Scattering matrices of multiport radio devices

Transcription

1 Gedj CAWKA Blystok Uversty of Techology do:599/4865 ctterg mtrces of multport rdo devces Abstrct Ths pper dels wth the mthemtcs models d propertes of three types the sctterg mtrces for rbtrry multport devces ormlzed to seprte resstve lods seprte complex lods d complex -port lod The troduced sctterg mtrces descrbe mtchg problem dfferet multport structures for exmple coecto of gve te rry wth multport bemformer These mthemtcs models my be used for lyss of sgl dstrbuto d ther dsturbces dfferet multport structures d for the desg of y optmum multport devces treszczee Artykuł przedstw modele mtemtycze włścwośc trzech typów mcerzy rozprosze dl dowolych ukłdów welowrotkowych ormlzowych do osobych rezystywych zespoloych obcążeń orz do obcąże welowrotkowego Wprowdzoe mcerze opsują problem dopsow różorodych welowrotkowych struktur (Mcerze rozprosze welowrotkowych ukłdów rdowych) Keywords: sctterg mtrces multport devces ew sctterg mtrx łow kluczowe: mcerze rozprosze ukłdy welowrotkowe ow mcerz rozprosze Itroducto Oe of the clssc problems te theory s desg of optmum equlzer to mtch rbtrry multport lod (te rry) to geertors [35] The soluto of ths problem my be crred out by use of dfferet sctterg mtrces ormlzed to dgol or multport etworks [-9] The pper presets mthemtcl models d propertes of three type the sctterg mtrces for gve multport devces (Fg): ormlzed to seprte resstve lods [] seprte complex lods [3] d complex -port lod The bss of the sctterg mtrx ormlzed to -port complex lod etwork s descrbed [4] Usg of ths mtrx for multport brodbd mtchg problem for the rbtrry te rry s preseted [7-9] These sctterg prmeters re bsed o egevlues d egevectors of the multport mtrces As show there s fte umber of the sctterg mtrces for gve -port complex lod (or vlue of coupled geertors) d ll of these descrbe the sme power dstrbuto gve multport etworks The sctterg mtrx ormlzed to dgol mpedce mtrx The sctterg mtrx coects vectors of cdet d reflected b wves of multport etwork t ll: b = There re dfferet determtos of the cdet d reflected wves d forms of the sctterg mtrx ormlzto ccordg to types of lods of the multport etwork (Fg): - ormlzto to resstve lods (Fg) [3]: () U b U I I ; the sctterg mtrx coected to mpedce mtrx : () 5 5 ( ) ( ) where = { } - dgol mtrx of geertor ressters; - ormlzto to complex lods (Fgb) [35]: ) b) c) z P Fg tructures of the multport etworks z z P (3) r U z I r b U z I where r = e z - rel prts of complex mpedce s z ; the sctterg mtrx coected to mpedce mtrx : (4) 5 5 g ( g ) ( g ) g where g = e g = e{z } rel dgol mtrx I the both cses totl verge power bsorbed by etwork for rbtrry cdet vector of seprte geertors (Fgb) s [3]: (5) P e( U I) ( ) ; superscrpt (+) deotes the complex cojugte trspose (lso clled the hermt cojugte) mtrx [5] A ew sctterg mtrx ormlzed to -port complex mpedce mtrx A soluto of the brodbd multport te mtchg problem for gve te ptter my be crred out wth bse of desg lgorthm wth use of ordry sctterg mtrx (Fgc) [-356]: Determto of mpltude-phse te dstrbuto for gve te ptter ythess of the correspodg brod bd dvder Alyss of prtl mpedce s for obted dstrbuto 3 ythess of optml mtchg etworks for determed prtl mpedce s 4 The t s chged: ) prtl mpedce s b) mpltude-phse te dstrbuto c) te ptter (m lobe d sde lobe) 5 Optmzto of the brodbd dvder Ths s terto process d soluto of mtchgptter problem my be very dffculty te feedg etwork (sgl dvder) b L o mtchg etworks A d) o - port source etwork o I U I L U - port te I rry U L 8 PGLĄD LKTOTCHICY I /6

2 ffectve soluto of ths problem my be crred out by use of ew multport sctterg mtrces ormlzed to - port etwork [47-9] A ew sctterg mtrx s costructed for gve lod mpedce mtrx L d o-dgol output mpedce mtrx o of the te-feedg etwork o (bemformer) d correspodg equvlet structure (Fgcd) Wth use equtos = U + o I d U = L I where U I correspodg vectors we hve for totl verge power P bsorbed by two-port (Fgd): (6) P e( U I) I L I where: L = 5( L + L + ) - rel prt of lod mpedce mtrx L I = ( o + L) - - curret vector For del mtchg cse we hve codto + o = L the cdet curret s I = o - where o = 5( o + o + ) - rel prt of geertor mpedce mtrx o (Fgd); the mxml verge power P mx bsorbed by two-port s (7) P mx 5 o The for rbtrry lod etwork (Fgd) reflected verge power P ref s: Pref Pmx P (8) 5 I [( o L ) o ( o L ) ( L L )] I After trsformto we hve: (9) Pref 5 I ( L o ) o ( L o ) I Therefore totl verge power bsorbed by multport etwork s gve by: P Pmx P ref () 5 o 5 I ( L o ) o ( L o ) I Further rel prt of output mpedce mtrx o hs rght- d left-stdrd fctorztos [47]: () o 5 ( o o ) H H Q Q where H Q H + d Q + - complex fctors d hermt cojugto of the mtrces ccordgly s (5) superscrpt (+) deotes hermt cojugte mtrx For the recprocl etwork o = o t therefore H = Q t d further oly rght -stdrd fctorzto wll be used The rel symmetrcl postve determed mtrx o hs orthogol expso [5]: () o t r o where = t = - rel orthogol egevector mtrx d r o > - rel egevlues of the mtrx o The fctor mtrx H hs followg expso wth use of complex rbtrry utry mtrx W WW + = : (3) H W r o I ths cse complex cdet d reflected b wves d complex ormlzed sctterg mtrx of the multport etwork (Fgd) ormlzed to the mpedce mtrx o of - port complex source etwork o re gve by: (4) H U oi ( L o) I H b U oi ( L o) I b If to sert these formule for cdet d reflected b wves () we hve kow equto for totl verge power bsorbed by multport etwork: (5) P Pmx Pref b b After comprso of equtos (-5) we mke mportt cocluso: there re fte umber of the cdet d reflected b wves d multport sctterg mtrces for gve multport complex lod through the rbtrry utry mtrx W But ll of these sctterg mtrces determe the sme power dsspto multport structure (Fgd) becuse hermt form s used for power clculto () d s result of further trsformto of formuls both mtrces W d W + dspper The sctterg mtrx my be expressed by mes of the mpedce mtrces: (6) ( H ) ( L o ) ( L o ) H where L d o - mpedce mtrx of the multport lod L d source etworks o (Fgd) It s possble to express terms of Y for the "ugmeted" - port: (7) C H ty H where: C ( H ) H Wt W C C s the complex utry mtrx determed by rbtrry utry mtrx W from (3); Y L o s dmttce mtrx of the "ugmeted" - port (Fgd) Thus the complex ormlzed sctterg mtrx s (8) t [ o t o (9) W r Y r W ] The elemets of re determed by mes of: () b j s j where re the elemets of cdet wve vector: () 5 H The determto of s j s my be mde from the codto of oe ozero elemet from vector d ll ozero elemets from : / ( h ) j j () j ( h j ) / h j where h h j re the elemets of fctor mtrx H (3) For determed sctterg mtrx d rbtrry exctto vector voltge d curret vectors re: (3) U [ o ( H ) oh t ] I [ ( H ) H t ] It s my be prove the totl verge power bsorbed by whole multport L s gve by (4) P e( U I) b b ( ) D PGLĄD LKTOTCHICY I /6 9

3 where mtrx D = - + med by dsspto mtrx The ormlzed totl verge power bsorbed by etwork L (ylegh rto) s lmted by the mmum d the mxmum egevlues of the dsspto mtrx D: (5) d m P / Pmx ( D ) / dmx D d d d d wht's more the ylegh rto s rrved egevlues d m d mx whe the cdet wve vectors re prllel to the correspodg egevectors of the mtrx D I ths cse the optmzto d the mtchg problem of the rbtrry multport etwork reduce to the mxmzto of the mmum egevlues of the dsspto mtrx D t the gve frequecy bd A ew sctterg mtrx for cscde coecto of multports Cosder the cscde coecto of multport etworks: exctto of - port lod from set of seprte geertors by multport couplg etwork (Fg) [78] Determe the complex cdet d reflected wves for the sctterg mtrx of the multport couplg etwork ormlzed to seprte complex terl source mpedce s z from ports d to - port lod etwork mpedce z o from ports The etre ormlzto mpedce mtrx correspodg to source etwork o (Fgd) s zo (6) o zo z o z for whch the frst geertors re ucoupled but the rest of these re zero (Fg) I ths cse the etwork structure Fg my be cosdered s the prtculr etwork of Fgd The for ports the complex ormlzed cdet d reflected wves re: (7) Ho U zo I Ho b U zo I where U d I re voltge d curret vectors of multport etwork for these ports (Fg); H o d H + o re dgol mtrces - fctors of the rel prt of dgol mtrx o of ucoupled geertors: o ( z z o ) / r H H (8) o o o where: h r z z o z z H o H o I I I U U couplg etwork U U I I I U U - port etwork z o lod (te rry) Fg The structure of cscde coecto of multport etworks Cosequetly cdet d reflected wves for these ports my be determed by the sme s () [3]: (9) r U z I r b U z I where r = e z The cdet wves exctly correspod to compoets of the vector : (3) / r For ports the cdet d reflected b wve vectors of the multport etwork ormlzed to the totl mtrx z o of - port complex lod etwork re gve by geerl reltos (9) [47-9]: (3) Ho U z o I H o b U z o I where H o - complex mtrx fctor of the rght-stdrd fctorztos of rel prt of z o s (): (3) o ( zo z o ) / H o H o The every compoet of cdet wve vector for ports s determed by etre vector = [ + ] t: / (33) 5 H o ; the codto of oe ozero elemet of s ll ozero elemets from totl cse: (34) / h j j j h j / h j where h h j re elemets of fctor mtrx H o (7) It s kow tht the symmetrcl rel mtrx o hs orthogol expso s () [5]: o r (35) t where = t = - rel orthogol egevector mtrx d r = r > - rel postve egevlues of the mtrx o Therefore fctor mtrx H o hs totl followg expso: (36) H W o r where W - s rbtrry complex utry mtrx WW = Cosequetly there s fte umber of the mtrces H o d vectors d b (7) (9) coected to rbtrry mtrx W for gve -port complex lod z o (Fg) The complex ormlzed sctterg mtrx of the etwork (Fg) s provded by the block reltos: b (37) b = b d for recprocl etworks: (38) t t t It s mtter of drect verfcto to prove the correlto betwee the block of complex ormlzed sctterg mtrx d mpedce mtrces from ports [3]: PGLĄD LKTOTCHICY I /6

4 (39) r z z r where - mpedce mtrx of the - port etwork cosst of - port couplg etwork d - port lod etwork (Fg); z - terl source mpedce s The etre sctterg mtrx my be expressed by mes of the dmttce mtrx [7]: (4) Co Ho C o Ho Y H ot Ho where: Y - dmttce mtrx of "ugmeted" - port : (4) Y Y Y Y z Y z o - blocks of mpedce mtrx of - port : (4) C o d C o re ll-pss ("phse") mtrces determed by ormlzto mtrces H o d H o : o o o (43) C H H r r (44) C H Ho Wt W o C o C o o o C t Accordg to (8) phse mtrx C o s equl to ut o mgry xs (43); but C o s whole symmetrcl utry mtrx determed by the complex rbtrry utry mtrx W (36) (44) The the blocks of the sctterg mtrx my be expressed by mes of blocks (4) of the dmttce mtrx Y of the "ugmeted" - port (Fg): (45) / / o Y o / o Y Ho Co Hot Y Ho The elemets of block wth (39) (44) my be expressed by the followg prmeters: z (46) s y r sj yj r rj z where represet the mpedce see lookg to port of sde uder mtched termtos by mpedce s z for j j j d lod etwork z o (Fg); y y j re elemets of block Y of the dmttce mtrx Y For block used (36) d (44) we obt: (47) W r Y r t t W The expressos (4) (4) d (43) re show tht block uequvoclly determed by terl source mpedce s z ; but there s fte umber of the sctterg mtrx blocks d the etre mtrx for gve - port complex lod through the rbtrry utry mtrx W All of these sctterg mtrces determe the sme power dsspto of the etwork (Fg) Power prmeters of ew sctterg mtrx Cosder exctto of the cscde coecto of -ports from sde oly: there re rbtrry vector d = (Fg)I ths cse voltge d curret vectors re: U (48) 5 o ( z o z 5 I ( o o ) U z o H ) I H o ot t I ths cse the totl verge power P bsorbed by - port lod etwork d couplg etwork for the gve exctto vector d = (Fg) s gve by P b b D (49) where: s determed by (6); mtrx D = - + med by dsspto mtrx [57-9] It s the hermet mtrx D = D + d the utry smlr to the dgol postve rel mtrx of the egevlues d [78]: d (5) D d d where s the complex utry mtrx (clled the modl) of the egevectors of mtrx D For del mtchg mxmum totl verge power bsorbed by whole etwork s: (5) D P mx The ormlzed verge power bsorbed by d for the rbtrry vector s the ormlzed hermet form of dsspto mtrx D (ylegh rto) [57-9]: P (5) P mx D The vlue of the ylegh rto s fucto of the cdet vector It s show tht the ylegh rto hs sttory quttes equl to the egevlues of the correspodg hermt mtrx; ths mes tht the ormlzed totl verge power P /P mx s lmted by the mmum d the mxmum egevlues of the dsspto mtrx D : P D (53) d m d mx Pmx d wht's more the ylegh rto s rrved egevlues d m d mx whe the cdet wve vectors re prllel to the correspodg egevectors of the mtrx D (5) As forms (49) - (5) the ormlzed verge power P /P mx bsorbed by lod etwork for the rbtrry vector s the ormlzed hermet form of hermt mtrx + (ylegh rto) d lmted by the mmum d the mxmum egevlues of ths hermt mtrx: P (54) d m d mx Pmx Cosequetly the optmzto d the mtchg problem of the rbtrry multport etwork s reduced to two tsks: - mxmzto of the mmum egevlues d m of the dsspto mtrx D (5) t the gve frequecy bd for the optmzto of the totl verge power P bsorbed by - port lod etwork d couplg etwork PGLĄD LKTOTCHICY I /6

5 for rbtrry exctto vector (the totl verge source power); - mxmzto of the mmum egevlues d m of the hermt mtrx + t the gve frequecy bd for the optmzto of the totl verge power P bsorbed by lod etwork oly for rbtrry exctto vector If - port couplg etwork (Fg) s lossless the vlues of the power re equl (P = P ) the utry codto for sctterg mtrx s provded: (55) D = - + = + d these two tsks re reduced to y oe Coclusos The cosdered mthemtcl models d the structures of three types the sctterg mtrces for rbtrry multport devces ormlzed to seprte resstve d complex lods d complex -port lod my by use for the evluto of the brodbd problems of rbtrry multport structures It s preseted structures d propertes of ew sctterg mtrx for geerl cse d for multport cscde coecto The clculto of the blocks of the ew sctterg mtrces d power prmeters bsed o the egevlues d egevectors of the multport etwork mtrces my be crred out by use the stdrd computer pckets As show there s fte umber of these sctterg mtrces for gve -port complex lod (or vlue of couplg geertors) d ll of these descrbe the sme power dstrbuto gve multport etworks The troduced sctterg mtrces descrbe mtchg problem dfferet multport structures for exmple coecto of gve te rry wth multport bemformer These mthemtcl models d prmeters my be used for the desg of optmum multport rectve equlzer to mtch rbtrry pssve multport lod to coupled geertors d the optmzto of multport te-mtchg etwork of the dfferet structures These mthemtcs models my be used for lyss of sgl dstrbuto d ther dsturbces dfferet multport structures d for the desg of y optmum multport devces oluto of these tsks s very mportt for MC d MD lyss of the rdo systems wth complcte trsmt d receve prts of whole system Ths work ws supported by the ector s Project r/w//5 Blystok Uversty of Techology Authors: prof dr hb ż Gedj Czwk Poltechk Błostock Wydzł lektryczy ul Wejsk 45D 5-35 Blystok - ml: gczwk@pbedupl FC [] Crl HJ The sctterg mtrx etwork theory I Trs o Crcut Theory vol CT- 3 (956) Mrch p [] Youl DC O sctterg mtrx ormlzed to complex port umbers Proceedgs I vol 49 (96) o7 p [3] Che WK Brodbd Mtchg: Theory d Implemettos World cetfc Publshg Co Pte Ltd (988) [4] ohrer A The ctterg Mtrx: ormlzed to Complex - Port Lod etworks I Trs o Crcut Theory vol CT- (965) Jue p 3-3 [5] ewcomb W Ler Multport ythess McGrw Hll Book Co ew York (966) [6] Czwk G Ltwńczuk x - chel brodbd homogeeous mcrostrp power dvder for commucto te rry X Itertol Coferece o Mcrowves dr d Wreless Commuctos MIKO Wrszw My 7-9 (4) p-3 [7] Chvk G Alyss of te rry usg ew sctterg mtrx 9th Iter Wrocłw ymposum d xhbto o lectromgetc Comptblty MC Wrocłw Jue -3 (8) p3-35 [8] Czwk G ow mcerz rozprosze welowrotkowych ukłdów teowych Przegląd Telekomukcyjy Wdomośc Telekomukcyje (4) r 6 str [9] Chvk G Grbruk M Alyss of power prmeters of twoport te usg ew sctterg mtrx publshed ths ssue PGLĄD LKTOTCHICY I /6