S-Domain Analysis. s-domain Circuit Analysis. EE695K VLSI Interconnect. Time domain (t domain) Complex frequency domain (s domain) Laplace Transform L

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1 EE695K S nterconnect S-Doman naly -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 Prepared by K

2 EE695K S nterconnect Krchhoff aw n -Doman t doman doman Krchhoff current law K 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 Sgnal Source n Doman t doman oltage Source: v t vs t t depend on crcut t vt t v S 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 Prepared by K

3 EE695K S nterconnect Tme and -Doman Element Model mpedance and oltage Source for ntal ondton Tme Doman t -Doman etor: v t t v t etor: nductor: d t v t dt v t t nductor: apactor: t v t d τ τ v v t t v apactor: v mpedance and oltage Source for ntal ondton mpedance wth all ntal condton et to zero wth wth v voltage tranform current tr anform mpedance of the three pave element Prepared by K 3

4 EE695K S nterconnect 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: apactor: v v dmttance and urrent Source for ntal ondton dmttance current tranform voltage tranform wth all ntal condton et to zero dmttance of the three pave element wth wth v Prepared by K 4

5 EE695K S nterconnect Example: Solve for urrent Waveform t ut t y K: etor: nductor: t t nvere Tranform: t e e u t forced repone natural repone Sere Equvalence and oltage Dvon et of rcut et of rcut K: Prepared by K 5

6 EE695K S nterconnect Parallel Equvalence and urrent Dvon et of rcut et of rcut K: Example: Equvalence mpedance and dmttance v t v t nductor current at t capactor voltage Fnd equvalent mpedance at and Solve for v t Prepared by K 6

7 EE695K S nterconnect 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 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 Prepared by K 7

8 EE695K S nterconnect Prepared by K 8 Example: Formulatng Node-oltage Equaton t S t doman S doman v 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 : Node : 3 S v Step 3: Expre element equaton n term of node voltage [ ] [ ] where 3 G G 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 :

9 EE695K S nterconnect 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: v S G G S v G G ero State when ntal condton ource are turned off ero nput when nput ource are turned off Solvng -Doman rcut Eqn. ont d Solve for node ung ramer rule: G S G S v G G v G G G ero State ero nput Prepared by K 9

10 EE695K S nterconnect Network Functon ero -tate epone Tranform Network functon 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 rcut n the zero-tate nput rcut Output or n the zero-tate or Out Out n n T T Out Out alculatng Network Functon Drvng-pont mpedance oltage tranfer functon: T Drvng-pont admttance oltage tranfer functon: T Prepared by K

11 EE695K S nterconnect mpule epone and Step epone nput-output relatonhp n -doman T X When nput gnal an mpule x t δ t T T mpule repone equal network functon H mpule repone tranform ht mpule repone waveform When nput gnal a tep x t u t G tep repone tranform gt tep repone waveform T H G g h τ dτ, dg t h t dt t nput T Output X rcut mean equal almot everywhere, exclude thoe pont at whch gt ha a dcontnuty Prepared by K