Review Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )

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1 Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 )

2 Volag/currn-division

3 Lcur 7 AC Circuis Sinusoids and Phasors

4 Conns Sinusoids Apliud, angular/cyclic frquncy, argun, priodic funcion Phasors Rcangular/polar/xponnial for, Eulr s idniy, and i-doain/phasor-doain rprsnaion Phasor rlaionships for circui lns R, L, and C Ipdanc and adianc rsisanc and racanc,conducanc and suscpanc Kirchhoff s laws in h frquncy doain Ipdanc cobinaions

5 Priodic funcions f ( x nt) f ( x) Sinusoids Sinusoidal: or in funcion Ti-varying () and argun, x, of (x) Apliud V Frquncy: angular (rad/s) and cyclic f (H): Priod T (s): T nt nt ( ) us in wih posiiv apliud V ( ) ( ) T f ( x n ) f ( ) ( ) ( x) v( ) V ( ) 0 T phas argun

6 Phas angl bwn wo signals Sa frquncy: Apliud ay vary Phas diffrnc: Exprss h in h sa for in and V > 0 Phas diffrnc : ou of/in phas lad/lag 5(x+ ) In phas : (x + ) 0 Phas v V ( ) v V ( ) a any i (x+/3) For a givn, V and ar iporan quaniis. lads (x) by /3 (x-/3) lags (x) by /3 Exapl: Calcula h phas angl bwn v = 0 (ω + /3) and v = (ω + /6). Sa which usoid is lading. () Sa for: () Copar: v 0 ( ) 0 ( ) 0 ( ) v ( ) 0 ( ) 0 ( ) v 0 ( ) 0 ( ) v lags v by /3

7 Phasors A phasor is a coplx nubr ha rprsns h apliud (V ) and phas () of a usoid. Coplx nubrs - hr rprsnaions Rcangular: Polar: Exponnial: Phasor rprsnaion - usoid v() ral par: v( ) V R V R V x y r r Rcangular Èxponnial V V V V V ω is iplicily prsn

8 On circl,, Priod T = i/circl = / Sinor: roaing phasor A circl of radius V Sinor: V on h coplx plan v() is h procion of h or on h ral axis. Th valu of h or a i = 0 is h phasor V of v(). Procion A coplx nubr: agniud and dircion - vcor counrclockwis

9 Phasor diagra Diagra/coplx nubr Magniud and phas V V ω is iplicily prsn

10 Ti doain and phasor(frquncy) doain To rprsn signals v(), i(): Ti doain: v ) V ( Ti dpndn Always ral Phasor (frqncy) doain: V V Ti indpndn Gnrally coplx is consan. Circui rspons dpnds on ( ) Exapl: Givn i () = 4 (ω + /6) and i () = 5 (ω /3), find hir su. i ( I ) I, i( ) I i I ( 6 3 4, I 5 ) 5( I I 4( ) 6? ) 5 5( ) 6 5 5( 6 in and I > 0 i( ) I ( ) 5 ) 6

11 Drivaiv and ingral in phasor doain v( ) V ( ) V V dv( ) d ( ) V V ( ) d d V ( ) V ( ) 3 V ( ) V ( ) V V V V Exapl: Ug h phasor approach, drin h currn i() in a circui dscribd by h ingrodiffrnial quaion. = di 4i 8 id 3 50 ( ) d 3 4I Frquncy doain 8I 3 I 50 3 dv ( ) d v() V V Usful in finding h sady-sa soluion: sa frquncy!

12 Phasor rlaionships for circui lns v and i in phas i lads v by / i( ) I ( ) I v( ) RI ( ) V RI RI Oh law holds v( ) V ( ) dv( ) i( ) C CV ( ) d CV ( ) V V I V CV I C i has a phas +/ CV v lads i by / i( ) I ( ) di( ) v( ) L LI ( ) d LI ( ) I V LI LI v has a phas +/

13 Ipdanc and adianc opposiion o h flow of usoidal currn V RI V LI V ZI Z V V I I C Ipdanc Phasor Volag Phasor Currn Coplx quaniy, bu no a phasor Z C Adianc I Y Z V S

14 Mor abou Ipdanc 0 Capacior, Inducor: Capacior, Inducor: Z C Z L Z C Z 0, L, an opn circui a shor circui 0,, a shor circui an opn circui Circui rspons dpnds on h frquncy! Z is a coplx quaniy Rcangular for (x+y) Rsisanc Z R X Y G Z Conducanc Racanc Currn lads volag X 0, capaciiv/lading racanc...( Z ) C X 0, induciv/ lagging racanc...( Z L) B Suscpanc No: G dos no always qual o /R

15 Kirchhoff s laws in h frquncy doain Boh KVL and KCL hold in h frquncy doain Ti doain v v... vn 0 i v R( V ) i i n R( V ) R( V )... R( Vn ) n R( V V... Vn ) n ( V V... V ) 0 R ( ) Vi ( i ) n v i Us h R of a coplx quaniy o rprsn h signal 0 0 Frquncy doain V n V... Vn 0 0 KVL KCL V V... V I I... In n 0 0

16 Ipdanc cobinaions Cobinaion of Ipdanc is siilar o rsisanc circuis. KVL V V Z Z q V I V... Vn Z I Z q Z I... Z n Z... Zn I Z Z... Z n I Volag/currn division holds. Y- and -Y ransforaions as wll

17 Exapl: Ipdanc Find h inpu ipdanc of h circui Assu ha h circui opras a ω = 50 rad/s. Z = Ipdanc of h -F capacior Z = Ipdanc of h 3- rsisor in sris wih h 0-F capacior Z 3 = Ipdanc of h 0.-H inducor in sris wih h 8-rsisor Z C Z L

18 Exapl: Solv h circui Drin v o () in h circui Sp : Phasor-doain quivaln Sp : Volag-division Z Z Sp 3: convr o i-doain

19 Appndix: Mahaics Trigonory A B B A C C B A B A B A B A B A B A B A an, whr, Coplx nubrs, * r y x r r r r r r y y x x y y x x

20 Lcur 8 AC Circuis Sinusoidal Sady-Sa Analysis