EE5 Fall 5 Mrolron Dvs and Crus Prof. Mng C. Wu wu@s.rkl.du 5 Suarda Da all SD - LTI: Lnar Tm-Invaran Ssm Ssm s lnar sudd horoughl n 6AB: Ssm s m nvaran: Thr s no lok or m rfrn Th ransfr funon s no a funon of m I dos no mar whn ou appl h npu. Th ransfr funon s gong o h sam -
Lnar Ssms Connuous m lnar ssms hav a lo n ommon wh fn dmnsonal lnar ssms w sudd n 6AB: Lnar: Bass ors à ass funons: Suprposon: Mar Rprsnaon à Ingral rprsnaon: -3 Lnar Ssms on Egnvors à gnfunons Orhonormal ass Egnfunon panson Opraors ang on gnfunon panson -4
LTI Ssms Sn mos prod non-prod sgnals an domposd no a summaon ngraon of snusods va Fourr Srs Transform, h rspons of a LTI ssm o vruall an npu s hararzd h frqun rspons of h ssm: -5 Eampl: Low Pass Flr LPF Inpu sgnal: W know ha: v os s s vo K os f s Phas shf Amp sal v v R s dv C d dv v vs RC d dv vs v d -6 3
Compl Eponnal Iz z θ φ z φ φ z z z z m φ Rz -7 Th Roang Compl Eponnal So h ompl ponnal s nohng u a pon rang ou a un rl on h ompl plan: os sn -8 4
Mag: Turn Dff Eq no Algra Eq Ingraon and dffrnaon ar rval wh ompl numrs: d d ò d An ODE s now rval algra manpulaons n fa, w ll show ha ou don vn nd o drl drv h ODE usng phasors Th k s o osrv ha h urrn/volag rlaon for an lmn an drvd for ompl ponnal aon -9 Unvrs of Calforna, Compl Eponnal s Powrful To fnd sad sa rspons w an h ssm wh a ompl ponnal Mag Rspons LTI Ssm f Phas Rspons A an frqun, h ssm rspons s hararzd a sngl ompl numr : f Ths s no surprsng sn a snusod s a sum of ompl ponnals and aus of lnar! sn os From hs prspv, h ompl ponnal s vn mor fundamnal - 5
Solvng LPF wh Phasors L s h ssm wh a ompl p: dv vs v d v s o s v f us o avod onfuson ral ompl s s - Eas!!! s Magnud and Phas Rspons Th ssm s hararzd h ompl funon s Th magnud and phas rspons mah our prvous alulaon: s an - 6
7-3 Wh dd work? Agan, h ssm s lnar: To fnd h rspons o a snusod, w an fnd h rspons o ωωωω and ωωωω and sum h rsuls: L L L LTI Ssm f LTI Ssm f LTI Ssm -4 on. Sn h npu s ral, h oupu has o ral: Tha mans h sond rm s h onuga of h frs: Thrfor h oupu s: odd funon vn funon f os f f f
8-5 Proof for Lnar Ssms For an arrar lnar ru L,C,R,M, and dpndn sours, dompos no lnar su-opraors, lk mulplaon onsans, m drvavs, or ngrals: For a ompl ponnal npu hs smplfs o: ò ò ò ò ò ò d d d d a L ò ò ò d d d d a L a ø ö ç è æ a -6 Proof on. No ha h oupu s also a ompl p ms a ompl numr: Th amplud of h oupu s h magnud of h ompl numr and h phas of h oupu s h phas of h ompl numr ø ö ç è æ a os ] R[ a ø ö ç è æ
Phasors Wh our nw onfdn n ompl numrs, w go full sam ahad and work drl wh hm w an vn drop h m faor ωωωω sn wll anl ou of h quaons. E ssm wh a phasor: Rspons wll also phasor: ~ f ~ f For hos wh a Lnar Ssm akground, w r gong o work n h frqun doman Ths s h Lapla doman wh s -7 Capaor I- Phasor Rlaon Fnd h Phasor rlaon for urrn and volag n a ap: dvc I ω C v C d v ω I ω C d d [ ω ] d C d ω ω C ω I ω ω C ω I ω C _ -8 9
Induor I- Phasor Rlaon l Fnd h Phasor rlaon for urrn and volag n an nduor: d v L d ω L d d [I ω ] LI d d ω ω LI ω ω ω LI ω ω L I I ω v ω v _ -9 Impd h Currns! Suppos ha h npu s dfnd as h volag of a rmnal par por and h oupu s dfnd as h urrn no h por: v Arrar LTI Cru v f I Th mpdan Z s dfnd as h rao of h phasor volag o phasor urrn slf ransfr funon f v f Z I I I f v -
Adm h Currns! Suppos ha h npu s dfnd as h urrn of a rmnal par por and h oupu s dfnd as h volag no h por: v Arrar LTI Cru v f I Th adman Y s dfnd as h rao of h phasor urrn o phasor volag slf ransfr funon I I f f v Y I f v - olag and Currn Gan Th volag urrn gan s us h volag urrn ransfr funon from on por o anohr por: v Gv Arrar LTI Cru f f I I f f G I I If GG >, h ru has volag urrn gan If GG <, h ru has loss or anuaon v -
Transmpdan/adman Currn/volag gan ar un-lss quans Somms w ar nrsd n h ransfr of volag o urrn or v vrsa v Arrar LTI Cru v J I I K I I f f f f [ W ] [ S] -3 Dr Calulaon of no DEs To drl alula h ransfr funon mpdan, rans-mpdan, w an gnralz h ru analss onp from h ral doman o h phasor doman Wh h onp of mpdan adman, w an now drl analz a ru whou pll wrng down an dffrnal quaons Us KL, KCL, msh analss, loop analss, or nod analss whr nduors and apaors ar rad as ompl rssors -4
LPF Eampl: Agan! Insad of sng up h DE n h m-doman, l s do drl n h frqun doman Tra h apaor as an magnar rssan or mpdan: m doman ral ru frqun doman phasor ru W know h mpdans: Z R R -5 Z C C Bod Plos Smpl h log-log plo of h magnud and phas rspons of a ru mpdan, ransmpdan, gan, Gvs nsgh no h havor of a ru as a funon of frqun Th log pands h sal so ha rakpons n h ransfr funon ar larl dlnad In EECS 4, Bod plos ar usd o ompnsa rus n fdak loops -6 3
Frqun Rspons of Low-Pass Flrs ωc Tω R ωrc ω /ω ωc ω RC Tω ω /ω Tω an ω /ω ω 3 ω [rad/s] f 3 ω π [z] -7 Frqun Rspons of gh-pass Flrs R Tω R ωc ω RC Tω ω /ω Tω an ω /ω ω 3 ω [rad/s] ωrc ω /ω f 3 ω π [z] -8 4
Eampl: gh-pass Flr Usng h volag dvdr rul: L L R R L L R -9 PF Magnud Bod Plo Rall ha log of produ s h sum of log Þ 4 Inras /dad Equals un a rakpon. - -3 5
PF Bod Plo dsson Th sond rm an furhr dssd:./ / / - -4-6 << >> - /d ~ -3-3 - 3 Compos Plo Compos s smpl h sum of ah omponn: gh frqun ~ Gan Low frqun anuaon - -4./ / / -3 6
Approma vrsus Aual Plo Approma urv aura awa from rakpon A rakpon hr s a 3 rror -33 PF Phas Plo Phas an naurall domposd as wll: p an Frs rm s smpl a onsan phas of 9 dgrs Th sond rm s h aran funon Esma aran funon: << 45-34 Aual urv >> 7
Powr Flow Th nsananous powr flow no an lmn s h produ of h volag and urrn: P v For a prod aon, h avrag powr s: Pav ò v d T In rms of snusods w hav P av T I osω ϕ osω ϕ v dτ I osω osϕ snω snϕ osω osϕ v snω snϕ v dτ T I dτ os ω osϕ osϕ v sn ω snϕ snϕ v snω osω I T osϕ osϕ v snϕ snϕ v I osϕ ϕ v -35 Powr Flow wh Phasors P av I os f f v Powr Faor No ha f φφ φφ vv ππ, hn PP aaaa -36 II ππ Imporan: Powr s a non-lnar funon so w an smpl ak h ral par of h produ of h phasors: P ¹ R[ I ] From our prvous alulaon: I P os f f v R[ I * ] R[ I * ] 8
Mor Powr o You! In rms of h ru mpdan w hav: P R[ I * ] Z R[ Z R[ Z * ] Z * ] R[ Z * R[ Z ] Z ] R[ Z] Chk h rsul for a ral mpdan rssor Also, n rms of urrn: * * P R[ I ] R[ I I Z] I R[ Z] -37 Summar Compl ponnals ar gn-funons of LTI ssms Sad-sa rspons of LCR rus ar LTI ssms Phasor analss allows us o ra all LCR rus as smpl rssv rus usng h onp of mpdan adman Frqun rspons allows us o ompll hararz a ssm An npu an domposd no hr a onnuum or dsr sum of frqun omponns Th ransfr funon s usuall plod n h log-log doman Bod plo magnud and phas Loaon of pols/zros s k -38 9