# s-domain Circuit Analysis

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1 S-Doman naly

2 -Doman rcut naly Tme doman t doman near rcut aplace Tranform omplex frequency doman doman Tranformed rcut Dfferental equaton lacal technque epone waveform aplace Tranform nvere Tranform - lgebrac equaton lgebrac technque epone tranform

3 Krchhoff aw n -Doman t doman Krchhoff current law K doman t t 3 t 4 t t t t t v t v t 4 Krchhoff voltage law K v t v 3 t v 5 t v t v t v t 3 3

4 Sgnal Source n Doman t doman oltage Source: v t vs t t depend on crcut t vt v S t S doman oltage Source: S depend on crcut urrent Source: t S t v t depend on crcut t v t t S S urrent Source: S depend on crcut

5 Tme Doman Tme and -Doman Element Model mpedance and oltage Source for ntal ondton t -Doman etor: v t t v t etor: nductor: v t d t dt v t t nductor: apactor: v t t d τ τ v v t t v apactor: v

6 mpedance and oltage Source for ntal ondton mpedance wth wth v voltage tranform current tranform wth all ntal condton et to zero mpedance of the three pave element

7 Tme and -Doman Element Model dmttance and urrent Source for ntal ondton Tme Doman t -Doman etor: t v t v t etor: nductor: t t v d τ τ apactor: dv t t dt v v t t t t nductor: v apactor: v

8 dmttance and urrent Source for ntal ondton dmttance wth all ntal condton et to zero current tranform voltage tranform wth wth v dmttance of the three pave element

9 Example: Solve for urrent Waveform t ut t y K: etor: nductor: t u e e t t t nvere Tranform: forced repone natural repone

10 Sere Equvalence and oltage Dvon et of rcut et of rcut EQ EQ K: EQ EQ EQ

11 Parallel Equvalence and urrent Dvon et of rcut et of rcut EQ EQ K: EQ EQ EQ

12 Example: Equvalence mpedance and dmttance v t EQ EQ EQ EQ v t nductor current at t capactor voltage Fnd equvalent mpedance at and Solve for v t EQ EQ EQ EQ EQ EQ

13 General Technque for -Doman rcut naly Node oltage naly n -doman Ue Krchhoff urrent aw K Get equaton of node voltage Ue current ource for ntal condton oltage ource current ource Meh urrent naly n -doman Ue Krchhoff oltage aw K Get equaton of current n the meh Ue voltage ource for ntal condton urrent ource voltage ource Work only for Planar crcut

14 Formulatng Node-oltage Equaton Step : Tranform the crcut nto the doman ung current ource to repreent capactor and nductor ntal condton Step : Select a reference node. dentfy a node voltage at each of the non-reference node and a current wth every element n the crcut Step : Wrte K connecton contrant n term of the element current at the non-reference node Step 3: Ue the element admttance and the fundamental property of node voltage to expre the element current n term of the node voltage Step 4: Subttute the devce contrant from Step 3 nto the K connecton contrant from Step and arrange the reultng equaton n a tandard form

15 Example: Formulatng Node-oltage Equaton S S t t doman doman 3 eference node Step : Tranform the crcut nto the doman ung current ource to repreent capactor and nductor ntal condton Step : dentfy N- node voltage and a current wth each element Step : pply K at node and : Node : S Node : v 3 Step 3: Expre element equaton n term of node voltage v G where G 3 [ ] [ ]

16 Formulatng Node-oltage Equaton ont d Step : pply K at node and : Node : Node : 3 S v Step 3: Expre element equaton n term of node voltage [ ] [ ] where 3 G G Step 4: Subttute eqn. n Step 3 nto eqn. n Step and collect common term to yeld node-voltage eqn. v G S Node : Node :

17 Solvng -Doman rcut Equaton G rcut Determnant: G G G Depend on crcut element parameter:,, G/, not on drvng force and ntal condton Solve for node ung ramer rule: S v G S G v G G ero State when ntal condton ource are turned off ero nput when nput ource are turned off

18 Solvng -Doman rcut Eqn. ont d Solve for node ung ramer rule: G S G G S G v G v G G ero State ero nput

19 Network functon Network Functon ero- tate epone Tranform nput Sgnal Tranform Drvng-pont functon relate the voltage and current at a gven par of termnal called a port Tranfer functon relate an nput and repone at dfferent port n the crcut T oltage Tranfer Functon T urrent Tranfer Functon n T Tranfer dmttance T Tranfer mpedance n T T nput rcut n the zero-tate rcut n the zero-tate Output or or Out Out n n T T Out Out

20 alculatng Network Functon EQ T Drvng-pont mpedance oltage tranfer functon: EQ T Drvng-pont admttance oltage tranfer functon:

21 mpule epone and Step epone nput-output relatonhp n -doman T X When nput gnal an mpule T T mpule repone equal network functon H mpule repone tranform ht mpule repone waveform When nput gnal a tep G tep repone tranform gt tep repone waveform T G g H h τ dτ, h t x t δ t x t u t dg t dt t nput T rcut Output X mean equal almot everywhere, exclude thoe pont at whch gt ha a dcontnuty

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