2 , 2 A? 4/4.: ’- 23s 1R2 (1 + 1’2) 1, .2 (31 * 42) .2 , q 1 e z 2 Z v binn AI - Z Z COSh AI (3.5.3) (B + R ) 1“2 1 2 1 2 Z, +2. B = arctan u j ——le~w§~ Cuth (2 A 2+) s ‘ 1 + 5152 ‘ AI “' ° For all w's, insertion loss can be given in one equation as follows: _ _ . 2 . 2 1 u 4- / + ” 2As 31“2 (1 ‘ ”152) n, .2 (31 “2) , 2 —| . - e :: _._._._...m.—.2- ,7 /,""‘"“"‘ 13.1.1111 PI ' ...... 7;" .-, ‘- C'J 5h PI ( 3 ° 1" ' Z ) (R1 + R2) “1‘2 “1“2 01‘ IV. DIS CU°SICTY OF TFE ¢ - FIT-1:231“): 01“ SFIEIEI‘RICAL FILTERS ' I)? ’1 LEWIS C11 11.11.11. PUULLL 1.115 In the preceding chanter, insertiun function is ref riulatei d1:- ferently from Zobel's decomposition. The inserticn less function, A”, :3 s so reformulated, permits an ease cf nvesti gaticn of As net possible with Zobel's formulation. Belevitch [17] has used this fermulati n in studying the AS function in rnly pass-band. In this chapter the complete discussi n of this f; mulatien for svnmle:rica] filters is c DSILC red. ‘Fhe rezcra b, I ......~ aura-»- Beleritch's p. (.1 properties ;Ut8129d fr:m this wider investi ’J'tiqn also incl: 0 results. These general esults are subsequently apslicd t. filter design. 4.1 TILE ¢ - 1771031011 111111115 CF 13110.3 Burners 3 Fir symmetrical, less es s filters w: th equal tezninating resistances ecticn 3.2 that the insertign 1” c3 ('1 U: at both terminal pairs, it is shown in in either band s(pass or block) can be g1 veri 0v the sinble ioimula -~--—-—-— Sinh‘ P (1.1.1) In inser ion loss theery, it has been fsund calvenient to let $32 = - [Lg-f— 511.112 P (4.1.2) The hyperbolic term in eq. 4.1.2 can be medified and expressed in terms of a ratio function, H, of the filter. Letting PI I: =- Tanh ~2~ (4.1-7) then, 2 D .... .oi-.h il l _ "£2 Substituting eq. 4.1.h ¢ : (11.1.5) - ‘2 1 . . . n In eq. h.l.p, C has two Iactors which are reelprocals, each 01 which is expressed in a different variable. it is sh wn that the first factor is effective in the seccnd is effective in the blecn—bana enly. h.2 A USEFUL COINORMAL MAPPING q AS 2H 2 <—-—-—--~> 2 are recipr cal in mathematical fern but }. l . ferent variables. erties of the ether. In this section we Consider in eq. h.l.5 and investigate its preperties. Consider the following function of a conplex w : (.1...1__Z.2. 2 22 If we let I (Z * E 1 k -- 2 W: 1,2 already noted, in eq. 4.l.5 the two factor Therefore, the study of one of In the following discussion the pass-band only, and ,1-222 1 s y““‘ - and z are expressed in dif— _ .-. ~...1 “,1, then yields tne r111- only the fi°st factor variable 2 (k.2.l) (h.2.2) A 4‘- [‘0 K) J V Therefore, we first consider the well—known functicn ef eq. h.2.2. In eq. h.2.2, let and iv (2.2.h) (4.2.5 _2u._ If the variable 2 in eq. h.2.2 represents the normalized inure im- pedance, the real part of 2 must be positive. Therefore; in the fallowing discussion the open left half of z — plane is not considered. Fig. h.2.l .art of real axis in the z — plane is mapped into the -—. » L The positive whole real axis of l - plane. But imaginary axis of the 2 ~ plane is mapped into the same part of the imaginair axis of x — plane as shown in fig. 4.2.2. Once the x — function is determined; W is given by the simple rela- tionship in eq. h.2.3. The mapping of the z - plane into the W - plane is also shcwn in fig. h.2.2. From eq. h.2.l and the fig. h.2.2 we can see that the positive real axis of the z - plane is mapped into the positive real axis cf the N - plane (but not in one-to-one correspondence) and the imaginary axis of the z - p ane is hopped into a portion of the negative real axis of the W a plane. Substituting eqs. 4.2.u and u.2.> into eq. 4.2.2, we can obtain the iol~ lowing relations between the real and imaginary parts or z and l. u=é~ (p-213) Cos‘c} 1 l ”I, a (4.2.3) V : -2- (p + 5) OLD o From eq. 4.2.6, the following properties of the A — function can be found: ”'1 I .2 (l) The circles (p = constant) in the 2 ~ plane are napped int; the ham focal ellipses in the X - plane. Their equations are given by (J q [H .__.____7_____-_.__.. + ...—......“— : h (El-.2.7) (0 ~ — (D + — The common focii of these ellipses are the (i) and (~i) points of the l ~ plane. (2) Two circles with the radii pl and 92 respectively areonapped into the same ellipse if pl 92 2 l. Therefere, the inverse funCtion, i.e., z 2 f (i) is not single valued. The l - plane, actually is a two-shoeted Riemann surface. The unit circle in the z — plane is mapped into the section of straight line between the points (i) and (~i) in the k ~ plane. Therefore, along this section cf straifiht line a cut can he made and the two sheets of Bi mann surface can be considered separately, each of which correSpends to either the inside or the outside region of the unit circle in the 2 ~ plane. (3) The straight lines passing through the origin in the z — plane (8 = constant lines) are mapped into homofocal hyperbolas in the l — plane. The focii of these hyperbolas are the same as those of ellipses. his mapping of eq. h.2.2 is illustrated in fig. h.2.l. rq w - plane Since 2 represents the normali e6 ”wage impedance of a lilter z is real and positive in the pass—hand and du‘o imaginary in the bloc: hand. - . . , l - ” Therelore, the main properties of the iactor (___7;_) , A c. as can be stated in a theorem. Thus: Theegfng. The factor (fmr an LC network) 2 l — 2 W (722*) is real and is always positive in the passuband and always negltive -26- and not greater than (-l) in the bl ch~band. At the cut—off fre- 'Q D quency, this factor increasis witnout limit. fiuw, we can c nsider the second factor in eq. h.l.5, i.e., 2H 2 /I " 1] ; (mm‘ {LP0203) Since this factzr is th. seal of the first factor in eq. h.l.5 ex~ { H O O *J 'U a cept for the difference in variables, the mapping in fig. h.2.l can be extended once more by taking the inverse of the W - function. In this case the napping from the H — plane into n « plane can be given as in fl' - . I $ h.2.3. q - plane ‘~ :» fig. h.2.3 Fig. h.2.3 indicates that the positive real axis in the 1 ~ plane is Inapped into the positive real axis of the n - plane. rhe imaginary axis ' 4.4.. _ q 7 p n 3 . a \ - ‘- .. n . ‘ (4 tne h — plane is mapped into the line segment (vl, O; on the nudgbwvb H) Ikeal axis of the q - plane. ’7- I... U! V1} 9 (8 *4 L4 The properties of the h — function are well—known [2). 1H8 the block— Considering the tiun, the two fL cf these twu 3"} U) inery band. ling in “Lid-3; :Lutp thflnt ):'rk..'pert A. zed and stated in the is real aid Viri es betqe e1 (0) and quency, q is zero. In order ts illustrate ures. fig. h.2.h and factors can es the filter and it is fig. h. ies 43 'V , a l‘3113 ()1 Thecrem II. The factcr (fir f)?” 2 Lg; 7: : (...- ) is always pesitive {-1) the fig. h.2.§. are sily be has a zero or pcle 2.3 Lf the a t and the properties isct r n, in eq. 4.2. at the Cult-Off frequency. if [1‘3 QI‘OLLI an LC—netwerk) . - as (‘ ’._.« O Furthei pr‘p ert es on the i.et-r (1) At the critical (2) At the cut—off (g) In the (matching pcints) of 2 ~ curve and the horize correspond to the zeros (A) In the blcck 2 respond Theorem I, these For tie values: C QI'I‘eS in thechloch—L , 'n tne pass properties CI'V'EO. frequencies to the maximum p: Z1 pen ding values . Y r.‘ 'u 1“ snax-Lakla and z x "" \IJ. cl 8.1’ 23 and infinite —bdnd, the absc ~ curve (imaginary) and the herizsntel lines, ,ints of W - c e nd, and '7 H of the frequencies; W has W. issa of the intersecti functieq, l psles curve. As it .3 U:lw 1’1 -band. At the cut-off cf the two fecters 3n e presented. Bethe from these figures. real and positive 5 can be nt:il line, .3 i .1 :3 3.1331” its Ere» 1.1.18 ‘ “C J‘ a. pass~band, the abscissa cf the intersectien rtiits r- 1'4 en points = + i, c s hnemn the same and equal to (~l). s such that 31 D In" l '. .49 l— W are th e 83: . .. l «.,_ r f l I J .L _‘ ,J irxlu x _ 1 ¥ 9. / _ z _ / 3L / (w\ / 2 V.\._/ _ z P. / d m / V" fly 1 //— Z 3 4 7 _ .. // 1 / // — [I {1 ¥ V I <4 I I III I I ll l I II I I I I l_l r x2 Ilirl 9% III/ _ 2 _ . I I L _ fl _ I — I q / l / / l l i _ — I IIIIIIIIIIIIAWIIIIIIITIIII “ ‘ — ”.1 , _ .IlrllAllulnlllll 2 X6 _ _ x 2 _ l X? a L _ e l r ....... e 4 _ V X _ _ _ l _ _ _ :2 g _ w _ _ z _ lll I‘ J H I u D l r O l A. .D l A. 0 . '1 ~ a ., 14.2% “29- Further properties cf the q - factor: (1) At the ritical frequencies (poles and zeros) of E, q has zeros. (2) At the cut-off frequenCV, q is zer;. . J" . i . i. . i .,. . , _ x (3) At the iniinite frequency, n has a liniting value oi. T __ _1_ 1. l i - 1+ - “-H' M-”.-. ”.... ---... 11 '=1 l where n is the number of sections in the filter and mi reprc~ sents the m- parameter of each individual section. As can be seen from the expression for We» , if the filter contains at least one constant—h Section, then “cw tends to infinity with . (D (in this case H (00) = l). (A) In the block-band, the intersection points of H - curve and the . horizontal line H1 = l Correspond to the poles of q — function. (5) In the pass-band, the abscissa of the intersection points Lf the H - curve (H is imaginary) and the horizontal lines HI = + i mw correspond to the minimum peints of the q - curve. From -neorem II; it is known that these minimums are the same and equal to (~1). (e) For the values of 51 and H2 of H, such that slug : i 1, the cor- responding volues of q are the same. 4 . o .-.-..1 m — .' .. 1 f" .M, “,5 ',~ ~ 4'- - ‘ ., ' s ° .. ,3 ihis ; indie cdn be deiiwee i: the is ution between HI and n is conSideied [2]. For an inage pe'ameter filter AI is the sum u: all ihsge attenuation Lunctions oi the individual sections and at iniinite frequency these indi— vidual lunctions have the 'dlue [2] or A.3 FORMULAS E\R rhE CfiARACEERI EIC POIflJS O? W AJD q CUhJES In the preceding chapter, general N— and n— runCtiens are considered and the general properties or these twe functions are obtained. For the Characteristic points or W« and q— curves such as, x1, x2, ..., x31, ha2, ..., in Ii5. 72.2, 0..., >111]- XI12, no.) X01, >182, 000} and X’I\1: X792; ... A. x21. u.2.u and fig. u.2.5, it is difficult to obtain a mathematical eXpressi n. But for particular z- and h- functions these rormulas exist and are given in the fellewing [7], [2] (LC network). A) z - function is in the form cf a gecmetric-mean variation ”(Eschebyscherf - approximation). Let the number of critical frequencies of 2 be n. Then, Critical Lrequencies (poles and zeros) of z recur at (bl ch region) 1 . __ . A Xi 1' *F-i—EE-T (l ‘ l) 2: .0.) n) (4'03-01.) snt~.-- n+1 L. .1 Unit values of 2 occur at (pass band) -- -_2.(.n-.‘.j.‘..) (:1 "* «I ._ . ) (3 X2, u Xu sn [ 2(nrl n (i — l, 2, ..., n + l) (4.).2) A Extremal points oi z uccur at (pass hanfl) (Xe, L O, Xe n+l L x“) L) H ~ functien is in the form sf a gceretric-mean variatien (”schebvscheff - approximation). Let the number of critical frequencies of H be n. Then, Critical frequencies of H occur at (pass band) _31_ real ,l I‘lllllllllllllll1ll -1 _h X: : an [:1._J-}.;..: :; (i :1 l) 2" ..." n) (4.3.);) Poles cf Al (H = 1) occur at (blue; band) = v l (i z 1, 2, ..., n + l (Q-B-D) “Pi *un 2 1-1 i an - “’~ A 2 n+l) xtrenal paints vf H occur at (block band) xv, . 3,11 = ““‘ (i = 0,. 1. n H) (hex) M 33:1 0.; 114-]- (v; : ~ ‘ ) x~ :. 0Q) \ n . ‘ Iii) ‘ 11A [1114.1 ~— 1— . 4-v FORMULAS FOR TEE lNoEfi"*' L05 8 AFD PHASE OF A SYMLETRICAL IMASE PAAALjfiflR FILE ER AT CUlv0FE FHE'QUEUCY The two factors, w and q, appearin5i in the exrression for ¢I in eq. . , - A . _ i 5 '2 _ 4.1.2, 51 ves an indeterminate :orm ior Q at cut-off frequency because the factor W tends to infinity while the ‘iactor q ends to zero when the frequency approaches the cut-off frequency. It is sh own in this section a 2 .5“ _- J— >n_q 3'13, o _s o q u that ¢ has a iinite value at the cut‘OLL frequency. inis prOLLCL lS dis- cussed by Belevitch [l9] and insertion loss and phase expressions at cut— off are given but for only the Special case where the image impedance of the filter is an m - derived tb'pe. In this section, we generalize Pele~ vitch' 8 results to a general iuage impedance case. In anticipation cf a detailed discussion of file next chapter assune now that terminatin5 sections are used in filter design. 'Sherefore, the ima5e impedance of the terminating section is the ima5e impedance Cl the filter. Using eqs. j.h.2O and 5.h.2l, and in those equat i ns letth" 'U n C_J F §< n Eip we have, i. i i i "-2 . ... 1 '2 Z: :_ m__,_,_-_- ”2.1 Jay-l -. (i301~ v (id) I ,2 2 Y2 v (h.4.l) On th other hand, the H functicn ef a syMnetrical image ; ranetcr filter can be fuund as fullews: including terminating in the cempgsite filter, Each half section sectians, has the fellewing h, functien II 'I :I ‘- O‘Y‘o J where Pr and H represent, respectively, the image transfer and rati: a ‘ “ functions or a 2 filter secticn. For the filter consists of n sections (includes terminating since P n P :- Z PIi and H = i‘anh ~5- Frem eqs. M.A.3 and n.u.u, we find :5: A H I 5 5'4 D v n 77 (l + min”) - H F" *J {I ‘O .3 r;- :5 II A TT 1. (1 + mino) + "1(1 - mihl) H F }. But since, in general, 3—1 I+ w, .* y. (D |+ .L. A l H \ 5 {Ti (1 : a1) ; '0 From eq. A.h.3, we have A 1- C p . .9- . ‘Ld V sections), (4.h.p) where A1 2 a1 + a2 + ... T an A2 = ala2 r ala3 + . . + and if we let H 5 :11 a . l l Q then eq. h.h.§ can be written as follcws: (l) n is even; I," + 4A... + o o o o *- ix L = £1 A r a ; 2 . 11 (2) n is odd; I 5 A1 + A3 + .... n A- u + :~ + A + .... + A n-l On the other hand Therefore, as x appreaches the value cf that H apprcaches the yellowing values: (1) n is even F. H , n-l l l , l .5; z; 04’?“ 3* :1".- ...... T ——-—~ 1‘ n i ’ If] m n ‘9 l 2 OT unity, eqs. L ‘1'. (h.h.e) (h.h.7) (h.h.8) (4.A.9) h.5 and h.h.9 shun -42.. . An N 1 :1 l -..- 7.. H ...... A e) .1. , l H" *' -‘ ‘f' I o o 'f' _- New, c nsidcr the value of Q — f) and h.h.2 fer normalized image impedance, P7 .. I .— n4 5 F______ ': V1.~ x2 P (x) *,1 1. an d L; ],_~_ JR:— 1 _. }'2 s. M.h.lO and h.h.ll can be written as, ”1353-; . 13—... .__.. T3" J""1 anC " 2 _ x , 1‘ 2:4 _ _' Q2(X) \/ with where n is the number of hating sections) while'v terminating half sectiens. Therefere, n > y’. For different number of n and ‘V , different types e functiens are obtained. However, tnere as follows: Case 1 - functinn at cut-eff. uron ego. is the number cf half ' are enly feur such different cas- (for V’ even) (4.h.lj) (fer n even (4.4.lh) (fer n add) (4.4.15) (4.2+.lt) full—sections in the filter (including the tcrui~ sectiens in ene cf the ,3 V3 2. ‘1 LithLL Al'— 7.3 ,c 2— .5 ‘96 can) (V Case 2 — (n even) ()) even) ( n even) (N) even) i ’—'—— z — h. 'Jl - x2 Pl(:) (‘V ’ 3 __.3L.i; ( i Vrza — l “€2( ) Case 3 ~ . 1 z = S- ---‘-M— P (X) ‘fi? Jl.— x2 2 \l‘o l I; -— ”rm-EM“ Q2(X) Case A - ,; l "' :1 ‘‘‘‘‘ W P 1‘ b {D \(l — x2 2( ) n = --‘->-{~—--* C»; (X) ( 11 08:13.) 12'- l 2 and H are su'stituted int; the eq. h.l.h, i ¢2=*< >2(1‘?Hs)2 . 3 . -2 and when x tenns tr unity, ¢ 1—33. 2s . - s m w -+'»? ‘ « u ts the Cases mensicned absve: (for . Q1 2 3 (Iur 06., will have the fellcwing values c rresgunding case—l) r: 35 .- 7 Ca». J) but, fro:.: eq. 3+.«'+ fact, enly tne (l) for }) add (2) For t) even I ‘2 9 (be) The ’J’ S alue cf 0 i h. ZCZIi‘VJTI we Oahu.) >v. :3 1:1 To calculate the value h.4.l and h.h.2. Since each in tends to unity, .8 lim (1 - x-—+l then Therefore, substituting eqs. h.h.l9 and 4.h.20 with u.u.21 and h.h.l§ yield Indeed, we have factor in eqs. h.4.l and h.4.2 can be written, when II A fry ; 5 3 , -~-r 7* (lul case-4) -‘LT 2 {Che p. ' 2 1 a ll ffur ‘i these 9 are not dijfcrcnt. In fillewing twe cases exist: (4.h.l7) k IT. 1:I K , 2 . . by" Pqu) (~1- . 4 . 13) ‘i ($.4.19) of P1 and P2 it is sufficient te cansider eqs. for P and P the fellewiigz l 2 \r 4 k _2) 2 ... .1- D k ) : mi m m ... m 2 ‘ 2 h 'V-l \ 1‘." m 1- fl (LL01; o 20) u 11:} o o 0 “Y m m ..... m 2 O r“ W“ fly (h.h.2l) 11 “‘3 o o o J: th int' 4“ w S S 11. . . . -s1 {:1 n . - I n ' I '23 3‘ (88.41.14) (£3330 Y (ICE) “7‘ 3 Y” Cm r ..._:.; :2 (1)) Q id]. I 2 i I _ Y L.. a o o 4111'; g.“ 1'. \J 15113 S S m ... ,m m (i33hfi)(neve \{ rel) I {I y p S :i S 3 3. I HI 0 o o - III _I:I 1"}1‘ = ( 6w) Q 1 I21 I- Y t I i edni ES.#.# bus SS.$.# .ape aniimdidadflfi .€.I.# bus I.I.$ .aps zebianoO :iie-Jua is aaeI n.idieani 103 Kevsd aw .anuideupe eaedd S S m ... m m "f I [I Y C J. l , (bbo \{ 1e?) TE EE; ' yn' W—;E~ j-' + I Id é = “A J- ‘Li I_ L 000 +3“ IA .1 C4 ($3.114) has 9 [I S “I o o o “I ‘f (nave V'xu?) 4L :3’ -~:%«—- Ii. 4; +.[ mi é = 1k . an I“i I YEJ ... Inl TH — a (as.A.A) ‘ ' is IeJIii TSJSMSTBQ egsmi me is easdq noidxeani end edsinoisa won 9w .voneupari lie—duo add ,#.S.{ .pe zeasdq neijteani and in? even aw (radii? Ieairdsmmva 5 re? ..e.i 83 4 I (BS.#.¢) ( q dan + q dnia --"- are = 8 l_I I 38 a .regedni as a: e suedw K(O 3 IA) k 8L I: J = I q (gonanperi lid-duo and 3A .edsriniedebni 3i BS.fl.$ .pe mi MTSJ Jamil and vine .sroiarsd? ,.e.i {33.A.# .pc ni are: Jamil and net nuiaeuoaib evnde and anidansfl (TS.$.$) ' q dnia —9 ----- i u) I 3 sonia- .llo—Juo is “Y 10 euiev end erIuoIso n59 9w = q dria AF To“? :3de ew I r 7 ' ‘ I _ 1-1 - I) K}; 1 i--.;‘._..._ . -.....” if; (14.4”..7“) 2... l _, c. 1' “ I ‘Vr . r‘- ‘ " f ‘ ~ - r: 1 \ ’ '> n r” .‘ v 1 I r A I * ' ' r -“ 3.1». i") Eran the ah ye discussien, tne values ei A ane I are inrwn. he-eirre. we can innediately attain the in r‘ q, 42 ( ) > 3: J -—~., 1.— (for 1/ an) O .::-. P A? 1 J. H. 1 r . q ,' . 5." AAAL o o o .u (_ n 1 .l . J- L “f . I A ¢ Y‘\(ll_ -.-_ ‘i' J .- 7.....l...vm.- L"-:~YWI- <‘ :7, \IZZ' Y CIPC‘1)<“I’.£I‘-2lf) -) .... J.\- 1 1r L11 0 o o H _. 14 . l J Y -E L— l I ,- . . . K I . " '1.‘r;eref<__;re, c..ns:.;1erin5 e0. 31.4.2; where C nah (13 x) > + l, at cut-- B = + arctan —- ‘_'—+-~—.~-~-——----—-- E -:- .31: (:.r V -:_:d«;_) 3'73 = + arctan 7:“ ‘“""-—' ”..-...“ Z ~--- 5p: (fjr V even) V“ A“ . m [— H L < I i—-’ , \‘\_ I" I }.J 1+.5 '1'1PP}~:OL-’;11-11(32) FQRIIULA FOR A‘ LI BEE L-LOCh—LAIED 0 Consider the Q — function in eq. %.1.2 in the blucL-bani, the emfressi O for Q“ is, J- D- where z is gurely imaginary. From Theorem I in sectian 4.2, the first rac— ter in eq. #.5.1 with negative sign in frent »f it (3 real ani . . , 2 pQSitive and net less than unity. Tneref re, the value a: the Q - func~ O p tivn is always greater than, or at least equal t7, the value mi 5 uh“ AI. _ 1+0 -. o ‘17“, \X"! f rmula f-r 13 find an at“- “inate A- 0 write, 211‘. .2 , 2 e s l + Q > 1 + Uinh AI ; Cnsn Al MH ‘ 3.1: ...‘D :ie_a ty ti 1n 2 ’1'“. -neref re, the insertien loss can be less than V .L , the h13ch~band. n the symmetrical filter desi restrictiens an A” the last term in eq. 4.5 “ O O . ‘v 4 ..4 '3 J = 0.65 nepers or 6.02 decibels .f‘ e; requirement fur As tlach~cand. r1, . an".-+ .: aw '~:~ - n , , e, .a. p 1 C “ :cr syhmetiicai iiiters, the I Ihula 4.3.3 in the b1;ch— 4- ,. . I) transier less is ad band, *ne can then ¥4 \ II the image transfer 19s EU, tn retain the ihgvsed .0 ,' 1 ‘. N . ‘1 ' ‘. 1 . —‘ is always c ns dare , 1°C ded tv the given Centaur _: 3 . .. , - ‘1 .. .Lb 14511."..113‘u1 dim}. (LITE, he htained Eran the a bel's decanpasition f rmula, eq. 2.1.h, i.e. ne— glecting the interactien term and c nsidering the minimum if an if the reflectign less which is very clsse te ~3 l 1...“, 3 o I V. FILTER TEMHITAT 7} General 5.1 Fig A law-pass ilter-terminating secti7n is he s m:3. It is a law “a1 — f) a meresy cf terminals, c nstant-h, at the ether pair of terminals, 9’" ‘ 1 " \ "c' '0 ,~0 ? 'r\’- ""1, ‘n‘\ . ordeal. 1118 due; \JJ. an liner-3e i..-.L:eda..1.:e higher unit values of z in the pass band. decibels, the fermula 4.5.J the image impedance is of the simplest ercr, SECTIQJS re represented symhilicallv LC diagram. ht sne fair ioeo; X‘Iitil - .. ° _, -2 - . .: . n lanG lm§€uaHCC 1% n1 .3 H on , . , - V is delined as a nu, er a" A sihple exaxnle ;f a T5 is the Bohel's A~derived half sectiwn sncun .. ’._:_ : V, .—, . r .‘ ,- 7‘- ‘ . _, - 4 h” ',. in 11;. y.l.l. At the pair vf t€“ulfl‘lo l—l , the crdei Vi tne lflabe ia— pedance is equal t; two. The image ihpedance at the terninals 2V2 is ’1 Censtant-L and a: erder one. " L 12 2‘11? “77‘ (309+ 1*(Zm1) (r’s2)_"' ‘_ (L01) ..2’ l"; fl2’ To be able to increase the order or ”he of the image imredances Vi a (”‘7‘ ' lo, babel used a transfcrmation and vbtained a set a: new higher crder de— 1.]! rived sections called MM', HE A ... types [20]. Each secti n i' U‘ C C I tained from the previcus she and has the fellcwing preperties. (a) An n—th erder derived secticn has enly one attenuatisn pole which correspends t.) the gurgletc: ;;. -— ml :32 1.1%. (b) The difference between the crders of the tug image iaredances *J. S 1.11.;itl. o (c) Bhere are two different sequences vf derived secticns. lhe image impedances of the secticns in one sequence are recipr cal to the ecrrespunding image imnedances sf the sectinns cf the ~ .- c a . :2 .1 second sequence With respect t; a csnstaht a : n — L\/t , where 0 'nd C_ represent, respectively, series inductance and 0 parallel capacitance of the eenstant—h half sectitn fr n which derived secticns are obtained. J. \ I O U ‘ Uh - Q 0 '1 F _ O The prsperty in (h) indicates that any n DTQCT derived hazel secti n »lf:‘ ul—rL‘o . w ‘~ . J a s *n. . 4-». ‘ '- ' (j ‘ x -.—. 2‘ a. ‘l‘ ' r" a f". - t ‘. can-st he ased in: a te-hina.ing secte n, is, elcegt ini bWC case there -.. - 1' . 0- - ‘ f) ' e x-.. -7 M -' fi- . ~~ ~‘ .— ~ . “ -.. ~‘ .-; 11 -~ 2. L GCJLUbLJ; i---r I". 2 L) 1.1. ”i1 .LS 9118 1145:;81 c -dLl lantge ..... l Such-1C; ( .1. then the trier of £12 is not a c nstant-k type image impedance. It is possible; however, to sbtain fer a To with n > 2, a c hstant—k r cascad d Label secti,ns tr a W {1' 0'3 (1‘ cf- } ‘ 't 1‘ 3 I" I 5 v -‘ ("V . , ‘ . image inpeaance by ecnoidexin T A -‘ _n ,-. 3’ ’ , ‘ o .3 ‘ "-‘._ +‘ f“. «L " ’ .L, e ' _- r‘ _. .‘ 7J4. .‘n ‘. matched baSls. The matching is seen tdat bfle last secti;n 2L th sea is an h-derived half sectisn fig. 5.1.2. [>3 \‘ \T l on. $1 ‘1 n —» 11 0-— This new secticn is, then, a T5. In srdcr tv make the dis- cussien clear; some of the higher crder derived secticns are given in liq. C l 1 As is seen iron fir. 5.1.3, each ‘erived section has two Lul- ’._L_.J. 3 branches) namely, series and shunt. KL mL 0 7 I . “01 C e i a C Constant—k J ""~' 7 f '1 .L'Q L1 “o 3.1.o_) (erj‘lb \4.) O2 "2! :1 "\ 7" . S : L: n1 2 de-ived ecti ns 2 , 2 1 1‘1” 2 k C . _ .... gm)“ LO cl =- fi-i-angw). C 5 , o l ..‘zl(l—ml ) 2 3' , % L ._. r a _. 1‘0“ 2 K C2 L2 = mlm2mj LO 2 2 < m l—m 1-m, mlm3(l_ml ) mlm2(l-m: T L : ___--_._ J 1 :_ Lu 1.: 3 m. o F‘ -2. "3 mlm (l-m22) 5 L2 k2 *2 C = 3 A C ...é .. , 2 m (l—m,d) Q 3 d E ’03 2 5 003 t' J “On l-mlg s Rec L = :“g1;“ “04 l l 5 “11:2 J V / L C = T1 P1 1" C .2 3 a 3 3 ‘fr‘ 2 M l , ~e<3 C>" T 443 -.-2 _ :12 ElhghQ—derived Sections Co henCL: ..I'th .1 5.2 A C]. u .-3 . . . . D ‘5’..- far aticn as follows: “Hi Z a can a higher hrder replaced '-3 these T5 will not USLEUL macuweflmc resonator t chieOli'L Censidercd the h‘ ”H AJ '3 7;:181‘ int; sections will be in la_der fer. network with branches cons? 5.l.la. iwnede nce out ha vi 5.3.11 ft'xel” Olnfit_c13 C_1uer's chains-u* anffgarati,n. The r fin.‘ w‘ 3 a olJWlL In pr 'I _ ”'5‘? _ agave chlgfl .. .L Ckicfl- A. C at here. -': " '-" hf»): . n‘ - 1' .L; 1ft; JQL‘ UL\L';.“1.L LU“ F“ QC ”v 0 1 v ”i-1C_V... 0-1.1 TC ‘ -.-. a ~~‘ ‘-\r\"1 q .. - ‘. "a f . ‘ Lunbluci a null Sbctlkh filth Frau the f ng the same tra ‘..:eth..d, each TD 01. but every {.431 13ft arbitrary allowing procecgyc W 1+«t an 1“" 0 *. -Ukb -..; we; ‘1 we...) 1 .3 5‘ .L a ‘ ~ w of a , ,\ 4 ‘ ‘_ ~’rg~--vAl ‘-"“.. L114 as.) J—-' ,. . _ ‘: .... , aCb‘ Ccil Livia .933... U. Int-a . 1‘) ~ ‘ ' ‘ 1 t '- ‘ I ‘ ’L 0 . '. .. ‘. 11.34.." 1.5. bi-[LCL‘ Ul‘kl.lo- impedances this secticn 59Ctivn(5); as in fig. 5.2.lh and c. Mlth nsfer function. 5.2.1 Procedure: 1. k2) It 7 w x ”12’ can be Shawn that, wx 2 Nb and w = W“. The expressi : Find the correspondinc symmetrical lattice which has the same C) image impedance Z s with the ratio functi n gf 25. ll 12) Idiltiply the series aims of this lattice by a censtant, s and divide the cross arms bv the same 1a ct r. In this P390333: the ratio funct£»n of the lattice tee -nes 25: but the 14330 in ye ance (z. ). Il ‘ 12 '13 ind the lzidder eQLivalent t; this lattice. Ladder net w-rhs ex- r. ist i1. and enly if, 0 < s x l. ConSequently, consider the half I secticn which has one image i1. pedance eq11al t Z 11 (ZI2), the other having higher order, say, WV (NV) with thc rati; function 53. A J New consider the half section in 3 and find an ther symnetrical lattice thich has WV (NV) as its image impedance with the ratic “ J ncti n f 25H. F] V Multiply cross arms by the same factor as in 2; i.e., s (O a s x and divide the series arms by s. The final lattice has the Sfime into- imleda nee, W" (fly), as in h, but the ratio function is new 23. In Find the ladder equivalent tv this lattice and c n5ie2r the half section. This final ha'f‘ secti:n(3) will have the image it’ P—c TN - ( IJI' z": ) "I ‘J ( 1" 1J ) Lilla ‘tlle l‘a.c '. ~ :1 I‘lnctj‘ C11 ‘1. E 1'» y a b H U} H) \. and W" are given as follows: J “ (l - 32) + (.2 141 49 Ll (1 ~ 52) + L: H . . - _ 'J. Y z t can be seen fr m the abcve fcrhulas thet. F“, .7]. '1": - ’-“- VJ W ; u, L The sectiwns in :ifi. 9.2.lb and c are derived what different vay and called h-derivatiens. .3 do not resrlt r netwer; consLsting e1 res nater an; s erhs. e1 that we want to ccnsiaer. Since the transf: sectiens i ihage impedances at bcth of the these sectiens with one Ur mere sectigns .btain a simple image imredance consider that fig. 5.2.la is a half '4’ Let us with the parameter m. Theref re, Z1 and Z T18 transformed sectien, 2 to a capacitor. tained frem fig. 5.2.1 and given in fig. 5.2.2. olnce by fiude [ Hence, {) at ene -f the terminal re Currespenis t; a h d whey simple L, C, tjlcy end C have is necess en a matched basis: ’4- I" herefore, 2j in Higher qrder h~dcrivatigns taraLler 7.18 111g;1ex‘ .1m; r ary t cascaue in ”:der te 3 th- f’nn Hunt M~derived section, I i‘ O. ‘u_..‘ VJ. resenatgr can be oh- the re msgLO (1-8 x .2 \I ll i nfl g! *2] i . ~50 5.2.2 7". .‘ thD iriu the formulas (5.2.3) and (5.2.4), we have 'T ‘,-' "W- '- --. .— ' 2 _ 2 '0 'h (2’ P ' tpr‘) ‘ As we glow fr.n the ab ve discussipn; this sceti n has the same trensfer functi.n as th uriginal sectien. In ereer ta reduce the erder _; W”; we cannect an thcr section in cascade en a matched basis t- this secti n. ' .'. ' " ’3 . ‘ " ‘. ,.-," ‘ V .7 1“ ‘ '3 *r r 1': 4‘ ' 1* '. .~, '. " . - .~ “\r 4" n lhls fLHal sect; n 1s blhy J an H~eeri ed Lull oCCtlmn with tne thiehetci u; sn. If we let (me < m 1) then the final T3 will be ehteined as in fig. .2. . Element values “5 this complete section are ‘0': - ,‘ .1 ‘3 -o g- . (A - .'~ -‘—‘ . ‘ ‘....'- -“ ..ven en tne IlQhJC in terms ,i the Lazahete.s v . '. ‘ 1. —— . ~‘ -. r.‘ A - 1 u‘ ‘ .- ‘ "\ . - -' .- .L ‘ 7,‘ a . ‘ ' t ,. ‘4 ml aau m2. image lmye'ances are else lCdLCiuefl en the SlLC 1i are. J -415- Zia ‘ ‘ ' ‘h ' l» “ Jl.— x2 “ 1; [l j» (l- - 4122) x2] 21“ .L c H nk‘l -Lif) C 1 F 2 e 1.11 - 1.12 r‘ 1 n ”2 ”l we mi (1 — meg) C ;_ ' ___ ....... C“ a 1 , , a v .112 (.31 1' 1.12) C ; m C Since this 26 is cbtaine‘ simply hv cascading two matched sectiens; its transfer function is tne sum of the transfer functi ns cf the "ndivldnel '1 "~ " " ‘y '~~ P ‘. "'1.Vr each a: W}1.LCL1 hua «:le A-1--'dt..l-‘-v ed half lhfis secti-n any be ebtained with a different preceQLre as QlTCJ eLsc— where [2]. 5.3 LADD“? TEPE TEMMIJATIEG SECIIOAS; $5. In the full wing discussisn. we shell restrict .U“¢elves ts a s1 "»el I ~, ~ ('1‘ - A .. :M-.- ,. .t.. n ..t.. '. .~~.- . " .- 1 "x "‘ .. "‘3‘ lelec" lo Vitn the can-ignratl1ns Diven ln rig. 5.3.la ani L. .rese lo 1- - .-. '~ ‘ "~ '. .M: , -1- , '1 .' - .: _,. " - — are assnmcd ts 0e attained by cesca‘lng he u- Ute m1d‘0614Cu cl .-e-sntnt ‘ "1'? ‘p " I r“ ‘ 1 ' ‘4 ~, . t ‘ d ‘ nr’ fi R' r- x '1 fi‘~nv"":~ 1"- "wfi --.‘u " Y‘-'.) 99 La 1—s,CLiJns en 4 misfia cne alsls o “26 lac;eas-nb ELe u_h-r -1 .nc cl . . a [I .’ ' I . u ‘. -. ,‘ ‘ FI-v .‘« ... fl . ‘-‘ f .< 4— » . ~\ «g - ‘ r the 1139e impelances l3 ldgqoblble l- netcnej 08Cu;ins are mafia [23]. .. w (‘11 l ‘ . .. 4.3 , . . .. ,.-. ,. .7 I... .1. ,.,7. - r1 .: .; ., 4.1 . 7w .7, rsr price eel phi Jses (e.g., m_nlmlu numuer l c ”ls In tn: lllte-) rm> deafl inero 1fi‘*~ rf‘r'the rdx‘-=hn"r"t"re “vs ‘~~en+’ "1 Tixwee t" 1' sec- . .. it... - ....u-. ... 4.9. v ,.1 et .3; 11.1 v ”1.1., 1-L~.\..L o-_u kw. . 1J.....-k. th....) 9.. . '.. \ '.' ‘ :A‘r fl . ' ." -'1‘. “ u. :1 7' "\‘fl Q ‘vtx. \ - ‘1 ‘ v '1'“ h... ," ' tusa ls tue dldl uf the other nil se.-es in; ence we “n,t the clehcnt r. I ‘ - values sf one cf then, these of the m her can be calculated easily L;J- A cascaded cgnnecti n e: simsle hid-shunt half sectiens with the same A cut—vff frequency; as in fig. 5.5.lh, generally d1es net lead tr a low I pass filter. This prculem is already discussed in reIerence 1231. In the tellewing discussisns, the T3 in fig. 5.3.1b us a unele is analysed and Swme mure remarks are made on the yrcperties .f this 23. ‘ Finflly, we smell Calculate the elcnent leues of this lo. .I ~DO~ 1- I \. ~ _ ‘v /\ \ ,“~ . “'7‘", , r , -‘-.‘ ‘vgr' ~er \-'(\ 5.4 PhCPEfiT Ea e? HID-odLnl TlPE TEAMIuAJIQM SECTIU.Q A hid-shunt tyye T3 is repeated in gig. 5.4.l, and the elements are \ v . q . _ _ . . _ -_ .- - ’ labelled. If a "wltdge driver, s1, is applied ts the ter inal pairs l—l ~ - o A‘ ~_ ~ _ 0“ o\ vr'u “ I \_ an“ a luafl resistance, R is Connected to the terminal gslis 2-2 , then L) the fellmving preyerties are Knevn [2h], [2]. Properties: 1- ‘rhe output voltage, E2, is sere at, and only at, the resonant frequencies, di, or the parallel resonaters (i.e., at the peles or transrer less). 2— At the resonance frequeneies er the resonators, “3’ the driving . . , . . . I _. _. psint impedance, Ad, seen at the terminal pairs l—l , l5 inde- pendent uf R hereiere L) v =7. =3. ”11 ocl scl and at w = d1, All, Zocl and Oscl have a pole. lhlS property ls valid only under certain CQDdlulunS. Inese CuDULLLURS are discussed in detail later in this Chapter. -51- 3- 2001 and Zsel cannot be equal at other than the resonant frequen— - . . 1 _ i.,_\-[. - eies, m., of tie rcovuatwrfl. l a‘ h— All critical frequencies a: 211 occur at Some or all of the res»- nant frequencies, a. of the resonators. inereforc, the number i’ of the critical frequencies of ET is less than, or a must equal L1 to, the number of resonat rs. Although the property as stated in (A) is or value, rurther clariii- cation is necessary. Such a clarirication is given in the following sec- ti on . 3.9 LEEFTEIWJ DISCUSSION OF 2:13:11 LOCAi‘IOZTS Or" ThE CnITICAL FREQUEr-CIES OF ILE RAGE IHPEDAHCES OF A IS ansider the IS in 11g. j.4.l. The mesn system or equatiens is written to include the voltage E , across the element R rather than H itselr. ihe a L _ ‘L system 91 equations is given by, P. 1 — 0.. ...—PI .1 PE 7 51 Z2 42 U 0 O l 1 -'7 a Z W]. ~53 ‘ I “2 2+ 3+ 4 4 O O 2 O -3 A L-+Z O I O O 4 4+ 5 C 0 j o . - o - Z 1 214-2 0 ; V~l O . 1 Z " z; —' 'I O 2v—2+"2>’-f 2? 52V: V -4 " iI -‘ -E “O O O 000 2 AeyLv;-L — 2 where (9.? 1) Z - iii. 2L p C21 ‘ 2i-l Z ..-..-4... 2i—l 2 ' l + -2- (7-9-2) w12 . 2 _ ' “i 1/ L214 C2i-l p = 3a) , i s 1,2, ..., 3) . -92- The image parameters for this is can be written in terms or the de- terminants or the coefiicient matrix or its sub—matrices in eq. j.y.l as follows [2]. by 1 7‘1 A 211 = ,;-—~§e~~'~~**— (9.9.3) 15‘- ll V+l,Y-‘-l _, A AV+l,V+l ll,v+l,v l A A _ n = tanh PI L A ligy+ljvtl ll V+l,V+l From eqs. 5.5.3, 3.7.4 and p.).j, we can obtain a relation among the image parameters: Z A 2 12 “2 1 V+l - i .3.— . o - H. > = 2,—me (MC) 11 7+1,V+l In the following discussion, eq. 5.5.o plays an important role. Eor a symmetrical determinant, since 2 . A - = A All Y+l,v+l Al,V+l “ii,V+1,v 1 then, iron eq. 9.9.), we have _ I = ......_..’_.._.i.'_.--.—. ‘\ h 1 j A_ A , (5-,.1) ll 7®1,V+l Let us calculate the determinants in eq. p.j.7. From eq. j.p.l, we have V — y 7 ' -. , 2V A1,‘V+1 ’ ('1) ("“2)('54) "' (Zev) ‘ ('1) 33; gel 1'y i4’ 1 ' z l in»- . .8 ‘9) i=1 021 (5 9 ) Ch the other hand, A , and A are the Special types o3 determi~ li Y+l,V+l «v v ’v (‘1 H - f ~ 4- " (”‘1‘ v- r'~- ‘~" ‘,\ “‘V ’ .‘ ' I » ‘ f' ‘ hants Called c,nthuano. iheg Can be -eaueed, by dSlnfi sflme useiul -‘ 4"" 0"] l I- ' ‘N N J ‘ . I ‘ , D -v- a‘ 1 r“ .~ '9 »- ‘ ‘ 4 ~ ~.‘ 4- h .. .r '— 7 ‘ .. traHDLQLMJLkwub lfltu a Simpler loin called simple Cgfit nuants L23]. r; ... ‘- L. ..- --.,: .4. V n “.1.“ - be ; , . '. x.-. i-q‘-,1r; use oi the pri,erties oi continuahts, all and A .1 A can ie in QSbmgwth‘; L _ . .. , . ...-’-,- -. ~...:. _-: .1 ‘Ub" lam-530:, 31‘ 391 V185 ELLE Iluo Ilfikt‘ufo’x... but for our fqr' 'm The iia -nal entries of L and A contain all rf ttc otter en» t ‘ U )Ll 9+l,\)+1 l c u i tries in these respective determinants. By definition, a determinant, A, of :rder n is the sum or all possible products or its entries taken n at a time with a proper sign. Therefore, in the expansion or A, tne term which . -- o c I 2 0 has the highest ordered denominator-polynomial in p 18 the product of all the diagonal entries of this determinant, e.g., for A/'1‘V l’ we have \T q i"- / p L ..-. ......_-._.l_.é + Fl.._c__. \' T... + 2 , _ C . 1+3“ p2/132 1“» pa 2 2 CD1 rd! 0 D2” 1 i'—l (l + '22) l 2 . r. . . . . 2 . . where, p (p ) indicates a polynomial runction of p oi order ‘V. There- Iore A and A can be exnressed in the following forms: ’ ll V+l,v+l ‘ ‘ Av+l,V+l : 'X' " ° '.-" .- an 4 . “ .. Y'- 1-. r ‘r r. - 'r \ PTlnCl:L€o oi Circuit syntheSJS oy a. o. zun and J. v. ie\ers n, thTflV— . - I1 1 .41“.". 'l ~; '1‘. [.' n ‘1' I“ .‘Q. ' . “ n‘fi'. _: r- -. filll blag to labs. ihis bOUn e-nta ns an appendix n the as licotiins simple continuants to ladder networks. n r‘ v‘r . wax ' - ‘ ‘ ~ ~C ‘ v > y'\ I“ J I --2 r‘ 'i-‘ ~e -- r ..' 1‘ V 7‘, - - . - 3‘ . aw - d ~~ ~-~. where ajaiu a},(p a.u m (9 ) ale u l nomial iinctiuhs in p oi oi- ders l) and ‘V — l respectively. Cancellations may occur tetween tne ‘ {1‘ numerato‘ and denominator polynomials in eqs. 3.4.0 and j.h.y. “his is considered later. Substituting eqs. 5.5.8, 5.5.; and 5.5.10 into eq. 5.5.{, we notain the followin; relation: 2 ‘{ 2 2 (J31 1:2 ‘1 = i-_ (5.).11) (A ~-.2 1 v2 C2 C2y 1).)(lv ) 3y_l (i) ) H l - . ,. . . . 2 .. In 5.5.11, Since the denominator is a Simple polynomial in p , then "2 . . (l - n ) can have only zeros at w = a:, 1:1,2 ..., v). On the other hand, wi's are the poles of transfer loss :unctien, a and since [2], I} l w H ...—..- AI' 1—1: I .a 772 .4- ‘ ‘ ‘3 " ~ then, (1 - n ) must have a zero at, and only at, these irequencies, e,. L From this snort discussi n we have the following conclusions: 9 (I) - moth Q»,(p2) and my (p‘) cannot have a factor of the form ‘52 (l + 4‘6) . (Ill_ -1 , . . . p 2 ‘ 2 , . . . p ‘ 7. J n2 W} both hare a Simple Iactor kl the iorm (l T ~é). e, cgurse: LL)"- f.) . 2 . . nd RY l (p ) may have common lactors other than " ‘ r""“v ‘1 1 ~. . " I v., -‘*'. v "r . ”‘ ." ‘ " ’h‘ ‘ P‘ T: r ‘ On the other Lane, a is purely lmCOanr& in the pass band and n has i s poles and zeros only in this region (pass band critical frequencies). 1 Since the poles of h also appear in the denominator of (l - H doubled, iron eq. 5.5.11 we have another Conclusion: \Z " E ‘ . v I ' 1 . f“ v/ 2 ~ (111) - fne piles c1 n ire stne ml tne zercs a; qx’\p ) or a '“. i (11’) . '-' . . . .\ '. ,:_. ,. .-‘ .. ‘ -. ‘H ' .. .- a? .',. .L“- -. :rr 3 in SCCLlun, one Vi tne requLCmCuto nere VElDQ 1c: an is uhab -’ ' 1 7‘ r‘" f ;‘ ‘ ‘t . ' . ‘ “ T. . “yr“ r~ .vr -,_ . .Sw n12 must nave ~ simple 1.rm, i.e., is a cnnstsnt-n image 1m)ednnce. 2 On the ctner nanc, Since LI als: nas the Inctcr \ 1 +‘c~5 in its numera- L. ‘5 .", '- ‘ 1, ': ywr‘z", a th ir_qr-() i, Q';r) a an -_‘ 'I’ I" ~- '3‘? a 1 -r —7- ,—-' L ‘Jr \11 (.181; lI-‘J---‘-.J ~41 ) e *AliLnby ...la; jvl.&w;’1Cw l Glut—L 2." (J10 {—JT 3 “LA. 3' ‘.'l th;‘\ I10 [I C. have tne :sctsr (l + 4.1) in its denominator. this case is ccnsidered later. Let us substitute eqs. 5.5.11, 5.3.3 and 9.5.; intc eq. 3.5.c. After making necessary simpliricaticnS; we obtain 2 2 . V I» 2 V v A (l -r .1.-.) 7T (1 + —-—--2) 77 (1 1 «~7- IZ’. _.__-_. [:1 i=2 “i H 1-.-1 “1d (5 5 13) v - ' -- "‘ " ""“"‘—-"—HQ' ~~ . 0 K thC') )2 where K H. U) ca 'd (I) } .. d k: 6 Q a m C+ 11‘.) :3 (J.- JTnc; e". 5.9.13 must be satisfied for the c nsidered 18, then we can 0 ’npeunnce when 212 is given 0? eq. 5.5.12. I ’J .“5 p 9; p—J (D H (D m d H 0 2 S a \ ) :5 S: 6 [\ H i _) I... A C A ..,_ _ _ . . . c , . . . rpm tn“ conclusiun (I), Since :1 (p ) and my (p ) cc nut 031.3313 ..‘\ .- ractcr with purer equal tc uni y. Since 310 is CDKSUn as in eq. 5.9.1 K. then tnis lactnr must appear in tne denominat r 01 41 . i.e., (a) 411 has a pole at w e Kl Q;nce lar a :8, 411 is TEQUIIQQ Bu be a higher ardered inage inneuance. .L we neXt investigate th maximun passitle order ior Lll' Frcm Cunclusisn (III), 11 my l(p ) nas a zero (tnat snculd me Simple) at any a&, tnen f‘ 2 . ..L. 1 . 4. ’ - ~. .. '..- " ‘2 ‘. . ,, -,- e\)(p ) can flflb nave a zera at tnls irequency, er, ll Q‘VKP ) nas a selc 2 . . at any'ufi} (P ) can not have a zera at tnis irequency. rnereicre, “Vol 2 2 x \ -\ "' ‘y I ,x ‘l , :A’ "‘ f: \ V‘ ‘U' ' ‘ ‘f ' O y assume tnat ncne c1 tne polynsmlals, n l (p ) anu §”Y(y ) “are a zexa at y- w = au's. uhen, loaning at eq. 5.5.11. 1 :1 we can conclude tnat (ale/AIL) can, in general, ne equal to a rational function with the sane degree, say n, or nunerat:r and denominator p0 gnomials. i.e., either 2 2 2 p. ‘ -3.” >,. -a-O-p —.‘ (1+~-2)(1 . X”2) (1 .2 ) “12 (n X2n—1 (- ) 7~~ = --~"“* "*“**‘* ‘t***"**"““‘ 5.5.14 s . 2 2 / IL ‘0 ‘32 T3 (1 + ---~)(1 + -5») (1+ --) 2 .Ll , t. ’32 - .. (1 + ff—)(1 +1“ (i ~+ ...g---.._,) (9.9.13) ' ' 7' O - a u - ‘ 2 where, Since 912 15 a Simple luage impedance, tne l‘ ’s, are tne :ercs .1. nd poles or L“l whicn are all real and positive numbers. lne ractprs n _ . q . . .. _. . . , (l + 25*) dc HUB appear eitner an tne leit nana Slue c; eq. p.;.lg or on its right nand sine; therefure, necessarily, these racturs in eqs. 5.5.14 and 5.5.1) cancel Such tnat the cegree or numerat r and den;;;nat;r FCmaln , 1:2 equal ta unity, Since (sIfi/nll) must centain the Iactar (l + wvé). ncnce, C. . (ul he have eitner, P2 212 l :72 _ _’.—.. '. ~—--°—’-°‘~‘— ( j ' ) 0 1L.) *1 i + .2... K22 OI‘ : l . E2 + .. -..... “12 ““12 (~ - 1") 1..--.. 9-7. 5 ‘11 l .112, (LL/2 2 fi2 0 For these cases, (1 +~Ilé) and (l + 42?) must be containeu in Q),(p‘). u. k2 inereiorc, considering eqs. 5.5.12, we have for 211 either 2311 = (27-9010) ) 01‘ (p2) has a zero at w = m. (A = 2: V—l i 3, ..., y’) then because 212 is as in eq. 5.5.12, the order or 311 cannot . . . . - 2 ‘lnererore, ir neither Qy (p ) nor R be increased beyond that indicated by eqs. 5.5.13 and 5.5.ly. Since the roots or 1 — h e O Oni's) and¢no are Known, rr>n a theorem whicn is stated 0y node [22], the n lunctlon can be determined uniqUely. Shererore, the two iorms of 311 in eqs. 5.5.10 and 5.5.ly canoct exist I 2 n - ‘ 1 i' o . D . - Simultaneo sly. I1 n contains the iactor l + 2.? in its numerator, 02-0 tnen ZIl as in eq. 5.5.18 is possible. In the ether case where the fac— tor does ppear in the denominator or n function, then ZII as in eq. 5.5.19 is possible. From the foregoing discussiin, finally we can see that in order t; .. )x' .. increase the order or All, keeping 212 as in eq. 5.5.12, Some or the sim- ;. ... - 2 p 2 . q, . , J- - ple zeros oi eitner Rp l(p ) or'sgy(p ) (out not both - see CGDClUSlUn lI) must coincide With Some or the poles or tne transfer less ”unctlon, i.e., with the ai's. then we nave tWo p SSibllitleS: I h. 2 ‘ .. " ' A) ouppwset C5R(p ) has Simple zeros at Someco, s. J- . . _ p2 - n this case, one er the rectors, (l + wwé) on the numerator or the w_. .1. right hand side in eq. 5.5.15, dqes n;t appear; but this ractsr still .‘x . I. ' 4 2 ' w ~ , ’ C appears in tne numerator oi l - n as double zergs. blncef'fiy 1 (p ) can not Contain tne iactor (1 +‘l£{): this faCtor must be in the denominatwr w-2 i ‘ '2 .° ,— ‘ \ -. 1 ‘ V - " Oi tne ratio (512/211). In other words, it must present a zero oL all. unereiore: O (h) Conclusmon: Ii Q))(p‘) has simple zeros at sgne c: the wp's, then these zeros are also the zercs or the ZIl image impedance. Suppose: R L L Y"]_ 2 x n . , ' a . _ this ase, tne iactor (l +.JL§) c~es not appear in the numerator w, :fl V 2 . , (p ) has a Simple zero at any m 2:31. l4 D o .2 , . _ , , . , . ., or 1 — n . out this iact r still appears on the right hand Side or eq. 5.5.13 as a double factor. hence, (ZIE/le) must contain this ractcr, i.e., a2 (1 + 1;?) must he a zero or this image impedance ratio or we can state i the iollowing: 'S, . 2 . . (c) Conclusion: Ir Ry’l(p ) has Simple zeros at some or there. then these zeros are the poles or 211 image impedance. From the above results, (b) and (c), we can see that to we able to increase the order or Z11, i.e., to be able to increase the number or zeros . . ,_ . . 2 - 2 . .ana poles ior 211, the polynomials Q~v(p ) and Ry l(p ) must have as many zeros as possible at the wl's. From a property or the image impedance [2], . , , . _ . , . . . 2 the degrees 01 numerator and denoninatcr polynomials in p , excluding the ‘)j" -.i, , . 2 factor . can diiier only by unity. rhereiore, only the Qw,(p ) o 2 . . ., . . or R l(p ) cannot have all the ”1'5 for its zeros. 'rnereiore. maximum Y obtainable order for 311 occurs if all zeros and poles or 211 (bloc: band critical irequenCies) occur at the poles of transfer loss. lo prove this, the above argument can be used, i.e., assuming that Z11 has more zeros and poles. In other words, the number 0i critical frequencies 0i 511 is larger than 1) . 'rhen, in (Ale/LII), we Will have some factors as (l + ~~§). Since these lactors do Hut appear either on the right hand side of eq. 5.5.13 or in the numerator or denominator of l - 12, they must cancel. Hence, the haximum obtainable order for tne LIL function is that one whicn will have allcbi's as its critical frequencies. The conclusion of this discussion follows: Let the ZI2 image impedance have a simple form (ctnstant-k type) ‘I as in eq. 5.5.12. In order to increase the order ci 211, it is [\1 be located at r) ;. necessary that the critical frequencies 11 the poles of transfer loss. Otherwise, the order of 311 cannot be increased. From the above discussion, because of 212 as in eq. 5.5.12, the simplest form of Z? is either as in eq. 5.5.13 or eq. 5.5.19, depending upon whether *1 t2 the factor 1 + A“? is in the numerator or denominator of the K — func- mo tion respectively. On the other hand, the form of the highest ordered 811, not only depends on the location of the factor 1 + £8" in H. but aISo on the oddness or evenness of the number ‘V. So far, no restrictions on the wi's are imposed. however, there is a reStriction an these - ~, .“ 4:1 and some other ai's are the poles or 271. then .L * _ . there should he some orders oetween these wi's. I! we now imp;se that (but it is not necessary) LU > CD > o o o > CO 1 2 V . o . o o _ I . .y: then two pQSsinilities :or all ex;st : If V is odd: } I I\ C ‘ “'0 . .12 ‘03....1 ( . ~— I. —.—--.—--—~ -~—.—— \I‘ O \J‘ O 79 \4/ 2 (l +:P:'é) ... (l +-p- ‘ \ 1 '9 L" V is even: These two f rms a; image impedances are used in the following discussions. Now, let us consider other simplest rurm IO? 312: i-€-: (5.5.22) Ir the foregoing discussion is repeated for this case, it can Le seen iron r ' I eq. 5.5.13 that 311 cannot have the factor 1 + $¥~ in its denuLlDaLOF. therefore, Z11 has the form of eq. 5.5.20. In addition to this, the rol— lowing conditions must hold: Ezker [24] has Studied different types or low pass sections including tne TS which we are considering here, wits the assumption w > up > ...)cbv and indicated that maximdm ordered image impedance can be dosaiaed if all the critical frequencies or this image inpedance occur atr s and s not known wheth ; Ho their element values, but also indicates that it such ladder sections exist if the order of image impedances is hi her than 202 or 2002. this problem is not discussed in this thesis, since we are concerned with the ES. Sut “einaps a disc 1531 on six lar to the one above c uld be used to clariiy the many points in this proulem. With a given set of image impedance and attenuatitn poles, it is possitle to rind symmetrical LC lattice networks out txeir ladder eo“v“lei s nay not »xist. hs Ielevitch indicated in his article L26}, the existence or a ladder equivalent to this lattice can be eneeked oy the Fmtjis=ta [27] criteri:n. This critericn is ei eided cy he:?ng vet and Selevitch L20}. It is wortnwnile to note that. for the ex::stenee o1 ladder mid-so Les tvpe lLW pass i'ilsers (i.e., ior pos tive element values) D.rlington gave two suificient conditions [3]. rhese two enndi ions are actually the sane conditions that we wanted to impose ior our T5. Thererore, the con- sidered is will exist (i.e., all elements have positive value) if is 212 a constant—«1 type image impedance or 311 has the highest vbtainable order. In this c nneetion, a general existence theorem ior ladder mid~series low pass networks is given by Fuji awa [2(J and these conditions are eXtended for band-pass 1a;der filters by Watanate [23]. Our pr;olei is now alniost clariiied. All Ie have to as now is to determine the element values of the TS as in fir. 5.3.1. ihis problem is considered next. CD 5.C E WEE T VALUES CF T In this section we consider the 93 as in iig. 3.1.1 with the fullawiny restrictions. (I) .1412 " (sot-:01) w? >03O > ... 2:2 3>cs_ (§.e.2) (III) A ...—J t Y A H i g» I LIL :. ----- (S.C.’4) '~-) . 2 w,?1 O (’1 :3. (1 pa ff With the above restrictions, the eqrivalent Foster forms of bh' circuit and (pen circuit impedances oi this T3 are given as in fig. 9.o.l. Where )1 is tahen as an odd numh-r. If'v is even: similar tw. terminal LC Foster forms for Z,C3 end Zsc and the reactance pattern can be given. 1 ;he imp dances Zocl and Zscl can he expressed in terms or Z11 and K as [2]: J :JhJ : /- ,« \b.‘.3) Since the w.‘s andcs) are kncwn, H can be unicu ly determined, and since L x: Zocl Del can he found ir,L eq. 3.3.5. In fifi. 9.0.lb, the only unknowns are the wa.'s (31-1 in nunher}. idese H ‘ . - .-‘r ‘ - -'.~',-~ \ ‘ ‘Q ' W ‘P‘l equenCies mt: ne inane as lullDMo. "f u12 fl? T L W .HTT { a ) .1. . .V .a U “l as Poles scl Lert l _ \V a w way-2 m U; 2 Poles 4 :Cl Jere; ) ( b 4 .V I.I.I.|.|unwul IIIIII / I 1/ lflhflhH.I| II II III/II / unnuius // flw / Ii/Ill II V l/ I, IIIIIIIluluIIL 9). _ \y u I’ll: l - ).V U V II Iwnn.l.l..|nH I. wO // III ‘/|- llllll 2 »a a 3 a. w .Y I «lllllll // a9. // 7lock—band Pass-band Fig. 5.6.1 C ..C‘.;.. ”' 3 W2 ~ " ‘ 7 t‘ f '= "' * 6 lb h * oincc L — “sol / JOCl, nen irom rig. ). . ,: we ave 2 2 2 p2 (l + _p___)2 (l + ’2'“)2 (l + ‘32“)2 w 2 (L. ,2 we. 2 31 a3 u-l -, x I, :_ h -_ ......é. -- 2 - 2 -..—- (we; (1 -+- 23) (1+ 13F (l + —-——g-~—)2 (13;: (409.2 way-2 . .2 ._ . , , - Since n muSt nave a unit value for my: «a, then irom eq. 5.6.6 we have the following W’ relation 2 73.2 p.2 n 0.2 ‘ ‘ *1 2 (1+23)(1+ 12')2. 0 (1+ 2 )2—hpc' (l+—°’-"2‘) ... (A; wa2 way-2 Gal 0.2 (i T mg» (i = 1,2, ..., V) (5.6.7) (Day—l where, as mentioned before, pi = jcbi. Actually, to be able to determine the H - function, the set of rela- tions 5.6.7 must be solved and.aéi's and h must be found. If )2 is a small number, e.g., 1) I 2 or 3, an attempt can be made to solve this system, but in general this is difficult since the system is nonulinear. Bode [22], has proved that H is unique. Therefore, there should be one solution to this system of relations 5.6.7. Now, suppose that we have found the H function, therefore from eq. .; . .7 v; r‘ 1- ") _. ,_ 1' q . JR! - 5.L.5 we can determine uOCl or ”sol since all is also nnown. one can con sider Darlington's method [3] of determining the element values of the mid-shunt ladder TS by use of either Zocl or Zscl. By this method, we can reall“ find all the element values if we know the H function. Eut J we only know that it has a unit value at w = d1. Therefore, L1 can be calculated without any difficulty. But to find the values of the other elements, we need the derivatives of H function since Darlinpton's method .3 requires a knowledge of the derivatives of Zocl or Zscl. Unfortunately, -66- as mentioned ab ve, although the H function is unique, it is difficult to . . . y, -0 .2 _ . . i . determine 1t iron the zeros o1 l - h and.mb. Obv1ously, if n cannot be determined, neither can its derivatives. Before starting to calculate the element values, we introduce a new parameter, 3 , as follows [3]: let p2=-..2=---1;— (968) j and then w,2 = “L" (5.6 9) .J. 5i n2 l ' 1+_L.é.=_-—-(3i- 5) (5.6.10) ”1 2 l - a>o 2‘; 1 1 2.2. e - ”.-.—59.... v (5.6.11) mo (no _] Equations 5.6.3 and 5.6. h, then, can be rewritten in terms cf ;3 as i‘Vw‘we 2} (32 2) ”‘(5v—1"97) 741$!) = ~ ~~~~~ (5.6.12) (:5 (11.. g) (5,, — ;) gem/7 (,72—2)...<,Iv -5> 1-010; (,Pl— ,2) (gm-,7) A new set of parameters, mi, are introduced in the foiledinb way: At “J (I and leQ) (5.6.13) 0) = wi, the factor 41 - 0302 appears in the expression of 215;) is des- ignated by I 2 21 J (DO 2 r \[l - (.00 J i = l - (1:5 = mi 2 mi (5.0.110 Therefore, the LOCI and ZSCl functions of the TS in fig. 5.6.la can be written in terxns of the j; - variable as follows: ' 1 Zoc1(é FV/:__—§ l ‘ + --“~.WWWW-~--_ ....-.-..... (5.6.15) L4 L A) ...: L212—1 . 1 w...“— . —--— ”—..—.- .3 _,Z ' C2)» Zscl has the same form in which only the last term (l / C 2v)does not appear. For 212, we have 030 mi'L/ -j' . 212: (5.6.16) ' \Il - w 0 Since at w = w. (or;; == 5i), 212 = LL = Zscg’ then eq. 5.6.16 wives 1 ka-Ji moi/‘V-ji 2 .... O '1 ) — ~9 - . (5.6.17) 2(15 All-mo g1 mi On the other hand, at“; == jfl, from fig. 5.6.la, we have 1 , l _ __ U -111” g Toercfore, equating equations 5.6.17 and 5.6.18 yields 2V di & . . , .2 . or, finally, Since “62 = l / LOCK and n = LO / C3, we have the general ex- V I pression for C , whether 1) is odd or even, that 2)) : " '1,” 021) InvCO (5.6.19) In order to calculate Ll’ consider the relation W2 ” 7 i “11 ’ ”ocl ' wscl Let t) be an odd number, then from eqs. 3.5.12 and 3.5.15, we have .;3. 12,32 (1 -woenge -:7)2 (97.2-1 - g>2 ~cou2! (; ~7’> )2 («IV " gr? l .7 L1 - gl"! . . Z.-\S ' Lev—1 1 ° _.3;. 4'Erw' C2w> ‘2” "~; 2le __| Multiplying; both sides of this equation by the factcr (Zl - 3 )2 and then taxing; the limit as j ——-> J 1" we have 7 _ 2 k2 (l—wo2gl)(!2~/l)2... (912—1. 97) 2 V --.. L (5.6.20) (1502(3 3 '- 51)?‘ °"°° (XV — 11)2 1 But, from eq. 5.6.lk, since 2 . 2 Z __ 1 (D0 0‘0 _. 1 2 _ 2 ,f ji " “,3 — LL ‘2 (13.2 " w.2 " w 2 (“.3 " Lni ) (50LJ021) o 1 ,3. o and - 2 V 2 J. - u) f . —- 11 o 1 1 eq. 5.6.20 can finally be put into the form 2 2 2 2 2 2 ml (m.-L - m2 ) (ml - 1112+ ) ... (ml - 11194) - L = ~ 0 (5.5.22) 1 (I? C. - m 2)(Y‘_ 2 _ M 2) ("I 2 - ... 2) A1]- 3 .1]. LAD: 00. All My (For V odd) In a similar way, if V is even, we can find 2 2 2 2 (PT. " 1.1 ) o o o (l) " I“ ) " V L. = l d 11 ~-—--------—— L. (5.6.23) m.I (1. 2 - 1.2,.2) (I; - m .1. 11 Q 11 V“)- and the definition of mi, we have o. «4‘...— C I I. m1 > m2 > ....... > m}, (5.6.24) For the following special cases, frcm eqs. 5.6.22 and 5.6.23, we can find that If V = 1 (taking me = 33 = ... = m3} = O) = . .6.2“ L1 ml L (5 2) If r): 2 (taki g m3 2 m4 = ...==*1v - O) n 2 - m 2 1 m1 3 If 1): 3 (takin31m+= m5::... =zny = O) r (“12 — m 2) L = 1 2 - (5 0:27) l 2 2 (m1 - mQ ) At .2 = 2 , in either case, i.e., )) odd or even, 311 ; Zoel ~ 25C1 = ... ’ .1. Therefore, from eq. 5.6.15, we have .3:L. - l — 0 (5.6.28) .. .9- + C 31-52 2 ". , Substituting the values of L1 from eqs. 5.6.22 and 5.6.23 intc eq. 5.6.23, and considering eq. 5.6.21, we have (m12 _ m32) (ml2 — m-2) ..... (m12 - m1?) 0 z _WV1. 2 2 5 2 2 (for 1)cdd) (5.6.29) In]. (”11 " mu ) no... (EJ1- " I3.1))”1) and 2 2 2 2 u (.1 "’ 1.3.: ) o o o 0 (U " I.“ ) C2 = lfinm:32f...si§-- X12 'fléi (for 1/ even) (5-6-30) For the following special cases, from eqs. 5.6.29 and 5.6.30, we find .J If >’= 1 (taking me = m? = shy = O) .-: r :1 02 ml 00 (5.6.,1) If ‘V=:3 (takimgxq‘2 m5 = ...==ldv = O) 11112-1112, C = ~~—~-————a¢—- C~ (5.6.35) [‘0 etc... If )J== l, the element values are determined by the foregoing, since L1 and C2 are given by eqs. 5.6.26 and 5.6.31 which corresponds to an M— derived simple half section. If ‘V = 2, we have only to calculate L3. This can be done by con- J 4 . '\ a . « (V "-.. ‘j ‘1, l' .. w I ’ ‘ 'I, " '-’ sidering the exp1e851ons cl all , “eel , ascl and the relation, all ~ “ocl . Zscl. but the following alternate argument nakes this calculation simple. From the eqs. 5.6.3 and 5.6.h, when p approaches zero, we find that L .-2 2 o ,- ,, C) n 1., ,- ‘Téilad see that {'3 On the other hand, zhen p approaches zero, from i5. 3.5.1 is inductive and apnrnaches the value of (L1 + L3 + ... + Loy 1) p, L _ and Z. is capacitive and approaches to the value of l / (C2 + C, + ... + 1 “g’ C ) p. Therefore, tie product Zocl . 25C} approaches to a constant value and by use of eq. 5.6.3h we finally have - + L + . . . . + L L CiL + C + a C2V«l : 69 (5.6.35) 2 h ° ° ° ° ' 2v 0 2, L can be found from eq. 5.6.35 as 3 I! Therefore, if >’ m2 , L3 = E; (m1 + m2) LO (D-O-JO) This section, i.e., 1} : 2, is the saw as in fig. 5.2.3. In order to calculate Cl for a BS, in toth cases, i.e., X) is odd er even, we can use the relation 2 __ n (1.1 — l/Ll bl Therefore, cansidering the value of L we have 1) (for v odd) 2 2 2 2 2 "ll (1 " I‘ll )(Hll "‘ 171:) ) o o o o (“11‘ " 111))—l) r. V,‘ c:l ' -~~--2-—\ 2 - — L (y. a . 3,) (fur )2 even) 83 far we were able to calculate only four elements of the general 15. In order to calculate the rest of the element values of the lo, unfortunately. we can not Continue this p‘ocess, i.e., considering only the relation “ocl z‘scl or Z from the Z: and h inis method actually pernits calculation or “ocl scl 1 function, and all the element values can be calculate from LL31 [3]. There~ fore, at this point, we have to change our techniques. A formal calculation technique of the element values for a T8 is described by applying to a I” with 'v’= 2, the method considered by need [2]. This technique gives a set of non-linear equations relating the element values of the T8, the latter of which are to be calculated. Even for ‘V = 2, too much algebra is involved; therefore, for )) = 3 it is more difficult since systems of non-linear equa- ions containing 6 unknowns are forued. But this technique can still be used little higher ‘9 '5 since in the previous discussion we have already calculated some of the element values. This will reduce the number cf un- knowns in the system of non-linear equations. Element values of TS ctrres- “ ‘ f ‘ '— “ A ‘-- ‘ r“ .’-‘r‘ h- s v \ .“'. -.~ -~ rA“ pending ‘V ~ 1, 2 agd 3 are given oz iaole 1. Hal practical pui.oses r7» .. Id" Taole can be employed. brow Table I we notice that the following relations >3 1).- 7 53: L , = L V. Kn ...! H H m [... I H O H l‘r }_1 r EL '4 l' t, H M O R) 1..-“. ll 0 L) H. II K .4 M ',- hold for )J = l, 2, an( 3. If we have shown the validity of this relation, the element values of T5 for ll: 3 could be calculated easily since the system of non-linear equations wguld only contain two unknowns. Io easy method has been found that eqs. 5.6.39 hold for ‘J > 3. In practice, the use of a terminating section with L): l, mostly gives satisfactory results. Hence, we end our discussion on filter terminating half sections here since more complicated sections have no general use in practice. mpedances at the terminal H. Note that the TS given in iable I have image pair 1 - 1’ of the form L 7 . z , z . 2 etc. (5.6.h ) .102 I w3 0)." I wS) .00. Ihe other forms, i.e.. I I'} [:7 002 , 203 ,Auub_, “05 , .... etc. (5.6.ui) Z as mentioned before, can be obtained if we consider the dual ladder to the ‘ TS considered here [3}. But dual the TS contain more inductances and on the other hand, as it is seen in Chapter II, the insertion function will be the same whether the image impedan‘e of the filter is in the form of ('7 Z . 0’2 I (400) 29"}- has-the image impedance of the form as in eq. 5.6.h1 is of or its recinrocal Z Z . L‘ &:29 ’ 0'92\J"l cal importance. l '7 ' . 'z—T Cl C3 T C5 T ( ) K ‘ I ‘ C Z , _ 1'; I I ' .3- . 1 C” 42 l + 2 (no , 2 T. 2 ., 2 , 2 , 2 ‘fll(ml .312”)- _ .nl (ml . .112)(.32 - m3 ) L = - -- L , L: _ L“, l 11112 - r332 0 " (1:112 - 211,2)(11111112 + 5132) -‘ m3(ml + ”3x“? + m3) 2 . 2 2 L - - L , l ~I.11)(m ~ m ) 5 1 m + m 2 O C = - 1 3 C n1 2 3 3 ( 2 _ 1‘ 2) L}; ”’1 m1 n2' 52 (l £)(mlmL + m 32) C~ = -~n~~-~— C . C = -”w-~"-m “C , C, = mi C_ A n 0’ o L _ Q 5. 5 m3 (m: + m3) (m2 + m3) a VI. A DESIGN PROCEDURE FOR SIMuEfRICAL LOW~PASS IXAGU PARAMEfEA FILE ER 6.1 GB: EAAL “" EV OF IMAGE PARAMEPER FILTERS If cascaded four terminal LC networks an a matched basis c nstitute a filter, then it is well-knawn that this filter has an iLa age trawl “ r func— tion equal tc the sum of the transfer functi:ns of the individual sectiens constituting the filter. Also, the image impedances of this filter are the same as the image impedances of the end sections. In filter design, generally, the insertion loss functien, AS, is given. In the effective pass- Land, usaallv it is desired that AS must be less than a given cc nstant value Acn' In the effective bch Len , Ac must be UL s) greater than a given contour of requirement which is generally ngt a her- izontal strais t line. If the filter works between its image impedances, insertiLn lvss and image transfer less are identical. hence; in the pass~hand, the insertien less is identically zero. Under such an operating conditicn, the design f imaee parameter filter is simple. In this case, when the transfer loss is desired tJ be flat in the LIQCA-hand, Caner [7; has given eXplicit farmulss fer the lwcation of the pc les (f transfer less fUflCtiJD. 0n the other hand, for the case of arbitrary transf I los s functizwn . the loca— tiens cf these pwles has ta be determined Ly cut-and—try metheds. The cut-and~try method invelved is 0425; Ceia L13; simplified Ly use Cf scme ‘D ‘1 V”. I ,. -,'1 ..L J. ‘ i 1 +1‘Liylul- - “mi/--‘jno’ e. 0/ \‘AthvJ-‘L it}- 1;..LCJ I (Ce--1 L [17} l 0,. "\,~ L. MA 1' 8- In n.rnul greetice, when a filter u rks betWecn tfir pure resistances, un1 th tvet een twe image impedances, the inserti,n less is hit identicel to the Lnece tra231e: loss. utncl s QCC‘IQ s tiJn f Imula is an expressicn the iqse tinn ftrcticn fer the gcneral mperating c«niiti»n. From this I’ “\ 'crmnla, it is cle'r dnnt in the 1‘ss band. if the image impedances of the filter can be made t» aptr-ximite the terminati n3 resistances, the inser— tion less will net differ much from the image trez.sfer less functitn in this region, i.e., A can be made as sxall as pcssihle, since Al is chn— (i: ticully acre in the pass-bend. Because of tzis idea, hotel intrcdnced the f c npcsite filters L31] which are actually image parameter filters but having two extra end sections called terminating half sectians, each if which has lijher order image impedences. " ‘ ‘ ': ‘ ‘t ‘0‘ I ‘ f‘ ‘. . ‘ 1“ fl ":1 ‘ ~.- : 1 ‘ n‘ ‘ V, I 1 0‘ ~q . I ‘- " r: :y int; cicinx the teiminating n11- sectlJns, lo, Mthh ere ciscus5ec in detail in Chapter V, the A“ function can be made very cl se ta AI in Q -q I I. ~- . x q n . the pass-bar a KAI = 0). cut, in the blcC4-pand, it may net be p ssiole ts mahe Ac and AI very clcse ta each ether, since, generally, at the critical 0 requencies cf the imare impedance of the ter nineting half sectim 3, AS has pcles but may net AI. As is shown in section 4.5, regardless cf the image impedance cf the symmetrical fil‘er, subtracting $.02 db. less Eran the inage trensferrless, 1.111 yield a lever bcund ta A, in the hleck- D A I) hand. The 11m a“:e transie losses :3 terminating half sections are else in- Cluded in AI. 31 is approx imatic is alwa7;s Lune since it censidcrwllr _~ --‘~-..-‘_o.--—.oq . .m—'_~o--~~ov-— "I W , n" .. A «. fl 1 _ -. , 3 , ... . Humpe°t, L., Uber cen hntxnzr- electriscner Jelleniilter nit chrescarle enem Letrieosverhalten,‘ Docturai dissertati n, rechnische Hechschule, hunich, Germany, 1347. rkgfi' 9-..3 -7g_ simplifies the image parameter filter design. In general, the commonly used approximate design procedure just de- scribed, although simple, necessitates the use of more sections than is necessary. This fact is demonstrated by a “precise design procedure" re- sulting in an economy of the sections to be used. Ehis precise design procedure is demonstrated in the following discussions. o.2 THE DESIGU PROCEDURE A low-pass filter operates between two resistances of value RT' The insertion loss is not to be greater than A between the frequencies 0 ~ f Os 1 (effective pass-band). It is not to be less than a given contour CS ror frequencies larger than 12 (effective block-band). rhese requirements on the AS - function are indicated in fig. 6.2.1. Low-Pass LC— RT Filter ////// // / _ ///////// m” ///////// ‘h 0 P1) H G Wu. C Fig. 6.2.1 (A) CHOICE Arm CALCULATION OF Tam-113111111211: HALF SECTIOI‘IS A cut-off frequency, i“, must be chosen. The difriculties in choosing \ f0 are well-known {2]. If f0 is close to L complicated terminating half 1} sections must be used. In this case, critical Lrequencies oi the image impedance or the terminating nal; section become numerous and are crowded very close to 10, in the block-hand. These critical frequencies produce some poles of As. On the other hand, it r0 is close to :2, then simpler terminating sec- tions can he used, with rewer critical irequencies or the image impedance. Correspondingly, only a few locations or poles or A” are iixed. rhis D ‘ gives more flexibility on the location or poles or us. But, if i.) is L really very close to L , then to pr vide a sharp cut-Orr, we may have 2 to use more intermediate sections than are saved by using simpler termi- nating sections. Since A? is given, we may checn Lirst from the Graph-l (at the end (I) oi this Chapter) to determine wnetner one, two, ... critical frequencies are required to assure that fl will correspond to Xuz, and from which the location or is is determined. After iixing the location of f0 and the type of IS, the critical frequencies or the image impedance or this ter- minating half section can be calculated irom the iormula (4.3.1). With the above information, the element values or T8 can be calculated completely. Fecause from the Graph—I, U is Known, i.e., fr]. (k2 u- .. k/RT = L /c ) (63.2.1) then k is determined. Also, since 10 is chosen L, c = 1/ w. 1' 2 (0.2.2) o o o Thereiore, a knowledge or the values or k, f0 and U yields "1-1., k 1“'1‘ - L ~“-~ = —--—-- (3.2.? 0 21:1? 211: I‘ ( J) o o l 1 . . C :; —--—.-_.... = fl --_1.._..-...... (00204) O 21‘ hiO 2;: 3' “.LI’ -79- From the fable-I in Chapter V, element values or 15 can easily be round. (*7) mammals: CF IZYI‘EIh-IEDIATE SECTIONS ‘ Since the expressian of insertion loss, in the lecL-band, is given by 2A5 1 ‘ 212 2 fl. .2 e z _ (-—~:-—) oinn Al the image attenuation, AI, can be round as A 2A 2 . AI v argsinh — (w—gE—é)2 e S (5-2-9) 1 - z The ES, already designed, determines z. Thereiore, the Lactcr 22 2 , - (...—- é) (L\O2OL') l - z E 2 is shown. fhis factor, in the blocs-band, is positive and varies between zero and unity. On the other hand, the requirement on As in the block~ . . . . . 245 band is also given oy Cs or rig. o.2.l. hence, e is known. rhererore, eq. 0.2.) gives the requirement on the AI — function in the block oand. Calculation to meet the requirement on AI can be simpliiied ii a digital computer is used. Indeed, first by Choosing a set or points on the given contour of requirement, Cs’ and using the Least-square routine, we can Lind an approximate polynomial for the equation of this antour. Since 2 is known, a program can also be written for the Lactor in eq. 0.2.0. Hence, in the blochiband, the whole expression, i.e., eq. o.2.j,can be found by including additional routines for argsinn and square root. An example is shown in fig. 6.2.2 to illustrate the requirement on nI. The requirement on the AI - function is indicated by tne solid line on this rigure. This AI ~ curve is ootained by calculation from eq. 6.2.l. \ 2'1. 2 \“ _ -..—... - 2 5“--- / l - Z --— X ow the problem is reduced to find the necessary number o: intermediate sections and the location of their transfer loss poles. ihis part of the design is considerably simplified if one uses Rumpelt's templet method. rhe description of this methfid can be found elsewhere [IT], [32], [33]. Ey using Relevitch's notations [17], on the transformed T—axis, the re- 9 quirement on AI is shown in fig. 6.2.3. Ncw, since the TS is known, the correSponding poles of A1 are given. These attenuation curves are drawn by using a templet. The symmetry axis of the templet curves, Ti, coincide with the critical points of the image impedance of the T’ on the new Y- axis (r1). In fig. 6.2.2, since we assume that the image impedance of the TS has only one critical frequency, there is only one such curve, Ti; ..- in fig. 6.2.3. Subtracting the curve T. from A the remaininq curve A 1 I’ L 11 is next to be obtained from the intermediate m - derived secti ns. It may happen that, when a complicated T5 is used, the All curve would be under the Y-axis. In such a case, we do not need to use intermediate sections and the filter ’ill consist of only the terminating halfusections. In the general case, the AI curve will have some positive portion in f'"~rv - n r‘ -. .... ~ -. ' ’7 ’ 7“} —" . RV , ~7 r '- ...-- some interValo of T as seen in fig. o.2.5. inereiore, L, using the temples, -30- A \ / I , Ii )’ I 1’ /4’ / \I \ 1A,'\ ‘t ,/ /’ \I ‘\ ,/ / \\ \ / ’1 ’// H \i I’, \ \ \x \ "" l \l I / ‘~‘* r l I." _ ‘— fl 1 = 00 x = 1 Y _ 0 Fl”. 6.2.3 it is easy to determine graphically how many curves, Ti’ must be used and where the critical frequencies of the Ti's are located (vi). If we con- sider the example in fig. o.2.3, the probable locations cf critical points, ri’ of curves, Ti’ must be very close to the abscissa of the maximum points of the All curve. After locating the necessary number of Ti curves on the Y—axis by this templet of Graph—II, we can directly find from ri the xi er mi. Hence, on finding the m,'s, the design procedure is com— pleted; because the element values of intermediate sections can be found in tenns of L , C and m. in Table-I. o o 1 6. 3 EXAMPLE A symmetrical, reactive (LC) low—pass filter is to satisfy the follow- ing insertion loss requirements AS 5 Aps = 0.0% db (0 5 x S xuz) A > A‘ = 32 db (XUA S X) with a H }_J O H \C) [U (fl ": \\\\\\\\\\\ &\\\\\\\ C- S 7/////7/////7' //////////// \\\\\ D h I. f ////////// * 0 d“ .3 ...b : I u d Fig. 0.3.1 ’ I q Ihe above requirements are also indicated in fig. t.3.i. From Graph-I (at the end of this chapter), it is evident that a ter— half—section with only one critical frequency could be selected. minating x.) this terminating half—section is the iirst one in Table-I. The approximate ‘alue of U can be found from Graph—I. On the ether hand, a more yrec se value may he obtained by the fellowing calculations. ch 1 's‘ a 'e . . a: . .a o‘ e refer ,; La , t hate Friw tie tatul ti n h lh l n1d h 1M 2 f the ’ ence 2] 'e r> “i“sn g ' 0891) 2 and since x 2 "h l ‘3 (l-‘J )2=~“—.+‘ "i then x.) .1. I; .312 Ehe ngrnalised image impedance of the filter is x l - x x - x [‘1‘ inerefcre. the fact¢r 5 given in fig. C.3.2. 1:. can be calculated easily. The correspondin' E curve From eq. 6.2.1. AI can be found. This has been dcne and the A - curve I" q q is given in fig. v.3.3. Now, by using the tehplet method. it is round tlat two intermediate sections having the parameters 32 e 0.21; m -, '--' O . .33 J rare, the filter contains two internediate secti-ns. - -.....— must he used. There If the approximate in the classical method discussed in sectien h.5 is H used for this example, A: 32 db, hence the image transfer loss in the black-band is taken as AI ‘ Abs + b = 33 db. From fig. h.lj.2 of reference {2}, more than 3 sectiins total are re- quired (terminating half secticns are included). Since the nethed pre- sented in this thesis requires exactly three sectiuns total to meet the requirements, the apprexinatien stated above certainly leads to at least one superfluous secticn. In this oarticular example, the image transfer l;ss, AI, is c nsidered as “flat-loss" in the block-band. “is. h.13.2 in reference [2] is then used te determine the necessary number of sections in this law-pass LC- filter. Tc satisfy the flat—lass pr per y, the 13C&tivn3 of the poles of L) A A re determined. On the other hand, we have alreac {a I yhalf sections to meet the given requirements fer the pass-band. Since these terminating half sections fix certain poles of AI, we have to check the following: I) It has to be verified whether one of the poles of flat—l‘ss A determined in the foregoing, is als; the required transfer pole :I the terminating half sections. In general, it will not be. If the AI-pole for the terminating section is not also a flat-loss hole, we may proceed as follows: 2) Shift the image transfer tcle closest t; the A «pale int; c inci- I dence. Since now the flat—loss property will nzt be valid, we have to check whether the blochnband requirement is still satisfied or Hot. In general, the block—band requirement an image transfer loss is riven by an arbitrary contour. Therefore, fig. 4.l5.2 in [2] cann t be used directly in this case. It can be emrlgyed if one replaces the given con— taur of requirement on AI by a horiz ntal line which is drawn at the maxi— mum point of this requirement. Although this can be done, the result is not so satisfactory since the number of necessary sections is unnecessarily increased. It is, of course, possible to avoid the unnecessary sections used. The f llewing remedies are ugicsted. I) By using a templet method a 330d approximation can be made and the number of necessary intermediate sections can be found from the block" band requirement. On the other hand, the terminating half sections will also produce soue additional transfer less in the blue -band cver that Prv" duced by the inte"mediate sections. Therefoie, the extra transfer l;ss '\ 1 when censidered senarately results in were sectiuns. n . . -. A . -' ,. ,n ma .. f" .. ~' --.~. 2) n neie precise meshed is as railsws. ny c nsiiezin; DAL tulml* .3 ‘ .-~a . 1" -‘. 3 a r - .-‘ 1.1 ’\-. -, ~ \' '~ ... s ,‘-__”- - , l~ .~ : H 1 .4. 1 n -. 1 natLA, half secticns, aid heir inn 3 transieé less. this ngS can ae r-a _ ._,,,. ‘ '~ .n-.‘., 4-‘ . _.-:.. .. . .. .' -_« 4. , ./ - ." 4--“ -’ .. . . . - .. subtle: ea ifbm ne .iven :equirement at AT tw determine a net :euuire— ..L g ' I I- ’ ‘I I I - .. j I V O J. 0 o .L. \ went. As. -his new A i~ then aroviued Hr litcsmediaue sectiuns; Lien a ’ i I .. .. .,,: .- . .. 4 .1. -1. : 4.x... .' .."1 H .. . ... 1, .°,._ 1.. 1 conclusiez, it can Le inleired tnat in all cases, aiscxssei 41 the n .. (N ‘ n . w _ c- ‘ . I. . ‘\n v o L 1‘ ‘ i “I‘m ' . . 7 '_ w a 0 If ‘1 _O +- VA: ‘ ‘ ,_ y . .- ..- : _ .1 » -‘ . -; - - ._ . - , "J '1 y. . ‘. v“ e , - . - \ y Le.a.QD...'\.,w_L 5.1L: ug. ...» ; u.-.1.k.'; [L ...D .. s- L}. in. __.. .J... {K LU' .,'._c.2-}"§, ‘v L-n)o) .LV nun] flu-t J. L g 0 fi ‘ g g _ u ,--‘1 ~ . kw ‘ x r‘ ‘e‘: ‘1‘ ‘1 3 't ‘1‘“) J-,\ ‘3 y - \ ~", 1‘ . 1‘ j-V , ‘ .t ~ ~ \" .‘.;- -¢ J-r L~‘ H L I...) |' ‘J JOJ "I’J—C’ ‘-' VLUL LJA.l'\4 15'.L.“...;o.‘ -.rl 1L~let).\;.'. \ f 08C, .1. I‘D 0 1.1.2: k~‘-~'\4 "a. Vlsi ,, V . 4.. .. ., -..' - 1., -° : . v. a, n ‘ . -,i ,th. -.. 1.4,. , , -.- etuyle ..LQ Ck“ hell with Lille Opticifuc 1-»-.‘.l'pc?uk; -4. CLEA..c.7lgrtrcLLllfl‘ 1):...” 2.) =1C].L'.o-- ————— _;= —— 2' db ” I JO l I I I I I I I I / / / / ' 2O / / / / / / / / / /// " 10 .r’ ==== T (m) (qu) Fig. 6.3.3 6.1+ SOME REE-NUS 01‘: THE IMAGE EMA: uL'l‘ER ZILEI‘ETOD On a close investigation or the fig. 6.2.2, we can see that at the critical frequency, XI, of the image impedance of TS, AI — of image attenuation of TS occurs at this frequency. This is a disad- vantage, since we are unnecessarily locating one of the largest values of AI at which the requirement on Al is zero. This always happens since We are using a particular type of £8: In Chapter V a class of TS is - 0. But a pale _,b- considered and it is found there that if one or the image impedances of this TS is a Constant-k type, to increase the order of the other image impedance of this T5, the critical frequencies of the latter image impe— dance must occur at the poles of transfer loss function. The above property of the location of the critical frequencies of the image impedance of terminating alr section indicates that no other termi- nating half sections with the mid-series or mid—shunt type of configura- tion could be found, such that its critical frequencies do not occur at the locations or the attenuation poles of these half sections. Let us keep the configuration of filter terminating half sections as mid~series or mid-shunt form and impose the condition that the critical frequencies of one cf the image impedances do not occur at the transfer poles of this section. This implies that the other image impedance cannot be taken as conStant-k type. Hence, this violates the definition of ter- minating section, i.e., the section will not be a terminating section as it is defined in Chapter V. fhe above problem was recently considered for only m - derived type of image impedance case [26], where Rowland's equivalent network trans- formations [341 are applied mostly to the mid~shunt type configurations. “rhese configurations constitute filter, whereas our discussion pertains to terminating sections. Since the procedure in {26} is not the exact answer to our problem, it will not he considered here. Returning to the earlier discussicn on terminating half secti ns, the disadvantage results in the increase of the number of elements in the filter. this disadvantage could be overcome by adopting a different de- sign procedure. However, the simplicity of the image parameter method is ”57” thereby lost. In practice, in most 01 the cases, only one critical Ire- quency for the image impedance or terminatinfl half section gives satis— factory results. Consequently, the aoove disadvantage or terminating half seetiuns is not so serious. If the disadvantage is considered formidable, some alternate methods are available: (l) Use of Darlington‘s method. ‘rhis method imposes a special con- tour or requirements on Ah in the olocn-oand. In this case, the following s can be done: Find the maximum point of the given Contour, Cs, and draw a horizontal line at this point. Choose this horizontal straight line as the centour of requirement. then apply the classical method. But in this case, necessarily more elehents in the filter will be used, since the degree or the Iilter is increased. (2) Use of Fromageot's method [5]. As indicated in the foregoing, this method involves searcning ror a ¢ - function which contains approxi~ mate requirements on the As — function. It is more general than Darlington's method, but does not differ from it when the element values are to be cal- culated. 6. '9 mama's momma It is considered of interest here to mention in this section iuttle's two mid-shunt type Zobel's sections which gives a ischebySCheri type of insertion loss charaCteristic in both eiieCtive pass— and blocn-band [35]. An explanation is given as to 'hy the extension or this problem to more than two sections does not give the same type of insertion loss charac- teristics. Consider 12m - derived mid-Shunt sections in cascade witn dirrerent parameters mi. the corresponding 9 ~ function will be round irom 2 l - 32 2 i.‘_2 ; 9 — - 2z ) binh PI (c./.l) where a .' . , x z = j:——--- With a 2 11/111m (0.9.2) 41.— x2 ‘ and - v i i PI 2: PIi (0.9.3) 1=l P- P . L H - I: e Il ~ l 111 ~ ianh —--2- - P"- — ml ‘o e Ii + 1 hence, PI 1 + mi 11‘ i _ .. .--...‘5. '4 ' e — i b. .4 l - m. h ( ) ) l o where H0 is the ratio runction oi the nal: prototype seeticn, i.e., Po y Ho T°nn “5 = *:——————”:'”" (Lao) x2 - 1 Consider PI 'PI . , ’ I, _ Sinh P = 5 (e - e ) \t.;.o) I and substitute eqs. 6.5.5 and 6.5.4 in eq. 6.5.6, we have V l + m n y l - n X s I“) P = .1. 77 ...‘f.___._i __ TT m... C ’ I 2 iel 1 -1m n =1 l +r‘ d, l 1) ‘ 4. \., or 0 2n 2 E) 2 [1+5 H + I +... r H ’ d +r an... -, 2v Ulw‘ ’ 2:,v o 01») 0 “3w 0 .v : binh — — - — -*-- —- ~---~--4—~——- I v 2 2 77 (l - r Ho ) ltl (c.5.7) where, if 1) is even, then 2n = 3), s = - l, 'V if a) is odd, then 2n = V-i, s = 1) , (j) \ Q I o :5 o, a is the symmetr ical function of'v variables, i.e., O'i’v ; :____-_- 3131 H132 no. IIIJV (LD'B'k‘j) 23132. ’ .JV ‘i,V On the other hand, from eqs. 6.5.2 and 6.5.5, we have hence, f') I ‘2 [(1 - a2) + a2 11 ‘] a (laJLiii)2 : O (L. .\ 22.4 1+ '12 (l - II 2) / “0 Substituting eqs. 6.5.7 and 6. 5. 9 intC: eq. 6. 3.1 yields 42 2 :2 2[ .2 gig-[01,61 3 ,s]2 2 - [(1 --. ) + a he] 1+ 02-2510 rooo:U2n1v-;10()Cj+f3vi.c +....Uva‘CI >3 = ,7 2(1-5?) 77—(l-m2 1-2“)2 --...- Q If 3) = 2, then substitutin: eq. 6.5.5 into eq. 6.5.10, we obtain (m1 + m )‘2 X2[(101- “2] [l-( + m m ) X2 2 2 _ 2 1 2 C , Q) — 9 2 2 2...2.__...____- (“5'1") a“ [l — (l — m_ ) :;]2 [l— (l - ”2 ) x ] Tuttle compared eq. 6.5.1 with the CD — function of a two section Darlingtoni i‘iltt r and showed the at the paraneters ml, mg, a, f'c can be L 3 determined uni’uely in terzs oi the oiia.eteis of 9D - function, (where ‘l f, is a iact or wish which the frequency scale to be multiplied). Indeed, the c: mpariszn 2f the C - and ¢D — functions gives four non-linear equa- tions of four variaol es (parameters) from which the parameters can be he» termined uniouclf. If V > 2, the mill: er of n:.2.rameters is "C“,liere‘o-r incre? oeCi tut the; now-- ber of non~linear equatiins increases :aster. i.e.. if“V u k > 2, the number of ungn wns is k + 2. but nu mber of n n-linear equations is 2: which is greater than R + 2 when h > 2. -90“ It can be shown that if a > 2, the nonlinear system or equatiuns is inconsistent. ihis implies that with the cascaded m — derived seetions, if ‘V > 2, it is not possible to attain Darlington's type cf insertion loss filter characteristics - "flat" in both pass and bloc? regions. VII. EFFECT OF DISSIPATION CY INSERTION LOSS AID PflASE O: A SIHHETRICAL IMAGE PARAMETER FILPER 7.1 G XERAL In actual practice, where the L and C elements in the filter are lossy, the insertion loss and phase oi the filter are slightly dirierent Iron the lossless form. In this chapter, the effect tr dissipation is considered and ‘iscussed thraugh the use of an example. This discussion is simplified by us= of a digital computer. rhe program which is written and used fur the example is general; that is, it can be used for any symmetrical low-pass image parameter filter. In the lossy case, tne formulas gar AS and ES functions are given by eqs. 3.2.1h and 3.2.1) in Chapter III. A "flow diagram" for the computer program is Shawn in fig. 7.1.1 at the end of this chapter. 7.2 EXAMPLE The example considered here is taken iron Reed's boah*. Data: A lew- ass filter cperates between two pure resistances, each of which is 75 ohms. ihe following are its design details: ”h--—.¢— .... .—- A ...-o- -—.——-‘—_-- - * Reference [2], pp. 197 - 207. LA. we have _\/=l_ = 555 he (to = 3.4;? x 10“ rad/sec) e 0.9940ph 1 = 1.1957288 ; i = 1.0137155 for terminating half secti3ns = 0.5u32568 = 0.1639233 ' \ : 0.13y7kt3657 g l = 0.137036383 fer intermediate sections = 0.27525272.) = 0.37805352h5 ; 2 kn [2]) L7] 'Jl ~ U~4 L . 2 » x h a ’ l = 1.06733063 Since the image impedance is of the 5cm? form, in eqs. 0.2.13 and 0-2-14, J the negative power for U must be used. Therefore, L o C o On the.data tape H H 0.201u12322s5 X 10'” henry 0.u03295h5 X 10-0 farad the following parameters appear: 75 ohms 0.20141232269 x 10'” 0.40s293u; x 10”d 3.40716764 X 100 (1/ QL) (1/ QC) -02.. I / n = (Number of divisions) Xfi a (Frequency interval) 2 1.3 ; 0.5us2500j9 ml 0.37006355h5 m 0 where QL and QC are the Q-factors or the L and C elements 01 the filter. In this example, QC is taken as infinity (dC = O) and for different values of 0L (= 50, 100, 200, 500, 20,000, 100,000,000) a set of insertion loss curves is obtained as shown in figs. 7.2.1-3. E #41,- / ADDI‘i‘ICJTEnL FOlfi-TJLAS RSQUIEUED FOR CALCUL’R‘BIOXS: A -: Z: A , , 5:3 :- 21: I.‘ (l - 1.1 a + m. b l o .L 0 mi "0 mi b_) , u k h. = arctan ---~—‘--,—--- + arctan ------—--- l 1 + mio‘ 1 - 111,1) L . l ’13 + c... t" F, [‘2 H H.m.> £2 \.0 + p P b b b F A d 4 u d a A1 d \A a O O O O ————————_——————————————————————————-fib H .OH flow rom I: 0+... ..om -9.“ 7- 14L, OH. - as ..D I oooaom , A@ P ud- GI Apev OH x m n < 1' OH ..ON ‘n r—4 C‘J x w L.“ w a _ A _ _ o _ _ __ __ __ :. :III _ a III/I 2* [III \Il'l \— // ’1 ll.\\ _ /... _ _ ..c 1‘ ii OOH J/ _ _ . _ :3 _ _ _ m, _ _ _ _ 3m - aw _ _.. _ _ >2 u 5i... _ _ . 33 w —>.'K')— GRAPE; I (U - l) 100'], AD” (5111) 20 142, x: 10"1+ 1 A -31 10 .3/, i 1-0 \ ‘\ \ \ ‘ , 1 2 18.9 .r 10”“ \ 2 4 l \ 1+.) X 10 \ \ \ \ ‘\\ \ \ 3 \ \ \ -1}. 0-2 0.113. x 10 1.01 1.03 1.05 1.07 1.09 1.11 1.13. » 1 n 1. AIDS :: 8.686 .011 ’2‘ (q + U) oo—o—4——H—.4+o or- '1 ET .1. -11-.-- r. C l . I )7 ...... --.—...- ,. i ‘¢l — mi2 T, 1 .L 'fi 73‘ P- infinity 2.30253 1.50040 1.20§97 0.91029 0.105;]-5 0.5103; 0.3;;07 {Di—”C O N)H‘C>C3C)(DC>C>CJC)C)C)C)C)C) . 0 \LJ CD '\] CWK"! 0.00000 infinitv GI qr- qt- .4- -h b db «1- «L .- D u- ~101~ COfCLJSICXS _‘1 A new formulation for inserticn loss and phase Kl a filter is con- sidered. The design procedure of lossless image parameter filters is described by using this new formulation. The discussion is focused an O the symmetrical low-pass filter. For hirh-pass and band—pass filter de— 9 J. sign, the well—finawn frequency trans? snations can be used and the prob— lem reduced to the low-pass filter design. Formulas for insertion f ncti a of a symmetrical image parameter \a ‘5 filter at cut-off frequency with a general terminating hall secti n are given. The terminating half sections are also considered in detail and Some formulas for the element values of terminating half sections are derived. In the classical procedures, the block-sand requirements on insertion loss function are reduced to the transfer-loss function by means or approximation f0 mules. 1his procedure results in allowing tolerances which, although sufficient, are by no means necessary. In this thesis, precise formulas are developed for reducing the insertion loss function to the transfer~loss function: Consequently, it is possiole to minimize the number of sections to be used in filter design. This new procedure is demonstrated through the use of an example. After designing the filter in which losses are not considered, the effect of dissipation on the filter elements is considered. A digital computer program is written for determining the insertion loss and phase characteristics for various values of QL and Q ‘Ihis computer program C. simplifies the investigation on insertion loss and phase characteristics. (l) (2) (12) (1) Q .U ~102- HIL‘LIWP‘XPH "'25 Zobel, O. 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J., ters," I.h.E., Vol. "Synthesis 0: Tchebyeheff Parameter Symmetrical Fil ”5’ NO' 42 APril; 1957; pp. kin-A73. Skwirz zynh ski, J. K. and Zdunek, J., "Note on Calculation of Ladder C Wireless Engineer, Vol. 29, K0. 3G2; March, 1952; p}. 00-79. T t O— efficients ior Symmetrical and Inverse.Impeaance Filters on a Digital Computer," I.R.E. 1930: pp. 328—333. 'fransactions on Circuit Theory, Vol. Ci-5, December, it 3&3’2343‘61 m "" (a- We. Ir. .._ L _ l: . - O' \ ‘~'-d9~'.~uc-w . ‘ ‘ A J hi my 11 uvriT .LJL &‘ ‘2: l L.,? L 34' "\ lll { E ‘3' K} i IIIUIWIIWIIH um iIIIWIH .3. 7 8875 1. 3 0 3 9 2 1. 3