Selection of zeros of reflection coefficient in the design of filters by insertion loss

sible o ms o J( ). Fo example, we canno ob ain he esponse o an impulse unc ion and canno e en hope o app oxima e o i easonably by an . . . o economical size. The same ype o commen applies, al hough less o cibly, o a s ep- unc ion inpu . These a e he penal ies o ha ing an incomple e se o da a, and canno be a oided. We can, howe e , ob ain he esponse o p ac ical inpu s in which he ise ime is ini e, e en hough he p ocess may be expensi e i he ise ime is small. Again, we mus emphasise ha his is he only known solu ion o his ype o p oblem in which we equi e he solu ion o a di e en ial equa ion we canno o m. Fo a nonlinea ci cui , we assume ha he nonlinea i ies a e in he lumped po ion o he ci cui and a e o he ype ha may be ep esen ed by ol age-dependen cu en gene a o s. I we knew he di e en ial ope a o y( ) we could w i e down a se o di e en ial equa ions desc ibing he sys em y( ) ( ) = J( )+ [ ( )] (2) whe e / is a ec o o known nonlinea unc ions. The solu- ion o eqn. 2 is gi en by he Vol e a-in eg al equa ion >(0 = «W( )+ J H( -x) [ (x)]dx (3) As be o e, y( ) is no explici ly de ined; all we know a e he alues o Y(co) a speci ic equencies. i0)( ) is ob ained, as in he linea case, by inding he i. . . . o Z(co)I(co), and H( ) is he in e se Fou ie ans o m o Z(co), and is he e o e de ined a disc e e poin s. Subjec o con e gence, »(() may hen be ound by a Pica d i e a ion, o ming successi ely i0), (1), «(2>,..., using he ecu ence ela ion (0 = i0)( )+ H( -x) [ U)(x)]dx . (4) whe e, a each s age, he con olu ion in eg al is e alua ed by o ming he . . . F(co) o / [ U)( )] and, o each co, p e- mul iplying his ec o by he co esponding Z(co) and, inally, e ans o ming, ia he i. . . ., in o he ime domain. This ype o i e a ion has, in ecen yea s, ecei ed de ailed conside a ion by his w i e and o he s1"4 in i s applica ion o inding he s eady-s a e esponse o a nea ly linea sys em. P esen conside a ions show ha he desi able applica ions o eqn. 3 a e wide han we e o iginally hough , since, o a wide class o ci cui s, i o ms he only known me hod o analysis. Howe e , in addi ion o he di icul ies ha ha e al eady been obse ed in he linea case, he e a e now ad- di ional di icul ies a ising om he limi ed egion o con e - gence o he Pica d i e a ion. I is shown4 ha , o easonably well beha ed unc ions/, con e gence o he i e a ion may be ob ained in o he wise noncon e gen cases by he in oduc- ion o a damping ma ix C, modi ying eqn. 3 o J H( -x) [ (x)]dx (5) whe e U is he uni ma ix, and, using he i e a ion, B(1>(0 = (2U-C)C i0)+C J H( -x) [C (0>(x)]dx (6) a+1)( ) = C iO>(0+(U-C) U)( ) + C j H( - x) [V<J>(T)] dx (1) In p inciple, again o easonably well beha ed/, i is always possible o ind a ma ix C such ha his i e a ion con e ges, bu , o g ossly nonlinea ci cui s, his esul is gene ally no a use ul one as he p og ess owa ds a solu ion may be so slow as o be uneconomical, e en i i is no comple ely los in he ounding e o s o he compu e . Highe -o de i e a i e solu ions such as he New on-Raphson i e a ion also become imp ac icably unwieldy o any bu he simples sys ems. ELECTRONICS LETTERS 18 h Ap il 1974 Vol.10 No. 8 Thus he p esen si ua ion may be summa ised by saying ha , wi h he echniques cu en ly a ailable, nonlinea ci cui s ha canno be modelled in e ms o lumped elemen s can be analysed in he manne desc ibed p o ided ha he nonlinea i ies a e no oo se e e. No o he me hod o ob aining such esul s seems o be a ailable. No me hod is known by which a solu ion can be ob ained economically in g ossly nonlinea cases, and u he wo k is equi ed in his espec . By using he gene alisa ion o nodal analysis desc ibed in an ea lie le e ,5 all ou ypes o dependen gene a o may be accommoda ed in his ype o analysis. T. B. M. NEILL 18 h Feb ua y 1974 Pos O ice Resea ch Depa men Dollis Hill London NW2 7DT, England Re e ences 1 NEILL, T. B. M.: 'Imp o ed me hod o analysing nonlinea elec ical ne wo ks', Elec on. Le ., 1969, 5, pp. 13-15 2 HEYWOOD, D. R., and MOORE, A. D.: 'Commen on an imp o ed me hod o analysing nonlinea elec ical ne wo ks', ibid., 1969, 5, pp. 269-270 3 NEILL, T. B. M.: 'Reply o commen on an imp o ed me hod o analysing nonlinea elec ical ne wo ks', ibid., 1969, 5, pp. 270-271 4 NEILL, T. B. M.: 'Spec al analysis o nonlinea ci cui s' in 'Ne wo k and signal heo y'. PPL Con . Publ. 12, 1972, pp. 122-131 5 NEILL, T. B. M.: 'Gene alisa ion o nodal and mesh analysis', Elec on. Le ., 1969, 5, pp. 365-366 SELECTION OF ZEROS OF REFLECTION COEFFICIENT IN DESIGN OF FILTERS BY INSERTION LOSS Indexing e ms: Fil e s, Ne wo k syn hesis, Poles and ze os I is shown ha a cha ac e is ic unc ion ha can be exp essed as he squa e o an odd a ional unc ion in w, wi h poles on he j<o axis and a in ini y, is always ealisable wi hou using ans o me s by he selec ion o he ze os o p(s), and his is he key o a il e design based on 2-po syn hesis heo y. I one wan s o inse an LC il e be ween a uni y-in e nal- esis ance gene a o and a load esis ance (bo h no malised), i s ansmission cha ac e is ic may be exp essed by means o i s inse ion-loss unc ion P20IP2, o i s cha ac e is ic unc ion F((o2), ela ed by De ining p(s)p(-s) = - = _** wi h = 4 /(l + )2; he il e inpu immi ance is1 Z^ o 7(5 = 777 1 +p{s) and i s ans e unc ion is E2 l + /(mlm2 — H(s) = (1) (2) (3) Fo a lowpass il e —o he band con igu a ions a e ob ained by a equency ans o ma ion— he ollowing exp essions2 gi e i s pa ame e ma ix: o — Z2i o -V Y2l = — Z22o Y22 = — n2 Z(mlm2-nl n2)(4) 133 I F(co2) > 0 o all co, and i s poles a e loca ed on he jco axis, aking om eqn. 1 he le -hal plane poles o p(s), eqn. 2 gi es an immi ance ha is a eal posi i e unc ion, and he pa ame e s om eqn. 4 will cons i u e an odd eal posi i e ma ix, which leads o a p ac ical ladde il e wi hou ans o me s, p o ided ha he Fialcow-Ge s 3 condi ions a e me . A his poin , he selec ion o he ze os o p(s) is o g ea es impo ance. Le N be an odd and M an e en polynomial in s sa is ying (a) deg ee o N > deg ee o M (b) highes -deg ee- e m coe icien o JV equals 1. Taking 2) - - — s = jco as he cha ac e is ic unc ion,* hen Pi S = JCO M2-N2 S = JCO and, om eqns. 3 and 1, and p(s)p(-s) = •-«! n2) = - -—M (m2-m1)2-(n2-n1)2 (l- )M2-N: M2-N2 (5) (6) Ob iously, om eqn. 6, he immi ance has a ze o o a pole a in ini y; he ea e i will be supposed o ha e a ze o, wi hou loss o gene ali y, and P(O) = ± (7) To o m p(s), he le -hal plane oo s o M2-N2 mus be ob ained, gi ing Mj+Ni. I may be easily p o ed ha (a) deg ee o N1 = deg ee o N (b) deg ee o N1 > deg ee o Mx (c) Mx(0) = M(0) (d) Nx highes -deg ee- e m coe icien equals 1 (e) he coe icien o a gi en powe o s in Mi ponding coe icien in M (coe icien s Mi M). The selec ion o he ze os o p(s) ha gua an ees he ul il- men o he Fialcow-Ge s condi ions by he ma ix o eqn. 4 leads o he co es- coe icien s p(0)M + N P(s) = whe e p(0) is gi en in eqn. 7, and, ese ing he symbol + o ob ain Z(s) and he symbol — o Y(s), is equi alen o aking Z(0) = and 7(0) = 1/ . F om eqns. 2, 4 and 5, he p oo ha his choice o p(s) is co ec educes o e i ying ha coe icien s—- ( ML — M I ^ coe icien s M 2 1 + } 1 + and coe icien s-—- {M + MI ^ coe icien s M 2 1 + J 1 + whe e he ac o 1/2 has been in oduced, so ha (8) n2 = has i s highes -deg ee coe icien equal o uni y, and o p es- e e consis ency wi h eqn. 6; e i ica ion o eqn. 8 is im- media e once i has been es ablished ha coe icien s M 5s coe icien s M. * Ob iously, we suppose he ze os o M o be on he jco axis 134 I is in e es ing o men ion ha , when M is a cons an , all il e ansmission ze os a in ini y, since he ze o-deg ee nume a o e m is independen o he choice o he ze os o p(s), he ul ilmen o he Fialcow-Ge s condi ions will no depend on ha selec ion. 22 nd Feb ua y 1974J. B. MARINO ETS Jngenie os de Telecomunicacion Uni e sidad Poli ecnica de Ba celona 331 A enida del Caudillo Ba celona, Spain Re e ences 1 TUTTLE, D. F.: 'Redes elec icas' (Dossa , Mad id, 1964), chap. 13 2 VLACH, j.: 'Compu e ized app oxima ion and syn hesis o linea ne wo ks' (Wiley, 1969), chap. 4 3 FIALCOW, A. D., and CERST, I.: 'The ans e unc ion o gene al wo e minal-pai RC ne wo ks', Qua . Appl. Ma h., 1952, 10, pp. 113-127 OBLIQUITY FACTOR FOR RADIATION FROM SOLID-STATE LASER Indexing e ms: Lase beams, Solid lase s The adia ed ield consis s o o wa d adia ion a an angle 0 and backwa d adia ion a n — 0 e lec ed om he dielec ic- ai in e ace in o an addi ional componen o ield a angle 0. The o al adia ion con ains a ac o ha simpli ies in o an exp ession ecen ly ob ained by mo e sophis ica ed me hods, and which is he e in e p e ed as he Huygens obliqui y ac o o he a angemen . A small co ec ion o he exp ession is also ob ained. Recen ly,1 by means o ma hema ical echniques in ol ing Fou ie in eg als and saddle-poin asymp o ic e alua ions, exp essions ha e been p oduced o some p ope ies o solid- s a e-lase adia ion, including he damped in e e ence wa es a he dielec ic-ai bounda y, and an exp ession leading o he obse ed sha pening o he adia ion-pa e n. The la e ac o can be w i en g(0) = whe e 2cos0 {cos0+V(£2-sin20)} {P/ko+V(e2-sin26)} (0 g(6) = adia ion-pa e n ac o addi ional o main adia ion-pa e n exp ession 6 = angle o -axis a which he ield is obse ed e2 = ela i e pe mi i i y o lase ma e ial /? = axial p opaga ion ac o in lase kQ = p opaga ion ac o in ai I is in e es ing o no e ha eqn. 1 can be ob ained by elemen- a y means, and ha i ep esen s, o he dielec ic-ai con- igu a ion, he o m aken by he Huygens obliqui y ac o .* Fig. 1 shows he a angemen , wi h an inciden su ace wa e o he o m1 Ey = Ei{x) exp (-j iz) z < 0 (2) whe e E,(x) is he ans e se- ield dis ibu ion o he inciden su ace-wa e mode. In egion 3 (ai ), he ield can be ound om he o m o Huygens's in eg al in cylind ical co-o di- na es:' 00 y/ = (8; £o)~*exp(-77 /4) dy/Q exp(-jko ) 8n V ~¥oIn- V? ]dX (3) * The Huygens obliqui y ac o is he adia ion pa e n o an elemen o a adia ing ape u e. I has he alue cos 0+cos 0 o adia ion a an angle 0 o inciden adia ion a an angle < >. I is slowly a ying, and is o en igno ed when calcula ing adia ion pa e ns om la ge ape u es ELECTRONICS LETTERS 18 h Ap il 1974 Vol.10 No. 8