CSE 245: Computer Aided Circuit Simulation and Verification

CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm

Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy Domain Analysis From im domain o Frquncy domain Corrspondnc bwn im domain and frquncy domain Srial xpansion of (si-a) - Lcur.

Oulin (Con ) Modl Ordr Rducion Momns Passiviy, Sabiliy and Ralizabiliy Symbolic Analysis Y-Dla Transformaion BDD Analysis Lcur.3

Lcur.4 Sa of a sysm Th sa of a sysm is a s of daa, h valu of which a any im, oghr wih h inpu o h sysm a im, drmin uniquly h valu of any nwork variabl a im. W can xprss h sa in vcor form x Whr x i () is h sa variabls of h sysm ) (... ) ( ) ( x x x k

Sa Variabl How o Choos Sa Variabl? Th knowldg of h insananous valus of all branch currns and volags drmins his insananous sa Bu NOT ALL hs valus ar rquird in ordr o drmin h insananous sa, som can b drivd from ohrs. choos capacior volags and inducor currns as h sa variabls! Bu no all of hm ar chosn Lcur.5

Dgnra Nwork A nwork ha has a cu-s composd only of inducors and/or currn sourcs or a loop ha conains only of capaciors and/or volag sourcs is calld a dgnra nwork Exampl: Th following nwork is a dgnra nwork sinc C,C and C 5 form a dgnra capacior loop Lcur.6

Dgnra Nwork In a dgnrad nwork, no all h capaciors and inducors can b chosn as sa variabls sinc hr ar som rdundancy On h ohr hand, w choos all h capacior volags and inducors currns as sa variabl in a nondgnra nwork W will giv an xampl of how o choos sa variabl in h following scion Lcur.7

Ordr of Circui n b LC n C - n L n h ordr of circui, oal numbr of indpndn sa variabls b LC oal numbr of capaciors and inducors in h nwork n C numbr of dgnra loops (C-E loops) n L numbr of dgnra cu-ss (L-J cu-ss) n 4 3 In a nondgnra nwork, n quals o h oal numbr of nrgy sorag lmns Lcur.8

Sa Equaions Linar sysm of ordinary diffrnial quaions dx d Sa Ax( ) + Bu( ) Inpu Oupu y( ) Qx( ) + Du( ) Lcur.9

Sa Equaion for RLC Circuis Th sa quaion is of h form C v& E v L & YT il E R il Or M x&() Gx() + Pu() - + Pu v : volag in h runk, capacior volag i l : currn in h loop, inducor currn. Y and R ar h admianc marix and impdanc marix of cu-s and msh E covrs h co-r branchs in h cu-s E T covrs h r runks in h msh analysis Lcur.

Sa Equaions Mx&() Gx() + Pu() If w shif h marix M o h righ hand sid, w hav x& () M - Ax() + M - Bu() L A M - G and B M - P, w hav h sa quaion x&() Ax() + Bu() Toghr wih h oupu quaion y () Qx() + Du() ar calld h Sa Equaions of h linar sysm Lcur.

RLC Nwork Analysis A givn RLC nwork L 6 g 3 C Vs C C 5 g 4 Dgnra Nwork, Choos only volags of C and C 5, currn of L 6 as our sa variabl Lcur.

Tr Srucur Tak ino r as many capaciors as possibl and, as lss inducors as possibl Rsisors can b chosn as ihr r branchs or co-r branchs L 6 g 3 C C /L 6 g 3 C C 5 g4 Vs C C 5 g 4 Vs Lcur.3

Lcur.4 Linar Sa Equaion By a mixd cu-s and msh analysis, considr capacior cu-ss and inducor loops only. w can wri h linar sa quaion as follows M Gx() + Pu() x&() Cu-s KCL Loop KVL Cu-s KCL + + 6 5 L C C C C C C 6 i v v & & & 4 3 g g 6 i v v g 3 - + Vs

Solving RCL Equaion by Taylor Expansion() Gnral Circui Equaion X AX + BU Considr homognous form firs X AX X A X and A A I + +! A! +... + k A k! k +... Q: How o Compu A k? Lcur.6

Solving RCL Equaion by Taylor Expansion () Assum A has non-dgnra ignvalus,,..., k and corrsponding linarly indpndn ignvcors Χ, Χ,..., Χ k, hn A can b dcomposd as AΧΛΧ L whr L M and Λ M L O M L L k [ Χ Χ ] Χ,...,, Χ k Lcur.7

Solving RCL Equaion by Taylor Expansion (3) Wha s h implicaion hn? AΧΛΧ A Χ A ΧΛ Χ Λ Χ whr Λ To compu h ignvalus: M L L L L O L M M k d( I A) n + c n n +... + c ( p )( + p + p )...(...) ral ignvalu Conjugaiv Complx ignvalu Lcur.8

Lcur.9 In h prvious xampl () () / / / i v X i v l r l c A + Χ ΛΧ X X A 3 3 j j + + whr Χ 3 3 3 j j j + Χ 3 3 j j hnc Χ Χ + A L crl, w hav Solving RCL Equaion by Taylor Expansion (4)

Solving RCL Equaion by Taylor Expansion (5) Wha if marix A has dgnrad ignvalus? Jordan dcomposiion! A ΧJΧ J is in h Jordan Canonical form And sill A Χ J Χ Lcur.

Lcur. Jordan Dcomposiion J + + J L J + + J! L similarly

Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Rspons in im domain Frquncy Domain Analysis From im domain o Frquncy domain Corrspondnc bwn im domain and frquncy domain Srial xpansion of (si-a) - Lcur.

Rspons in im domain W can solv h sa quaion and g h closd form xprssion Th oupu quaion can b xprssd as No: * dnos convoluion Lcur.3

Impuls Rspons Th Impuls Rspons of a sysm is dfind as h Zro Sa Rspons rsuling from an impuls xciaion Thus, in h oupu quaion, rplac u() by h impuls funcion δ(), and l x( ) w hav h() y() Q A B Lcur.4

Soluions in S domain By solving h sa quaion in s domain, w hav x(s) (si-a) - x( )+ (si-a) - Bu(s) y(s) Qx(s) +Du(s) Q(sI-A) - (x( ) + Bu(s)) +Du(s) Suppos h nwork has zro sa and h oupu vcor dpnds only on h sa vcor x, ha is, x( ) and D, w can driv h ransfr funcion of h nwork y( s) H(s) Q(sI-A) - B u( s) Lcur.6

Lcur.7 Ky Ky Transform Transform Propry: Propry: Bilaral Bilaral Laplac Laplac Transform: Transform: ) ( ) ( ) ( ) ( ) ( s Qx s y s Bu s Ax s sx + ) ( ) ( ) ( ) ( ) ( Qx y Bu Ax d dx + d dx sx(s) d x s x s ) ( ) ( x() Frquncy Domain Rprsnaion

Sysm Transfr Funcion sx( s) Ax( s) y( s) Qx( s) + Bu( s) Exprss y(s) as a funcion of u(s) y( s) Q( si A) Bu( s) Transfr Funcion: H (s) Lcur.8

Transfr Funcion Tim Domain Impuls Rspons Frquncy domain rprsnaion u(s) H(s) Linar sysm y(s) H(s) u(s) Tim domain rprsnaion u() y( ) h( τ ) u( τ ) dτ h() Linar sysm Th ransfr funcion H(s) is h Laplac Transform of h impuls rspons h() Lcur.9

Corrspondnc bwn im domain and frquncy domain W can driv h im domain soluions of h nwork from h s domain soluions by invrs Laplac Transformaion of h s domain soluions. Sa Equaions in S domain sx(s) x( ) Ax(s) +Bu(s) y(s) Qx(s) +Du(s) Invrs Laplac Transform Sa Equaions in im Domain x() L - [(si-a) - x( ) + (si-a) - Bu(s)] L - [(si-a) - ]x( ) + L - [(si-a) - ]B*u() y() L - [Q(sI-A) - (x( ) + Bu(s)) +Du(s)] Q L - [(si-a) - ] x( ) + {QL - [(si-a) - ]B +Dδ()}* u(s) Lcur.3

Corrspondnc bwn im domain and frquncy domain Soluion from im domain analysis Soluion by invrs Laplac ransform x() L - [(si-a) - x( ) + (si-a) - Bu(s)] L - [(si-a) - ]x( ) + L - [(si-a) - ]B*u() y() L - [Q(sI-A) - (x( ) + Bu(s)) +Du(s)] Q L - [(si-a) - ] x( ) + {QL - [(si-a) - ]B +Dδ()}* u(s) (si-a) - A muliplicaion of u(s) in s domain corrsponds o h convoluion in im domain Lcur.3

Srial xpansion of (si-a) - Whn s w can wri (si-a) - as (si-a) - -A - (I sa - ) -A - (I + sa - + s A - + + s k A -k + ) Thus, h ransfr funcion can b wrin as H(s) Q(sI-A) - B -QA - (I + sa - + s A - + + s k A -k + )B Whn s w can wri (si-a) - as (si-a) - s - (I s - A) - s - (I + s - A + s - A + + s -k A k + ) Th ransfr funcion can b wrin as H(s) Q(sI-A) - B s - (I + s - A + s - A + + s -k A k + )B Lcur.34

Marix Dcomposiion Assum A has non-dgnra ignvalus,,..., k and corrsponding linarly indpndn ignvcors Χ, Χ,..., Χ k, hn A can b dcomposd as A ΧΛΧ L whr L M and Λ M L O M L L k [ Χ Χ ] Χ,...,, Χ k Lcur.35

Lcur.36 Marix Dcomposiion Thn w can wri (si-a) - in h following form (si-a) - in s domain corrsponds o h xponnial funcion A in im domain, w can wri A as (si-a) - (SI XΛX - ) - X - (si Λ) - X X - n s s s.. X A X - n.. X