7.3 The Wilkinson Power Divider
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1 //9 7_3 Th Wilkinn Pwr Dividr.dc / 7.3 Th Wilkinn Pwr Dividr Rading Aignmnt: pp Th Wilkinn pwr dividr i th mt ppular pwr dividr dign. It i vry imilar t a ll 3dB dividr, but ha n additinal cmpnnt! HO: THE WILKINSON POWER DIIDER Q: I dn t hw th Wilkinn pwr dividr dign prvid th cattring matrix yu claim. I thr any way t analyz thi tructur t vrify it prfrmanc? A: Y! W imply nd t apply an dd/vn md analyi. HO: WILKINSON DIIDER EEN/ODD MODE ANALYSIS Jim Stil Th Univ. f Kana Dpt. f EECS
2 //9 Th Wilkinn Pwr Dividr 73.dc /3 Th Wilkinn Pwr Dividr Th Wilkinn pwr dividr i a 3prt dvic with a cattring matrix f: j j a j j j S = j j j b Nt thi dvic i matchd at prt ( S = ), and w find that magnitud f clumn i: b b 3 a a 3 3 = S S S Thu, jut lik th ll dividr, th incidnt pwr n prt i vnly and fficintly dividd btwn th utput f prt and prt 3: P P = = 3 = 3 = P S P P S P But nw lk clr at th cattring matrix. W al nt that th prt and 3 f thi dvic ar matchd! S = = S33 Likwi, w nt that prt and prt 3 ar ilatd: Jim Stil Th Univ. f Kana Dpt. f EECS
3 //9 Th Wilkinn Pwr Dividr 73.dc /3 S = S = 3 3 It th (narly) idal 3dB pwr dividr!!! Q: S jut hw d w mak thi Wilkinn pwr dividr? It lk a lt lik a ll 3dB dividr, nly with an additinal ritr f valu btwn prt and 3: Thi ritr i th crt t th Wilkinn pwr dividr, and i th ran that it i matchd at prt and 3, and th ran that prt and 3 ar ilatd. Nt hwvr, that th quartrwav tranmiin lin ctin mak th Wilkinn pwr dividr a narrwband dvic. Jim Stil Th Univ. f Kana Dpt. f EECS
4 //9 Th Wilkinn Pwr Dividr 73.dc 3/3 Figur 7. (p. 3) Frquncy rpn f an qualplit Wilkinn pwr dividr. Prt i th input prt; prt and 3 ar th utput prt. 3dB html Jim Stil Th Univ. f Kana Dpt. f EECS
5 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Evn/Odd Md Analyi f th Wilkinn Dividr Cnidr a matchd Wilkinn pwr dividr, with a urc at prt : Prt Prt Prt 3 T implify thi chmatic, w rmv th grund plan, which includ th bttm cnductr f th tranmiin lin: Prt Prt Prt 3 Jim Stil Th Univ. f Kana Dpt. f EECS
6 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Q: Hw d w analyz thi circuit? A: U EvnOdd md analyi! Rmmbr, vndd md analyi u tw imprtant principl: a) uprpitin b) circuit ymmtry T hw w apply th principl, lt firt rwrit th circuit with fur vltag urc: 3 Turning ff n pitiv urc at ach prt, w ar lft with an dd md circuit: Odd Md Circuit 3 Jim Stil Th Univ. f Kana Dpt. f EECS
7 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 3/ Nt th circuit ha dd ymmtry, and thu th plan f ymmtry bcm a virtual hrt, and in thi ca, a virtual grund! = 3 Dividing th circuit int tw halfcircuit, w gt: 3 Nt w hav again drawn th bttm cnductr f th tranmiin lin (a grund plan) t nhanc clarity (I hp!). Jim Stil Th Univ. f Kana Dpt. f EECS
8 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Analyzing th tp circuit, w find that th tranmiin lin i trminatd in a hrt circuit in paralll with a ritr f valu. Thu, th tranmiin lin i trminatd in a hrt circuit! = Thi f cur mak dtrmining trivial (hint: = ). Nw, inc th tranmiin lin i a quartr wavlngth, thi hrt circuit at th nd f th tranmiin lin tranfrm t an pn circuit at th bginning! A a rult, dtrmining vltag i narly a trivial a dtrmining vltag. Hint: = = And frm th dd ymmtry f th circuit, w likwi knw: 3 = = Nw, lt turn ff th dd md urc, and turn back n th vn md urc. Jim Stil Th Univ. f Kana Dpt. f EECS
9 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 5/ Evn Md Circuit 3 Nt th circuit ha vn ymmtry, and thu th plan f ymmtry bcm a virtual pn. I= 3 Dividing th circuit int tw halfcircuit, w gt: Jim Stil Th Univ. f Kana Dpt. f EECS
10 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 6/ 3 Nt w hav again drawn th bttm cnductr f th tranmiin lin (a grund plan). Analyzing th tp circuit, w find that th tranmiin lin i trminatd in a pn circuit in paralll with a ritr f valu. Thu, th tranmiin lin i trminatd in a ritr valud. Nw, inc th tranmiin lin i a quartr wavlngth, th ritr at th nd f th tranmiin lin tranfrm t thi valu at th bginning: in ( ) = = ltag can again b dtrmind by vltag diviin: = = Jim Stil Th Univ. f Kana Dpt. f EECS
11 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 7/ And thn du t th vn ymmtry f th circuit, w knw: 3 = = Q: What abut vltag? What i it valu? A: Wll, thr n dirct r ay way t find thi valu. W mut apply ur tranmiin lin thry (i.., th lutin t th tlgraphr quatin bundary cnditin) t find thi valu. Thi man applying th knwldg and kill acquird during ur chlarly xaminatin f Chaptr! If w carfully and patintly analyz th abv tranmiin lin circuit, w find that ( if yu can vrify thi!): = j And thu, cmplting ur uprpitin analyi, th vltag and currnt within th circuit i imply fund frm th um f th lutin f ach md: Jim Stil Th Univ. f Kana Dpt. f EECS
12 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 8/ j j = = = = = = 3 = 3 3 = = = j = = 3 Nt that th vltag w calculatd ar ttal vltag th um f th incidnt and xiting wav at ach prt: ( = ) = ( = ) ( = ) z z z z z z P P P ( = ) = ( = ) ( = ) z z z z z z P P P ( = ) = ( = ) ( = ) z z z z z z P 3 3 3P 3 3 3P Sinc prt and 3 ar trminatd in matchd lad, w knw that th incidnt wav n th prt ar zr. A a rult, th ttal vltag i qual t th valu f th xiting wav at th prt: Jim Stil Th Univ. f Kana Dpt. f EECS
13 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 9/ ( ) ( ) z = z = z = z = P P j ( ) ( ) z = z = z = z = 3 3 3P 3 3 3P Th prblm nw i t dtrmin th valu f th incidnt and z z z = z ). xiting wav at prt (i.., ( = ) and ( ) P P Rcall hwvr, th pcific ca whr th urc impdanc i matchd t tranmiin lin charactritic impdanc (i.., = ). W fund fr thi pcific ca, th incidnt wav launchd by th urc alway ha th valu at th urc: ( z = z ) = z z=z Nw, if th lngth f th tranmiin lin cnncting a urc t a prt (r lad) i lctrically vry mall (i.., β ), thn th urc i ffctivly cnnctd dirctly t th urc (i., βz = βzp): And thu th ttal vltag i: = z = z z = z z=z =z P in ( P ) ( P ) ( ) ( ) = z = zs z = z = ( z = zp ) P Jim Stil Th Univ. f Kana Dpt. f EECS
14 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Fr th ca whr a matchd urc (i.. = ) i cnnctd dirctly t a prt, w can thu cnclud: ( z = z P ) = ( z = zp ) = Thu, fr prt w find: ( z = zp ) = ( z = zp ) = = = Nw, w can finally dtrmin th cattring paramtr S, S, S : 3 S ( P ) ( ) z = z j j = = = z = z P S S 3 ( = P ) ( = ) z z = = ( ) = z z P ( = P ) ( = ) z z = = ( ) = z z P Q: Ww! That md lik a lt f hard wrk, and w r nly 3 f th way dn. D w hav t mv th urc t prt and thn prt 3 and prfrm imilar analy? Jim Stil Th Univ. f Kana Dpt. f EECS
15 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / A: Np! Uing th bilatral ymmtry f th circuit (, 3, 3 ), w can cnclud: j S3 = S = S33 = S = S3 = S3 = and frm rciprcity: j j S = S = S3 = S3 = W thu hav dtrmind 8 f th 9 cattring paramtr ndd t charactriz thi 3prt dvic. Th rmaining hldut i th cattring paramtr S. T find thi valu, w mut mv th urc t prt and analyz. Prt Prt Prt 3 Nt thi urc d nt altr th bilatral ymmtry f th circuit. W can thu u thi ymmtry t hlp analyz th circuit, withut having t pcifically dfin dd and vn md urc. Jim Stil Th Univ. f Kana Dpt. f EECS
16 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / Sinc th circuit ha (vn) bilatral ymmtry, w knw that th ymmtry plan frm a virtual pn. I= 3 Nt th valu f th vltag urc. Thy hav a valu f (a ppd t, ay, r /) bcau tw qual vltag urc in paralll i quivalnt t n vltag urc f th am valu. E.G.: Nw plitting th circuit int tw halfcircuit, w find th tp halfcircuit t b: Jim Stil Th Univ. f Kana Dpt. f EECS
17 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc 3/ Which implifi t: And tranfrming th lad ritr at th nd f th wav lin back t it bginning: Thu, Finally, w u vltag diviin t dtrmin that: = = Prt = Prt 3 And inc th urc i matchd: ( z = zp ) = ( z = zp ) = = = Jim Stil Th Univ. f Kana Dpt. f EECS
18 //9 Wilkinn Dividr Evn and Odd Md Analyi.dc / S ur final cattring lmnt i rvald! S ( = P ) ( = ) z z = = ( ) = z z P S th cattring matrix f a Wilkinn pwr dividr ha bn cnfirmd: S j j j = j Hi wrt handut vr! S, what d ya think? Oh n, I v n much wr. Jim Stil Th Univ. f Kana Dpt. f EECS
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