s-domain Circuit Analysis

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1 Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preered c decribed by ODE and heir Order equal number of plu number of Elemenbyelemen and ource ranformaion Nodal or meh analyi for domain cc ariable Why? Soluion ia nere aplace Tranform Eaier han ODE Eaier o perform engineering deign Frequency repone idea filering MAE40 inear ircui 3

2 Elemen Tranformaion olage ource Time domain S i depend on cc Tranform domain S S i depend on cc S _ i urren ource i S depend on cc i S MAE40 inear ircui 33

3 MAE40 inear ircui 34 Elemen Tranformaion cond onrolled ource Shor cc, open cc, OpAmp relaion Source and acie deice behae idenically onrain expreed beween ranformed ariable Thi all hinge on uniquene of aplace Tranform and lineariy g g i r ri i i " " µ µ i P N P N O O S S

4 MAE40 inear ircui 35 Elemen Tranformaion cond eior apacior nducor G G i i i i i i d i d di " # # Y Z Y Z Y Z i i d i d d i " # #

5 eior apacior i Elemen Tranformaion cond i 0 _ Noe he ource ranformaion rule apply MAE40 inear ircui 36

6 Elemen Tranformaion cond nducor i0 i0 i _ i 0 i 0 MAE40 inear ircui 37

7 cc behaior A Example 0 T& p 456 Swich in place ince, cloed a 0. Sole for. 0 niial condiion 0 A domain oluion uing nodal analyi domain oluion ia inere aplace ranform A e u c A 0 MAE40 inear ircui 38

8 Sole for i A u _ K around loop Sole i Example 0 T& p 457 A A i 0 A i0 0 i 0 A # i 0 " A & % $ ' ner # i A " A e" i 0e " & % u Amp $ ' MAE40 inear ircui 39

9 mpedance and Admiance mpedance i he domain proporionaliy facor relaing he ranform of he olage 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 olage acro he elemen wih iniial condiion zero mpedance i like reiance Admiance i like conducance MAE40 inear ircui 40

10 Baic rule ircui Analyi in Domain The equialen impedance Z eq of wo impedance Z and Z in erie i Z eq Z Z Same curren flow Z Z Zeq Z The equialen admiance Y eq of wo admiance Y and Y in parallel i Y eq Y Y Same olage Y Y Yeq Z Y Y MAE40 inear ircui 4

11 Example 03 T& p 46 Find Z AB and hen find by olage diiion A B _ Z eq A _ B Ω & Z # & # $ Z $ $ % eq " % " MAE40 inear ircui 4

12 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 ubdiiion 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 MAE40 inear ircui 43

13 Example 06 T& p 466 The wich ha been open for a long ime and i cloed a 0. Find he zeroae and zeroinpu componen of Find for A ma, H,.5KΩ, /6 µf A 0 MAE40 inear ircui Z eq A z zi Z Z eq eq A A A A A 44

14 Example 06 cond z zi Z Z eq eq A A A A A A Subiue alue 6000 z " z 3e "000 " 3e "3000 [ ] u.5 zi " zi "0.75e "000.5e "3000 [ ] u MAE40 inear ircui 45

15 Example 0 T& no in 4h ed Formulae node olage equaion in domain _ 3 x µ x MAE40 inear ircui A B D _ 3 x 0 0 µ x 46

16 Example 0 cond _ 3 x Node A: Node D: A 0 A B D µ x 0 D µ x µ Node B: B " A B " D B B " " 0 " 0 0 Node : MAE40 inear ircui " B [ G 3 ] " 0 47

17 Example 06 T& no in 4h ed Find O when S i a uni ep u and 00 A B D S _ O oner o domain A B D S _ O 0 MAE40 inear ircui 48

18 A B D O S _ Node A: 0 A S Node D: D O Node : 0 Node B: G B G S 0 Node K: B GO 0 Sole for O G % % & G # O " S & # & " S G # & # & ' # $ ' $ Example 06 cond Nodal Analyi ner T MAE40 inear ircui "& & ' % " # # $ e O u 49

19 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 inere aplace Tranform i a um of exponenial The exponenial mode are gien 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 MAE40 inear ircui 50

20 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 hae no pole in e>0 mpedance/admiance of cc are Poiie eal or energy diipaing MAE40 inear ircui 5

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