ECE 570 Session 7 IC 752-E Computer Aided Engineering for Integrated Circuits. Transient analysis. Discuss time marching methods used in SPICE

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1 ECE 570 Sessio 7 IC 75-E Compuer Aided Egieerig for Iegraed Circuis Trasie aalysis Discuss ime marcig meods used i SPICE. Time marcig meods. Explici ad implici iegraio meods 3. Implici meods used i circui aalysis 4. Implemeaio of discreizaio i SPICE Supplemeal readig: ladimirescu, Te SPICE Book, Capers: 6, , 0.4

2 . Time marcig meods Two classes: explici - iexpesive per sep, limied sabiliy, o for circui aalysis implici more expesive per sep, beer sabiliy, suiable for circui aalysis. Example of circui C d = E d R d = d E R = 0 Ω C = pf = 0 sec

3 Noaio: ( ) exac (eoreical) soluio approximae (umerical) soluio a = We wrie e differeial equaio a ime d ( ) = ( ) E ( ) d = We ca approximae e derivaive usig a) d d formula or b) d d formula. ( ) ( ) ( ) = = i e form ; = Forward Euler (F-E) ( ) ( ) ( ) = ; = Backward Euler (B-E) 3

4 Usig F-E formula we obai ( ) ( ) ( ) E ( ) Compuaio of e soluio o e basis of e above formula, from e iiial =, ca be described by e differece equaio codiio ( ) o o For a cosa exciaio, ( ) differece equaio is Te aalyic soluio is = E A E ( ) =, ad zero iiial codiio e soluio o e A =. ( ) = A e ad i approaces e level A as ime icreases. 4

5 Te umerical soluio,, will ed o e same level iff < <.. Explici ad implici iegraio meods Discussed explici F-E meod as sabiliy problem Sep size is limied by sabiliy. Te explici meods are o suiable for circui aalysis, were we wa o be able o compue wi large seps we e soluio cages slowly (i.e. we e accuracy does o require small seps). Backward-Euler (implici meod) d d ( ) ( ) is is e firs order meod. Tis meod will be applied o e equaio of circui d ( ) = ( ) E( ) d. 5

6 Cosiderig e fiie differece approximaio we ave ( ) ad e differece equaio: = E = 0,,, Assume: 0 0 = a be =, [ ] E = E = = cos a e soluio is 0 5 = be0( a a a ) were a = ( ), b = ( ) a < () 5 ad also 5 because iegraio sep. wiou ay resricio o e 6

7 Ass.: 0 = 5, E = 0 Numerical approximaio yields a = Te soluio is: = a 0 a < 5 () STABILITY CONDITION < THE METHOD IS ABSOLUTELY STABLE Te imporace of absolue sabiliy 7

8 3. Implici meods used i circui aalysis B-E: T-R x = x x -rs order meod xˆ dx = d = (defaul i SPICE) x = x ( x x ) -d order meod THESE ARE ONE-STEP METHODS, ALSO CALLED SINGLE STEP METHODS. 8

9 EXEISE- Apply T-R meod o e circui ad obai differece equaio for e circui. Hi d = E d d = d For = we ave ( ) = ( ) E( ) For = we ave ( ) = ( ) E( ). Te T-R formula is = ( ) 9

10 Liear Mulisep Meods (LMM) Geeral formula for LMM k x = α x β x k i k i k i i k i i= i= 0 represeig k-sep meods. k We β 0 = 0, e meod is explici ad i is o suiable for circui simulaio. We β0 0, e meod is implici ad suiable for circui simulaio, e meod is suiable for solvig siff problems. 0

11 THE GEAR S METHODS k x = α x β x k i k i k 0 k i= IN SPICE k =,3,,6 (for k = we ave α = ad 0 β =, wic resuls i B-E) 4 Example of -sep meod k = β 0 = α = α = 3 3 EXEISE. Derive differece formula for e circui usig Gear s meod wi k=. Ceck e meod sabiliy ( ζ < ). 3

12 Hi: Applicaio of -sep Gear meod yields 4 x ( γ ) x x = γ E were γ = τ = τ Ceckig e sabiliy. Te caracerisic polyomial is 4 ζ ( γ ) ζ = A geeral soluio o e omogeeous differece equaio is x = C A ζ CB ζ Sabiliy Codiio is ζ < ζ < erify a e differece is always sable regardless of.

13 A example of formula for 3-sep Gear s meod k = 3 β 0 = α = α = α 3 = x = α x α x α x β x Noe a = 0,,, k i= α = i wic is a expressio of e meod cosisecy codiio. 3

14 Exercise 3 Formulae B-E equaio for e circui below R I d Q D Ass. : Q D -jucio cap. oly Q D = Q( ) - Resul: d I = I ( e ) d s = s R R C( ) I ( e ) E dqd C( ) =, = d d = 0,,, EXEISE 3a. Apply F-E ad ge e differece equaios for e circui i e Exercise 3. 4

15 4. Implemeaio of discreizaio i SPICE (Example of T-R meod) Te capacior i C = ( ) T-R formula - Te circui relaios d i = C = i d C Usig e followig oaio for e umerical approximaio o e curre I i( ) I i( ) ad e circui relaios we obai = ( I I ). C 5

16 Te explici formula for e curre is C C I = I g I or usig e oaio for e coefficies I = g I C were g = ad C I = I Te circui ierpreaio yields e followig equivale circui for e capaciace I. g 6

17 Te Iducor Te circui relaios L I di = L d di = d L Te T-R formula: I = I ( I I di ) were I =. d Usig e circui relaios we obai I = I L L g I 7

18 Te circui ierpreaio of e formula yields e equivale circui I g Replacig e capaciors ad iducors by eir equivale circuis we obai COMPANION NETWORK Exercise: Use B-E formula o ge equivale circuis for e capacior ad iducor. 8