EECS 5 SPRING 24, ctu ctu 2: Fquncy domain analyi, Phao EECS 5 Fall 24, ctu 2 Announcmnt Th cou wb it i http://int.c.bkly.du/~5 Today dicuion ction will mt Th Wdnday dicuion ction will mo to Tuday, 5:-6:, 293 Coy You can go to any o all dicuion you lik. ab will tat Fb 3. Rading aignmnt fom th txt will tat nxt wk Th i a homwok t du Wd. /28
EECS 5 Fall 24, ctu 2 Contxt In th lat lctu w cod: how cicuit can b modld a lina cicuit by dign o appoximation. How to cont a lina cicuit into a t of diffntial quation. In thi lctu, w will co: How to u complx analyi to ol cicuit by conting th diffntial quation in th tim domain into algbaic quation in th fquncy domain. EECS 5 Fall 24, ctu 2 ina Cicuit modl t of lina diffntial quation ito i dc ic C di Th wi cony th aiabl (oltag and cunt) btwn th quation (componnt), by applying Kichoff law. Fo th low pa xampl: i out in in i i c + c inducto c + i out capacito 2
EECS 5 Fall 24, ctu 2 Diffntial quation fo low pa: W a going to tak th output cunt qual to zo, fo implicity, o: out in dc ir icr RC c + out out dout + RC EECS 5 Fall 24, ctu 2 { } a in + c out 2 + b d Equation Fo any lina cicuit, you will b abl to wit: in { in } + b + in( t) + c2 in( t) + c3 in( t) + {}, {} 2 2 d 2 2 in H pnt ina opato, that i, if you apply it to a function, you gt a nw function (it map function to function), and lina opato alo ha th popty that: { a f + b g( t)} a { f } + b { g( t)} 3
EECS 5 Fall 24, ctu 2 It now jut mathmatic, and thfo ay! Onc w tablih a lina modl fo a cicuit, by dign o appoximation: W can dictly u th powful mthod of lina analyi fom mathmatic. W can dlop ou intuition a to what will happn, allowing u to dign. EECS 5 Fall 24, ctu 2 Foui Tanfom On impotant lina analyi tchniqu w will u i th Foui Tanfom: Th Foui tanfom tat: f ( t F( ω ) ) j ω ω t f F( ) dω Notic that what thi ay i that infomation that i xpd a a function of tim (oltag o cunt fo xampl) can b compltly xpd a a function of fquncy: F(ω) 4
EECS 5 Fall 24, ctu 2 inaity t u function of fquncy F(ω) (oltag and cunt) ath than function of tim a ou nw aiabl. Th Foui lationhip how u that if w can find th function of fquncy, w can thn cont thm into th oltag and cunt a a function of tim that w want. W can do thi in y quation fo ou lina componnt. j ω ω t f F( ) dω W jut ubtitut thi fom into ach of ou quation EECS 5 Fall 24, ctu 2 Rito: i dω i dω 5
EECS 5 Fall 24, ctu 2 Capacito: dc ic C d ic dω C c dω ic dω C( jω) c dω EECS 5 Fall 24, ctu 2 Inducto: di( t) d dω i dω dω ( jω) i dω 6
EECS 5 Fall 24, ctu 2 In ach of th, w can liminat th intgation o fquncy, and th contant, to gt: i i C( jω) c c c ic C( jω) i ( jω) EECS 5 Fall 24, ctu 2 Conion of lina cicuit to algbaic quation Somthing wondful jut happnd: ach of ou imultanou lina diffntial quation w jut contd to algbaic quation (jut multiplication by a contant fo th xampl), and th am thing happn to y lina cicuit. Of cou, th am thing happn to th lationhip did fom Kichoff law (thy a lina too; adding oltag, fo xampl) 7
EECS 5 Fall 24, ctu 2 Th adantag of changing diffntial quation into algbaic quation com at a mall pic: th contant that w a multiplying by, and th function of fquncy fo both oltag and cunt, a now complx numb. No matt how complicatd th cicuit, if w di th cicuit with a al function in (t) whn w find th output by uing th in Foui tanfom, it will b al a wll, a wll a any oltag o cunt at any nod, at all tim. EECS 5 Fall 24, ctu 2 Complx numb It i impotant to think of complx numb a jut an xpanion o th dfinition of al numb. Fo xampl, if A, B, and C a complx: A( B + C) AB + AC AB BA A + B B + A Thi m tiial, but th i only on* oth dfinition fo numb which oby th popti**: *Th oth on i quatnion **finit, but not countabl. 8
EECS 5 Fall 24, ctu 2 If you ha a calculato which can handl complx numb you can jut plug thm in. Othwi, you can u th did ul to gt ult fo complx numb fom th ul fo al numb* A a + a j B b + b j i i A + B ( a + b ) + ( a + b ) j i i AB ( a + ai j)( b + bi j) ab aibi + a bj j + aib j j i ud by lctical ngin fo th imaginay contant to aoid confuion with i fo cunt which got th fit. *jut a w handl al by appoximating with intg calculation and kping tack of dcimal point EECS 5 Fall 24, ctu 2 Notic that unlik th al, th i a complx numb that, whn multiplid by itlf, gi ngati on: A + j AA + + Almot all function ha xtnion o th complx numb, and in om way complx numb m mo complt in that th in of function xit, oot can alway b found, tc. 9
EECS 5 Fall 24, ctu 2 http://mathwold.wolfam.com/ A complx function i aid to b analytic on a gion R if it i complx diffntiabl at y point in R. Th tm holomophic function, diffntial function, complx diffntiabl function, and gula function a omtim ud intchangably with "analytic function" If a function i analytic in a gion R, it i infinitly diffntiabl in R. A function may fail to b analytic at on o mo point though th pnc of ingulaiti, o along lin o lin gmnt though th pnc of banch cut. A ingl-alud function that i analytic in all but poibly a dict ubt of it domain, and at tho ingulaiti go to infinity lik a polynomial (i.., th xcptional point mut b pol and not ntial ingulaiti), i calld a momophic function. EECS 5 Fall 24, ctu 2 Why intoduc complx numb? Thy actually mak thing ai On inightful diation of ix Conid a cond od homognou DE '' y + y in x y co x Sinc in and coin a linaly indpndnt, any olution i a lina combination of th fundamntal olution
EECS 5 Fall 24, ctu 2 Inight into Complx Exponntial jx But not that i alo a olution! That man: jx a in x + a2 co x jx a in x + a2 co x jx jx 2a in x But a numb minu it complx conjugat gi an imaginay numb o i pu imaginay. Diffntiating: a j jx + j jx 2a co x And a 2 i al, o: a ja 2 jx a( j in x + co x) And inc j a( j in + co) a EECS 5 Fall 24, ctu 2 Th Rotating Complx Exponntial So th complx xponntial i nothing but a point tacing out a unit cicl on th complx plan: ix co x + i in x i t ω + iωt iωt 2 iωt
EECS 5 Fall 24, ctu 2 Magic: Tun Diff Eq into Algbaic Eq Intgation and diffntiation a tiial with complx numb: d iωt iω iωt Any ODE i now tiial algbaic manipulation in fact, w ll how that you don t n nd to dictly di th ODE by uing phao Th ky i to ob that th cunt/oltag lation fo any lmnt can b did fo complx xponntial xcitation iωτ dτ iω iωt EECS 5 Fall 24, ctu 2 Complx Exponntial i Powful To find tady tat pon w can xcit th ytm with a complx xponntial i t TI Sytm ω i( ωt+ φ ) H H At any fquncy, th ytm pon i chaactizd by a ingl complx numb H: Thi i not upiing inc a inuoid i a um of complx xponntial (and bcau of linaity!) i inωt H (ω) φ p H (ω) ωt 2i iωt i coωt + 2 Fom thi ppcti, th complx xponntial i n mo fundamntal ωt Mag Rpon iωt Pha Rpon 2
EECS 5 Fall 24, ctu 2 Rmmb ou low pa filt? in C (t) + _ i c (t) out EECS 5 Fall 24, ctu 2 PF Exampl: fquncy domain t look at a ingl fquncy fom th ouc: d ( t) + RC V o V j( ωt+ φ ) V al complx V V + RC jω V ( + ω τ ) V V j V V ( + jω τ ) 3
EECS 5 Fall 24, ctu 2 Magnitud and Pha Rpon Th ytm i chaactizd by th complx function V H V ( + jω τ ) Th magnitud and pha pon: H V V + ( ωτ ) 2 p H tan ωτ EECS 5 Fall 24, ctu 2 Why did it wok? Th ytm i lina: R[ y] (R[ x]) R[ ( x)] If w xcit ytm with a inuoid: V coω t V R[ If w puh th complx xp though th ytm fit and tak th al pat of th output, thn that th al inuoidal pon ] V o o co( ωt + φ) V o R[ j( ωt+ φ ) ] 4