s-domain Circuit Analysis

Transcription

1 Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preerved c decribed by ODE and heir I Order equal number of plu number of Elemenbyelemen and ource ranformaion Nodal or meh analyi for domain cc variable Why? Soluion via Invere aplace Tranform Eaier han ODE Eaier o perform engineering deign Frequency repone idea filering 3

2 Elemen Tranformaion olage ource Time domain v v S i depend on cc Tranform domain S v S v S _ i I i depend on cc urren ource I i S v depend on cc i S v 33

3 34 Elemen Tranformaion cond onrolled ource Shor cc, open cc, OpAmp relaion Source and acive device behave idenically onrain expreed beween ranformed variable Thi all hinge on uniquene of aplace Tranform and lineariy g I gv i ri ri v I I i i v v!!!! " " µ µ v v I i v P N P N O O S S!!!

4 35 Elemen Tranformaion cond eior apacior Inducor G I I Gv i i v i v i I i I i d v i d di v ! " # # Y Z Y Z Y Z v i v i v I v I v d i v d dv i !! " # #

5 eior apacior i v Elemen Tranformaion cond i v I I v 0 I I v _ 0 I v 0 I v 0 Noe he ource ranformaion rule apply! 36

6 Elemen Tranformaion cond Inducor I i0 I i0 i v I _ i 0 I i 0 37

7 Example 0, T&, 5h ed, p 456 cc behavior A Swich in place ince, cloed a 0. Solve for v. 0 I v I v 0 38

8 Example 0 T&, 5h ed, p 456 cc behavior A Swich in place ince, cloed a 0. Solve for v. 0 I v Iniial condiion v 0 A domain oluion uing nodal analyi I I domain oluion via invere aplace ranform A! v e u c A I v 0 39

9 Solve for i A u _ Example 0 T&, 5h ed, p 457 i A _ I _ i 0 40

10 Solve for i A u _ Example 0 T&, 5h ed, p 457 K around loop Solve i I A A _ I _ i 0 A! I i0 0 i 0 A # i 0 " A & % $ ' Inver # i A " A e" i 0e " & % u Amp $ ' 4

11 Impedance and Admiance Impedance i he domain proporionaliy facor relaing he ranform of he volage acro a woerminal elemen o he ranform of he curren hrough he elemen wih all iniial condiion zero Admiance i he domain proporionaliy facor relaing he ranform of he curren hrough a woerminal elemen o he ranform of he volage acro he elemen wih iniial condiion zero Impedance i like reiance Admiance i like conducance 4

12 Baic rule ircui Analyi in Domain The equivalen impedance Z eq of wo impedance Z and Z in erie i Z eq Z Z Same curren flow Z I Z I Zeq I Z The equivalen admiance Y eq of wo admiance Y and Y in parallel i Y eq Y Y I Same volage Y Y Yeq I Z I Y Y 43

13 Example 03 T&, 5h ed, p 46 Find Z AB and hen find by volage diviion v A B _ v A B _ 44

14 Example 03 T&, 5h ed, p 46 Find Z AB and hen find by volage diviion v A B _ Z eq v A _ B Ω & Z # & # $! Z $! $ % eq!" % " 45

15 Superpoiion in domain cc The domain repone of a cc can be found a he um of wo repone. The zeroinpu repone caued by iniial condiion ource, wih all exernal inpu urned off. The zeroae repone caued by he exernal ource, wih iniial condiion ource e o zero ineariy and uperpoiion Anoher ubdiviion of repone. Naural repone he general oluion epone repreening he naural mode pole of he cc. Forced repone he paricular oluion epone conaining mode due o he inpu 46

16 Example 06, T&, 5h ed, p 466 The wich ha been open for a long ime and i cloed a 0. Find he zeroae and zeroinpu componen of Find v for I A ma, H,.5KΩ, /6 µf 0 I A v I A I A 47

17 Example 06, T&, 5h ed, p 466 The wich ha been open for a long ime and i cloed a 0. Find he zeroae and zeroinpu componen of Find v for I A ma, H,.5KΩ, /6 µf I A 0 v Z eq I A z zi Z Z eq eq I A I A I A I A I A 48

18 Example 06 cond z zi Z Z eq eq I A I A I A I A I A I A Subiue value 6000 z " v z 3e "000 " 3e "3000 [ ] u.5 zi " v zi "0.75e "000.5e "3000 [ ] u 49

19 Example Formulae node volage equaion in domain v _ 3 v x µv x v 50

20 Example Formulae node volage equaion in domain v _ 3 v x µv x v A B D _ 3 x v 0 v 0 µ x 5

21 Example cond _ 3 x Node A: Node D: A v 0 A B D µ x v 0 D µ x µ Node B: B " A B " D B B " " v 0 " v 0 0 Node : " B [ G 3 ] " v 0 5

22 Example Find v O when v S i a uni ep u and v 00 A B D v S _ v O onver o domain 53

23 Example Find v O when v S i a uni ep u and v 00 A B D v S _ v O onver o domain A B D S _ O v 0 54

24 Example A B D O S _ Node A: v 0 A S Node D: D O Node : 0 Node B: G B! G S v 0 Node K:! B! GO! v 0 Solve for O # G & # % G O " % G " & % S % $ % ' $ S ' Nodal Analyi Inver T # " & % % " $ ' v e O u 55

25 Feaure of domain cc analyi The repone ranform of a finiedimenional, lumpedparameer linear cc wih inpu being a um of exponenial i a raional funcion and i invere aplace Tranform i a um of exponenial The exponenial mode are given by he pole of he repone ranform Becaue he repone i real, he pole are eiher real or occur in complex conjugae pair The naural mode are he zero of he cc deerminan and lead o he naural repone The forced pole are he pole of he inpu ranform and lead o he forced repone 56

26 Feaure of domain cc analyi A cc i able if all of i pole are locaed in he open lef half of he complex plane A key propery of a yem Sabiliy: he naural repone die away a Bounded inpu yield bounded oupu A cc compoed of, and will be a wor marginally able Wih in he righ place i will be able Z and Y boh have no pole in e>0 Impedance/admiance of cc are Poiive eal or energy diipaing 57