Synthesis of Transformerless Active N-Port Networks: By LW, SANDBERG. ‘Ti losing Soom pr: Thecems An acnary apneic NN mati ows he comple freqerney al rational ane ne (a) can ein! te soe rue mrs uf oy N poet bear rainy ony veils, ces, fant N mgasve RC impedence, ad (D1 ean, te general be read a Ue slinen un oN pos wetoor mda Pairs, mee vcore erste td 3 wegitine Hope if “Phe suc ad nfo centions fer the fvatance- mat ea sion of Uraatunnsien ante of capo mel cgatoe rence ain an ss nw eee. [0 eit tea wrie coal el hy perwntny wyondry oe Be etiam (an asiiary NN shart en batiance males 20a anbcneed ravraecon ase RC netorek raping ne mre thn Ned sures The passe RO aetna intresting property tat on lag be ration on (BN ot Bernal eter of eter én lene wih cana reece ne ana na itrnad wes, Th ative uta eon anny ie rain ith negtieimpatance eo m alaulctice, aire, rks. “Tar cevrlopment of he tensistrhes povided the network ynthesit oth effin Inyeoon aot slereut and bas stimulated eonsdeeuble fegos in the theory selie neunks dang As ht dete, ‘Several tehizncs ie ets moped Tor re rn hanmere weive RE reaiston of taste al Srving pin! Sanesiona! "Tan Lak, Anew eashod hat ang wal aie Fert (i the simp f sneeney variable: un Iw aie the tenner ov ving tion of ¢ transience ertive AC nerons coating one aeive ele then’. Te parielay, Liavl's Techie! has bet eh bes for rl of The tte work, FO nH wen svar voewnteAL semnesAL, MAY IE a ok ww 1 T secak nes | cack ne tw en] “un Es seoently Hew shea? Hh aa bite WX nts of a ritoaalfnetons eon bo eid asthe shorter wdttsce matric ffs minaformeries W part tive AC nevronk contig‘ eantroled werent goo ub eset ae ei Hs Pots cv sages i eel of stay te Toren ld Er the abetael to tae papas. The proof, presley ae nest #62, fs based on feck qe developed in an ester oper for Saccring logs of matrix coffin: polynomits ina seal Variable. For the #3 Clad opse 8) = 1, one eet redoee to that af spre? * We also proset in Sovsion Ia procedure for t1e raliation of a scbsinay NN short-cinait adzsittance matric oe ex unbalanced tetve HC network nqpiring no ote toa A consnlld sees. The oye pave RC nelwrk has the inereting priperty’ Mit i ca lsayebe waliend ana (Nb 1) lormina network nf bowrlerminl ane Padaneee with commion aefeeee wade an un sterual nodes. ‘This Tenlt set only disphces she halgoed setwrlsassumption Semple "he poof even in Ret 1, but i er decals tenes i ts cm rg Cansiger a 24-poateatwonk ef sso au sapien chentenzed iy the shure-cvesitsdittnce sues Fd gp that 9 ative He adres — is rome spool AEP = Len 8 own in Pgs. Tbs unvenostto pation Fa fall LASSROIMTRLARS Serv ROU SHWoLS Tt voy a KN a Ya Ya lw ‘he shortsiteit wdmstnnge mate's ¥ abating the voltages and eur reat at ports Fh — 1257 ra avy bestow coe Y= ¥y Was ~ eg uur ery Ye & wor the amsensvit Cadieates mates sranspastion, Weasmunie het ¥ = (1/D)(M, fem ebitury prose esrmele WC ¥ rales eed vatinnal fuses, where iva mans of ely omnis aad 2 is 4 common denominator palynoe al. The gt hese Technique srs faut the tars subnet 2) Te deere that Fi wciahie as travformerless RC network aad thet the ele rents i dig faye) P0 HC arsine elias "Te mates ea he expressed a5 res yeas when Ky esa Ky ane teal spinels svefisont matron: ana che yy fee rel ad iy Dance ae Saw w 1 ie wel ow that f te ovficent marion in (8) are “denna ‘ngonal” nates ¥ canbe re el eansemerie baled eC fevnurk.” Our objective fo deowoine si ebrnateiere fn {20 that Yewtalice the dormian-digosal condition To simplify thy dees in eased thn to bogey 2a The CComsilor the eas of arsine Yay Voy Yor and lig Cy pae aa) stisfng (2) seh that Yo and [¥_— ¥en At a ep in tsning sight Sat the lation prablem we reveite (2) i the fling fo AYSIE= YuP Mae = Ke eu nie ca) Tok dew anisingona mss 3 as ene a amity nc ZT FOL tue eae sxeren mecutenn soensady aay te eis convenient to employ th foiewine uetaten: weeded bau, lik — Do oy shore By By tl Ruane NC N amtins of polvnomils and gis a omic: denerinator polyoma “Feo tat end (3), Dee a= eigen. 7 “Ve let-hnad se of (7) sa be writen before esucliaton of eaton fnctors nein mari of rl raion neti wth common denecninatar polynomial 4d P. Since the poles af the right ae deo (7) are se= flee! tn He dstnet end on he negs'seel ash, Xe rs le chuses fo Va the leas, ecmcon dawnnaie pelystial of as cin of Feviana: fmetione hee oy rene ist are cts ad on he mga ve ral axia. To seis hia emi, se employ a mle polyno! Fartoriatins terbicue desea in a ruler piper” Spoeifealy, se iE ehown Sn Appetais A thst, ven Y, ¢ realizable eubmateie Yy— (19) ea chosen «the (a) dog aie = deg q@ — NToGE = 1-0), where Ly — max sun deg Wy gD (Qe) le vifalisgnal aerator pelynomioln,t8 7) as nny et of real ecynomialseousstent wita ys — aad dag oy (eo, hae ony etree auton thas satiny the dann al veaieu wth the ney ign (Gi she mater polynomial P ened in (6 of Sogree® dex a} La can he writen 2 the roduet PD, of two mario palvtamia Pal Be {with AXA! mate eneticincs) of degrees respcetively ey and Ly {fo} det P does tans aeutelys ond (1) the sntce piyosunal By Tae the praperiy Ha det Pa, poly sonal af ogee Wy, hae oly divine aegsive en] aegce that arn Aiea thee of 4 Tat yl ch “alfouss we sal! assume that ews (a) chro 1 sce ai dy oro with (ae (4), te hab the Sefthand side of (7) can have only distinct negatiee-rel ples Xf ehven tn be (a) Sikere essay mnaoeo vel outs, fr hen 7) eee 10? b Low SSRI al By = Fam ne Caio S ty king, with chi ela 2 af Ny Yo 38 2ezular av innit fee fond i: cbereone, vung be migsitude of slit ere Wis cowaye posse on {0 tiny te domestica eit fete ft noe of ¥. Hee It Yo = LP a o 1 remains to identify Yes dhe yee hae the dominent diagonal ‘owe ca be sats Tho as 30s of "The et-hnd id w (S) aso is togue a inalty since te reins cog D+ dee P. 4 sage) Pe dex @ + Ny 10) redines a eng DS ms deg a rom i), ade TL toe ai here isa nonger real cumin AF = eg d+ Na, Th sew of (10) cud (12), (8) 04m be item a ~ dine (a teens) = Ee a wane o- <w a the A re agtantiecwfflant mateo. TL sea fom (13) That eae tlie tort fn Ye is real tothe rrsespning se the Hghthand side andl that ag (GesunenDoseeces aera) ~ Rg (yen) : wy - ing Kn oe tH Mig Cased = ag thaaseae eed ~ Hag lane fe wt BOW = 128 he Baca db Hed a lo An (aaa = eine hao th 1 Ete oe 1 dim bem dome . 10 = Spt cn om Mate fan ae Fa ‘ohio the atresia nay) ane cose to san’y the fdonsinantaiagoraleamditon i the let aoe of BH Ue matric is relaubin ao 4 trausiormelen tlineed 2N-port Cnet fir alleymmrtrie 47x N sateees ¥ of eral tional funtion. ‘Dae alization of a aphteery aymettic npei-sirit inpeanne smstiix Z von be treated se folows Ihe elemeite of matrie R diag C04) so be ekosea nennogucive and s.tbrintly ange cha: ¥" "2 R] eit ‘Theccfre, 2 eam be relied by neat ng a Etumegave} ers sere sev il al pont 8 = To A oF knee harap 2% ‘The peonf rlsting to the swesity of AV wngasive- aairaneee follows @reetiy trom ¢ moro geucral resule developed previously." + ee th had a Th AR pn te CRS al ue itor) a to "The tenis poset in his soon bear bey om the problems of relirig uniilinerd traisformesless N-port active ZC networks These ecarid-asios ar treat tel the felling tetion We euler 46 + stermival 42 nese to while i cornered we temminals F(R = 1200-207) a the sonsin freee ele UGA | Ti-termizal setive network se shown in Fig. 2, Dente by Boy Ey, Bey hey ey ssid Eile flowin cokinn mettiows of vakages and outcest 7 Bea Binge e-/®|, 2 -|2], Baw By Be 8 + a n “ine nel ine | fn he dts convenient ve surtiow B te shrs-ceenit admitcanes matris af WS un wan oman aueusucses soma, ae at the GLY terminal nero, aloe: xox oy Yo Ya Ta] fejx me ely us [ea "Th sett neon in asus tap the eit 1 --Ar, | B= BE, uw where Asad Bae XX A coeciaat mito. enor difeult to desive The following expression for the eher-citeuit admittance acti ¥ relat Jig By andy, the voltages and cureats at the A eceesible para i Fig. Y= Ya Ya Ya JAYaB a aM | Hel UF ANH. (em; {esha dilly a eigen Ieuan ug Loa the mastees and Fan ey Aw ov, B=0U, en whete U isthe identity matrix of erder 3 oud aed bare real easton us thal ab > 0, The eyutbedsteesque does ot furier meet The ‘Anns of ad 8 en that the (247 — L)-trminalaetire tvgnh can Salvage Waleed rh 2 voltegeincersinn or cuneat-iverson ewolivedinpedane converters by chou rpociiely «— 1, > 0. teh Iya 0, Rute therfore at the retliasie. cam aay be Secomplisbed wih N wore eo ete We shall vosider eset 1he one i whith 9, 2 > 0 andl ents “eesti sacoseary ria The weaning Dar abjctive ie & prove ow all eerie mati: ¥ ala $ oan be rutin’ c2 6 (QV -} jemainal nergark nf tie tern Sopa Ch tens ad cli fr arhevig this Le vezi ate tat the eae ate wae Fe ee ‘Ag aud By iulteduoed a Bectiw 220 TANS) IOHERTSS APT RON NECWUREE im fa meal ymmeirie otinent-digocal trons dh npestive of secs ters, | Sp Sa Se Taal 2b, 1s clear tha al ingen terest in Far rel to banegetive tC firing int cliitasee tretions. Vor sinpliity we aume the & fest hr haven poke a tity OK 6 She Hoatzoion Perique ur nota identi to at ued in the proeaing Seon Yu= P= bs Dey Ye = Xe, By poralling the delepment fn Section 2.1% and sing (20), (21 aud (2) oy obit ae Se ae ee We again assume that Yy is chosen ths (a {ty Beetin 2.1) anes. ssn in alton thst theo igo erm in Yo ane else to be negative Re ecng-point aitianee fusions Fee tL Next et wu + aK! — a ‘zene and wwe worsen tea paraseters to be chow in ascondsnce Sith tae essucsion bb? aud By ie nomsingiler watrit of 3° pry omits choses hs ee ety Su {/3P. ga uei cving point litem ass that fe monsera wt Hee vrs and Enice a Binoy It ese tied P= ‘We condor the matries Yon By. Eom (2 cod 28) se Sd 570 re wenn suena enue goby aK Hd ‘at y [Pi ie] 4 aia aa or too a * veo goteofrl da] Supyus hal? mgs > Nite nyse dog, — cy Bs = dei fore acs 8. aati sal wh Unt ed deen Yond Yu ist yee R? divi ois odmisance funtion Tie ‘Cha hate fl sat le cn uly Tage wri the ding erin the et 8 owe Neth pin Ue ste ees tte deren th states Ya, Yn and Yo 2 Datrmivotion af Yo Yo 0% Ya 1 pps = ay — ve ~ uve + vet on BBE = ely — Ye ue = Bh hare gdotes at hee +). 1 ean ewily he shown that the lefthand side of (28) is rgular atin nit. Hence ites be write En fo -Ee,.-Eu whee the Fy are val a wesseal nonsymmetsi ccffiiont marries, Can kn Smo Sa tnd the clomens in Ge and Ha, are sonnegacive Tie oon fr (28) athe esynanetay ie Remi be abso ter the torn “ee + EY! Re enti sanieyeomctr pcs 2 ‘ocho antigymauctsic puto! 1281, we oan Dee -'y * o.-6!)-m =m co a a ee salen ie ey bea by ati: lene