Lecture 3: Phasor notation, Transfer Functions. Context

Transcription

1 EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of dffrntal quatons nto th frquncy doman, a st of algbrac quatons. In ths lctur w wll covr: Analyzng crcuts wth a snusodal nput, n th frquncy doman, a sngl frquncy at a tm How to smplfy our notaton wth Phasors Solvng a coupl of xampl crcuts How to prsnt nformaton about th crcut drctly n th frquncy doman usng dagrams of ampltud and phas at dffrnt frquncs Bod plots

2 EECS 5 Fall 3, ctur 3 { v t} av n c out t b d dt Equaton For lnar crcut, you wll b abl to wrt: v n { v n t} t b t vn t c vn t c3 vn t {}, {} d dt v n Hr rprsnt nar oprators, that s, f you apply t to a functon, you gt a nw functon t maps functons to functons, and lnar oprators also hav th proprty that: { a f t b g t} a { f t} b { g t} EECS 5 Fall 3, ctur 3 Sngl frquncy approach Anothr way to look at th stuaton s that snc th crcut s lnar, w can vw nput as th sum of sn wavs at varous ampltuds, frquncs, and phass. ach pc of th Fourr transform If w can undr th crcut for an arbtrary snusodal nput, w thn can fgur out what th crcut wll do for an arbtrary nput, or nputs. Just brak all th nputs up nto snusods, put thm through on at a tm, and thn add th rsults for ach back up at th nd, and that s your answr!

3 EECS 5 Fall 3, ctur 3 Snusodal stmulus W ar gong to analyz crcuts for a sngl snusod at a tm whch w ar gong to wrt: vn t sn t φ But w ar gong to us xponntal notaton v v v n n n t t t [ sn t φ φ φ t ] t φ C. C. t φ t t φ / / Complx conugat sam as frst trm, but wth - whrvr t occurs EECS 5 Fall 3, ctur 3 Sn n Sn at vry nod! It s spcally ntrstng bcaus voltag or currnt n our crcut, f ths s th only nput, must also b snusodal wth th sam frquncy, and so can also b wrttn n ths form. v t [ t [ I φ φ ] ] t t C. C C. C Bcaus our quatons wll b lnar, th sam thngs wll happn to th complx conugat trms as happn to th frst trms, so thy wll ust tag along 3

4 EECS 5 Fall 3, ctur 3 Dffrntatng or ntgratng Ths form s partcularly usful bcaus t s asy to dffrntat or ntgrat wth rspct to tm φ t v t [ ] C. C. v t [ v t dt [ d dt φ ] φ ] t t C. C. C. C. EECS 5 Fall 3, ctur 3 Phasor notaton As you may hav notcd, w ar gong to wrt xprssons of ths form a lot, so t s vry common to tak th followng shortcuts n notaton: Tak th constants to nclud th phas, thy wll bcom complx constants Snc all xprssons wll nclud th complx constants, stop wrtng C. C. vrywhr. Snc -t appars vrywhr, or ts C. C. n th C. C. trms stop wrtng t as wll Don t wrt th ½ thr All of ths, togthr, ar calld Phasor notaton 4

5 EECS 5 Fall 3, ctur 3 Phasors Each of th voltags btwn nods, and ach of th currnts, can thn b rprsntd by a sngl complx numbr rmmbr, ths s for a sngl frquncy nput of a partcular phas and ampltud φ t v t [ ] C. C ˆ { t [ I ˆ φ { ] t C. C Iˆ Î EECS 5 Fall 3, ctur 3 Trcky Bts: Phasor notaton s vry convnnt, but thr ar som trcky parts to look out for: You can not us phasor notaton wthout addd prcautons f you nd to multply voltags and currnts such as n a powr calculaton, bcaus that s not lnar! Anothr way to look at phasor notaton s that nstad of addng th CC and dvdng by, you tak th ral part, whch gvs th sam rsult. Howvr, you must not tak th ral part or add th complx conugat bfor you put back n th tm dpndnc -t 5

6 EECS 5 Fall 3, ctur 3 Solvng nar Systms usng Phasors Any lnar crcut bcoms a lnar quaton: { v t} { vn t} &, {} hav th form d d { v t} av t b v t b v t c v t dt c v t dt dt dt For our complx xponntal nput t ths s: t t d t d t t t a b b c c dt dt t t t t t a b b c c t t c c H a b b Whr H s ust som complx numbr at EECS 5 Fall 3, ctur 3 Notc that lnar oprators actng on all of th othr voltags or currnts ar also a complx xp tms a complx numbr: { v H t} t t a b b c c So w ar now prpard to calculat our crcuts rspons at frquncy usng algbra, nstad of dffrntal quatons! 6

7 7 EECS 5 Fall 3, ctur 3 Complx Transfr Functon Exct a systm wth an nput voltag v n Dfn th output voltag v to b nod voltag branch currnt For a complx xponntal nput, th transfr functon from nput to output or voltag or currnt can thn b wrttn: ust multply top and bottom by t suffcnt tms 3 3 d d d n n n H EECS 5 Fall 3, ctur 3 Th ampltud of th output s th magntud of th complx numbr and th phas of th output s th phas of th complx numbr: cos ] R[ H t H y H y c c b b a Hx y H t t p p

8 EECS 5 Fall 3, ctur 3 Impdanc Suppos that th nput s dfnd as th voltag of a trmnal par port and th output s dfnd as th currnt nto th port: v t t Arbtrary TI Crcut v t t t φ t I Th mpdanc Z s dfnd as th rato of th phasor voltag to phasor currnt slf transfr functon φv φ Z H I I t I t φ v EECS 5 Fall 3, ctur 3 Admtanc Suppos that th nput s dfnd as th currnt of a trmnal par port and th output s dfnd as th voltag nto th port: v t t Arbtrary TI Crcut v t t φ t t φ t I Th admttanc Z s dfnd as th rato of th phasor currnt to phasor voltag slf transfr functon I I φ φv Y H t I v 8

9 EECS 5 Fall 3, ctur 3 oltag and Currnt Gan Th voltag currnt gan s ust th voltag currnt transfr functon from on port to anothr port: If G >, th crcut has voltag currnt gan If G <, th crcut has loss or attnuaton v t t Gv I G I Arbtrary TI Crcut I I φ φ φ φ t v t EECS 5 Fall 3, ctur 3 Transmpdanc/admttanc Currnt/voltag gan ar untlss quantts Somtms w ar ntrstd n th transfr of voltag to currnt or vc vrsa v or t t Arbtrary lnar Crcut t or v t J I I K I I φ φ φ φ [ Ω] [ S] 9

10 EECS 5 Fall 3, ctur 3 Drct Calculaton of H no DEs To drctly calculat th transfr functon mpdanc, transmpdanc, tc w can gnralz th crcut analyss concpt from th ral doman to th phasor doman Wth th concpt of mpdanc admttanc, w can now drctly analyz a crcut wthout xplctly wrtng down dffrntal quatons Us K, KC, msh analyss, loop analyss, or nod analyss whr nductors and capactors ar tratd as complx rsstors EECS 5 Fall 3, ctur 3 PF Exampl: Agan! Instad of sttng up th DE n th tm-doman, lt s do t drctly n th frquncy doman Trat th capactor as an magnary rsstanc or mpdanc: tm doman ral crcut frquncy doman phasor crcut ast lctur w calculatd th mpdanc: Z R R ZC C

11 EECS 5 Fall 3, ctur 3 PF oltag Dvdr Fast way to solv problm s to say that th PF s a voltag dvdr, usng phasors: Z C H o C s ZC Z R R RC C EECS 5 Fall 3, ctur 3 Bggr Exampl no problm! Consdr a mor complcatd xampl: Z ff H o s Z ff H R ZC Z C ZC R Z C Z ff Z C R R Z C

12 EECS 5 Fall 3, ctur 3 Buldng Tnts: Pols and Zros For most crcuts that w ll dal wth, th transfr functon can b shown to b a ratonal functon Th bhavor of th crcut can b xtractd by fndng th roots of th numrator and dnomnator Or anothr form DC gan xplct H G H z H p K n n d n3 d d 3 z p z z p p G z p K z, p, EECS 5 Fall 3, ctur 3 Pols and Zros cont Th roots of th numrator ar calld th zros snc at ths frquncs, th transfr functon s zro pols Th roots of th dnomnator ar calld th pols, snc at ths frquncs th transfr functon paks lk a pol n a tnt z H p z p

13 EECS 5 Fall 3, ctur 3 Fndng th Magntud quckly Th magntud of th rspons can b calculatd quckly by usng th proprty of th mag oprator: H G K G K z z z p p z p p Th magntud at DC dpnds on G and th numbr of pols/zros at DC. If K >, gan s zro. If K <, DC gan s nfnt. Othrws f K, thn gan s smply G EECS 5 Fall 3, ctur 3 Fndng th Phas quckly As provd n HW #, th phas can b computd quckly wth th followng formula: K z z p H p G K p G p p z p z p p p Now th scond trm s smpl to calculat for postv frquncs: π p K K Intrprt ths as sayng that multplcaton by s quvalnt to rotaton by 9 dgrs p p p 3

14 EECS 5 Fall 3, ctur 3 Bod Plots Smply th log-log plot of th magntud and phas rspons of a crcut mpdanc, transmpdanc, gan, Gvs nsght nto th bhavor of a crcut as a functon of frquncy Th log xpands th scal so that brakponts n th transfr functon ar clarly dlnatd EECS 5 Fall 3, ctur 3 og ratos and dfnton of Th frquncy rspons can vary by ordrs of magntud rapdly W can xpand rang by takng th log of th magntud rspons dcbl dc - - log s s

15 EECS 5 Fall 3, ctur 3 Why? Powr! Why multply log by rathr than? Powr s proportonal to voltag squard to mak ratos n powr and voltag com out th sam, for powr ratos us, for voltag ratos, us log log s s EECS 5 Fall 3, ctur 3 Exampl: Hgh-Pass Fltr Usng th voltag dvdr rul: H R R R H H H H τ 5

16 EECS 5 Fall 3, ctur 3 HPF Magntud Bod Plot Rcall that log of product s th sum of log H τ 4 Incras by /dcad Equals unty at brakpont. - EECS 5 Fall 3, ctur 3 HPF Bod dsscton Th scond trm can b furthr dssctd: / τ / τ / τ << τ >> τ τ - /dc ~ -3 τ - 3 6

17 EECS 5 Fall 3, ctur 3 slop At brakpont: / τ s 3 Obsrv: slop of sgnal attnuaton s /dcad n frquncy / τ s / τ s 4 6 EECS 5 Fall 3, ctur 3 Compost Plot Compost s smply th sum of ach componnt: Hgh frquncy ~ Gan ow frquncy attnuaton./ τ / τ / τ