5.4 The Quarter-Wave Transformer

Transcription

1 4//9 5_4 Th Qurtr Wv Trnsformr.doc / 5.4 Th Qurtr-Wv Trnsformr Rdg Assignmnt: pp , 4-43 By now you v noticd tht qurtr-wv lngth of trnsmission l ( λ 4, β π ) pprs oftn microwv ngrg prolms. Anothr ppliction of th λ 4 trnsmission l is s n impdnc mtchg ntwork. HO: THE QUARTER-WAVE TRANSFORMER HO: THE SIGNA-FOW GRAPH OF A QUARTER-WAVE TRANSFORMER Q: Why dos th qurtr-wv mtchg ntwork work ftr ll, th qurtr-wv l is mismtchd t oth nds? A: HO: MUTIPE REFECTION VIEWPOINT Jim Stils Th Univ. of Knss Dpt. of EECS

2 4//9 Th Qurtr Wv Trnsformr.doc /7 Th Qurtr-Wv Trnsformr Sy th nd of trnsmission l with chrctristic impdnc is trmtd with rsistiv (i.., rl) lod. R Unlss R, th rsistor is mismtchd to th l, nd thus som of th cidnt powr will rflctd. W cn of cours corrct this sitution y plcg mtchg ntwork twn th l nd th lod: Mtchg Ntwork R In ddition to th dsigns w hv just studid (.g., - ntworks, stu tunrs), on of th simplst mtchg ntwork dsigns is th qurtr-wv trnsformr. Jim Stils Th Univ. of Knss Dpt. of EECS

3 4//9 Th Qurtr Wv Trnsformr.doc /7 Th qurtr-wv trnsformr is simply trnsmission l with chrctristic impdnc nd lngth λ 4 (i.., qurtrwv l). R λ 4 Th λ 4 l is th mtchg ntwork! Q: But wht out th chrctristic impdnc ; wht should its vlu?? A: Rmmr, th qurtr wvlngth cs is on of th spcil css tht w studid. W know tht th put impdnc of th qurtr wvlngth l is: ( ) ( ) R Thus, if w wish for to numriclly qul to, w fd: ( ) R Jim Stils Th Univ. of Knss Dpt. of EECS

4 4//9 Th Qurtr Wv Trnsformr.doc 3/7 Solvg for, w fd its rquird vlu to : ( ) ( ) R R R In othr words, th chrctristic impdnc of th qurtr wv l is th gomtric vrg of nd R! Thrfor, λ 4 l with chrctristic impdnc R will mtch trnsmission l with chrctristic impdnc to rsistiv lod R. R R λ 4 Thus, ll powr is dlivrd to lod R! Als, th qurtr-wv trnsformr (lik ll our dsigns) hs fw prolms! Jim Stils Th Univ. of Knss Dpt. of EECS

5 4//9 Th Qurtr Wv Trnsformr.doc 4/7 Prolm # Th mtchg ndwidth is nrrow! In othr words, w ot prfct mtch t prcisly th frquncy whr th lngth of th mtchg trnsmission l is qurtr-wvlngth. But rmmr, this lngth cn qurtr-wvlngth t just on frquncy! Rmmr, wvlngth is rltd to frquncy s: v p λ f f C whr v p is th propgtion vlocity of th wv. For xmpl, ssumg tht v p c (c th spd of light vcuum), on wvlngth t GHz is 3 cm ( λ.3 m ), whil on wvlngth t 3 GHz is cm ( λ. m ). As rsult, trnsmission l lngth 7.5 cm is qurtr wvlngth for signl t GHz only. Thus, qurtr-wv trnsformr provids prfct mtch ( ) t on nd only on signl frquncy! Jim Stils Th Univ. of Knss Dpt. of EECS

6 4//9 Th Qurtr Wv Trnsformr.doc 5/7 As th signl frquncy (i.., wvlngth) chngs, th lctricl lngth of th mtchg trnsmission l chngs. It will no longr qurtr wvlngth, nd thus w no longr will hv prfct mtch. W fd tht th closr R (R ) is to chrctristic impdnc, th widr th ndwidth of th qurtr wvlngth trnsformr. Figur 5. (p. 43) Rflction coffict mgnitud vrsus frquncy for sgl-sction qurtr-wv mtchg trnsformr with vrious lod mismtchs. W will fd tht th ndwidth cn crsd y ddg multipl 4 λ sctions! Jim Stils Th Univ. of Knss Dpt. of EECS

7 4//9 Th Qurtr Wv Trnsformr.doc 6/7 Prolm # Rcll th mtchg solution ws limitd to lods tht wr purly rl! I.E.: R + j Of cours, this is BIG prolm, s most lods will hv rctiv componnt! Fortuntly, w hv rltivly sy solution to this prolm, s w cn lwys dd som lngth of trnsmission l to th lod to mk th impdnc compltly rl: z R, β r r possil solutions! Howvr, rmmr tht th put impdnc will purly rl t only on frquncy! W cn thn uild qurtr-wv trnsformr to mtch th l to rsistnc R : Jim Stils Th Univ. of Knss Dpt. of EECS

8 4//9 Th Qurtr Wv Trnsformr.doc 7/7 R R λ 4 Ag, sc th trnsmission ls r losslss, ll of th cidnt powr is dlivrd to th lod. Jim Stils Th Univ. of Knss Dpt. of EECS

9 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc /8 Th Signl Flow Grph of Qurtr-Wv Trnsformr A qurtr wv trnsformr cn thought of s cscdd sris of two two-port dvics, trmtd with lod R : λ 4 R S x S y Q: Two two-port dvics? It pprs to m tht qurtrwv trnsformr is not tht complx. Wht r th two two port dvics? A: Th first is connctor. Not connctor is th trfc twn on trnsmission l (chrctristic impdnc ) to scond trnsmission l (chrctristic impdnc ). Jim Stils Th Univ. of Knss Dpt. of EECS

10 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc /8 I I + + V V Port Port Rcll tht w rlir dtrmd th scttrg mtrix of this two-port dvic: S x This rsult cn mor compctly sttd s: S whr nd + + Th signl flow grph of this dvic is thrfor: x x x x Jim Stils Th Univ. of Knss Dpt. of EECS

11 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 3/8 Jim Stils Th Univ. of Knss Dpt. of EECS Now, th scond two-port dvic is qurtr wvlngth of trnsmission l. W know tht it hs th scttrg mtrix: j y j β β S Flly, lod hs scttrg mtrix of: R R + S 4 λ y j β y y y j β R

12 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 4/8 Of cours, if w connct th idl connctor to qurtr wvlngth of trnsmission l, nd trmt th whol thg with lod R, w hv formd qurtr wv mtchg ntwork! R λ 4 Th oundry conditions ssocitd with ths connctions r likwis: y x x y y y W cn thus put th signl-flow grph pics togthr to form th signl-flow grph of th qurtr wv ntwork: x x y j β y x x y j β y Jim Stils Th Univ. of Knss Dpt. of EECS

13 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 5/8 And simplifyg: x j β x j β Now, lt s s if w cn rduc this grph to dtrm: From th sris rul: x x x j β x From th splittg rul: x j β x Jim Stils Th Univ. of Knss Dpt. of EECS

14 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 6/8 From th slf-loop rul: x j β x Ag with th sris rul: x j β x And flly with th prlll rul: x x j β + So tht: x x + j β Jim Stils Th Univ. of Knss Dpt. of EECS

15 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 7/8 Q: Hy wit! If th qurtr-wv trnsformr is mtchg ntwork, shouldn t?? A: Who sys it isn t! Considr now thr importnt fcts. For qurtr wv trnsformr, w st such tht: R R Insrtg this to th scttrg prmtr S of th connctor, w fd: R R + R + R + ook t this rsult! For th qurtr-wv trnsformr, th connctor S vlu (i.., ) is th sm s th lod rflction coffict : R Fct R + Sc th connctor is losslss (unitry scttrg mtrix!), w cn conclud (nd likwis show) tht: S + S + Sc,, nd R r ll rl, th vlus nd r lso rl vlud. As rsult, nd, nd w cn likwis conclud: Jim Stils Th Univ. of Knss Dpt. of EECS

16 4//9 Th Signl Flow Grph of QurtrWv Trnsformr.doc 8/8 + Fct ikwis, th trnsmission l hs λ 4, so tht: π λ β π λ 4 whr you of cours rcll tht β π λ! Thus: jβ jπ Fct 3 As rsult: j β + And usg th nwly discovrd fct tht (for corrctly dsignd trnsformr) : And lso r rcnt discovry tht : A prfct mtch! Th qurtr-wv trnsformr dos dd work! Jim Stils Th Univ. of Knss Dpt. of EECS

17 4//9 Multipl Rflction Viwpot.doc /7 Multipl Rflction Viwpot Th qurtr-wv trnsformr rgs up n trstg qustion μ-wv ngrg. z z R R λ 4 Q: Why is thr no rflction t z? It pprs tht th l is mismtchd t oth z nd z. A: In fct thr r rflctions t ths mismtchd trfcs n fit numr of thm! W cn us our signl flow grph to dtrm th propgtion sris, onc w dtrm ll th propgtion pths through th qurtr-wv trnsformr. Jim Stils Th Univ. of Knss Dpt. of EECS

18 4//9 Multipl Rflction Viwpot.doc /7 V ( z) + jβ ( z + ) V ( z) R + jβ ( z + ) R λ 4 pn n j j Now lt s try to trprt wht physiclly hppns whn th cidnt voltg wv: j z V + ( z ) β + ( ) R R rchs th trfc t z. W fd tht thr r two forwrd pths through th qurtr-wv trnsformr signl flow grph. Jim Stils Th Univ. of Knss Dpt. of EECS

19 4//9 Multipl Rflction Viwpot.doc 3/7 Pth. At z, th chrctristic impdnc of th trnsmission l chngs from to. This mismtch crts rflctd wv, with complx mplitud p : p R R So, p. j j Pth. Howvr, portion of th cidnt wv is trnsmittd () cross th trfc t z, this wv trvls distnc of β 9 to th lod t z, whr portion of it is rflctd ( ). This wv trvls ck β 9 to th trfc t z, whr portion is g trnsmittd () cross to th trnsmission l nothr rflctd wv! R R p Jim Stils Th Univ. of Knss Dpt. of EECS

20 4//9 Multipl Rflction Viwpot.doc 4/7 So th scond dirct pth is p j9 j9 not tht trvlg β 8 hs producd mus sign th rsult. j j Pth 3. Howvr, portion of this scond wv is lso rflctd () ck to th trnsmission l t z, whr it g trvls to β 9 th lod, is prtilly rflctd ( ), trvls β 9 ck to z, nd is prtilly trnsmittd to () our third rflctd wv! R R p3 whr: p ( ) 3 j9 j9 j9 j9 ( ) Jim Stils Th Univ. of Knss Dpt. of EECS

21 4//9 Multipl Rflction Viwpot.doc 5/7 j j Not tht pth 3 is not dirct pth! Pth n. W cn s tht this ouncg ck nd forth cn go on forvr, with ch trip lunchg nw rflctd wv to th trnsmission l. Not howvr, tht th powr ssocitd with ch succssiv rflctd wv is smllr thn th prvious, nd so vntully, th powr ssocitd with th rflctd wvs will dimish to significnc! Q: But, why thn is? A: Ech rflctd wv is cohrnt wv. Tht is, thy ll oscillt t sm frquncy ω ; th rflctd wvs diffr only trms of thir mgnitud nd phs. Thrfor, to dtrm th totl rflctd wv, w must prform cohrnt summtion of ch rflctd wv this summtion of cours rsults our propgtion sris, sris tht must convrg for pssiv dvics. pn n Jim Stils Th Univ. of Knss Dpt. of EECS

22 4//9 Multipl Rflction Viwpot.doc 6/7 It cn shown tht th fit propgtion sris for this qurtr-wvlngth structur convrgs to th closd-form xprssion: pn n Thus, th put rflction coffict is: Usg our dfitions, it cn likwis shown tht th numrtor of th ov xprssion is: ( R ) ( + )( R + ) It is vidnt tht th numrtor (nd thrfor ) will zro if: R R Just s w xpctd! Physiclly, this rsults surs tht ll th rflctd wvs dd cohrntly togthr to produc zro vlu! Not ll of our trnsmission l nlysis hs n stdy-stt nlysis. W ssum our signls r susoidl, of th form xp( jω t ). Not this signl xists for ll tim t th signl is Jim Stils Th Univ. of Knss Dpt. of EECS

23 4//9 Multipl Rflction Viwpot.doc 7/7 ssumd to hv n on forvr, nd ssumd to contu on forvr. In othr words, stdy-stt nlysis, ll th multipl rflctions hv long sc occurrd, nd thus hv rchd stdy stt th rflctd wv is zro! Jim Stils Th Univ. of Knss Dpt. of EECS