EE5900 Spring Lecture 4 IC interconnect modeling methods Zhuo Feng

Transcription

1 EE59 Spring Parallel LSI AD Algoriths Lecture I interconnect odeling ethods Zhuo Feng. Z. Feng MTU EE59

2 So far we ve considered only tie doain analyses We ll soon see that it is soeties preferable to odel in the coplex frequency doain, s Priarily when circuits are linear or even weakly nonlinear LTI ckt We ll first consider analysis in the real frequency doain. Z. Feng MTU EE59

3 Analog circuits are generally analyzed in the frequency doain ( j) Transient response is of short duration copared to periodic response of interest: ( t ) t transient. Z. Feng MTU EE59

4 9 cos t f 9 8 5x F H j j R Steady state response will be of sae frequency but different ag & phase. Z. Feng MTU EE59

5 Focusing on the steady state solution we can analyze the ag and phase responses.667. H = radians/sec - H = atan radians/sec.5 Z. Feng MTU EE59

6 Under sall-signal assuption, nonlinear eleents are linearized i about dc operating pt. First step is to solve for nonlinear dc solution Ex: siple coon eitter ckt R R c c RL R R e.6 Z. Feng MTU EE59

7 D gain establishes bias pt for the aplifier Linearized odel at final N-R iteration represents sall-signal signal ac odel and soe dc bias currents i c v ce.7 Z. Feng MTU EE59

8 ac analysis for analog ckts. Solve for dc operating pt. via N-R. hange RHS vector ters to generate sall-signal ac odels fro final N-R linearization. Asseble coplex ipedance equations Y j. Solve for agnitude and phase at discrete frequency points for variables of interest.8 Z. Feng MTU EE59

9 Sall-signal assuption: input voltages are so sall that t nonlinearities iti are negligible ibl Not really true for all analog circuits and all responses of interest Nonlinearities cause distortion of analog signals which are often design constraints Later we will consider nonlinearities in the frequency doain via other analyses.9 Z. Feng MTU EE59

10 Exaple:. Solve for dc bias point R R c R R e. Z. Feng MTU EE59

11 To siplify we ll use an Ebers-Moll odel for the forward active region B F I F I F E. Z. Feng MTU EE59

12 Nonlinear dc solution for this ckt is a -D proble R R R R e I F I F. Z. Feng MTU EE59

13 i d Load line Last N-R iteration Diode odel EQ I EQ v d I EQ. Z. Feng MTU EE59

14 . Build ckt equations for ac sall signal analysis id EQ EQ v d. Z. Feng MTU EE59

15 . dv s dt j dc operating pt. is used to specify capacitance values for nonlinear s too.5 Z. Feng MTU EE59

16 ac Analysis Model: R R I I F j bc jj c IN j R EQ j be jj js R L Re je.6 Z. Feng MTU EE59

17 Forulate nodal equations: Y ( j) ( j) I ( j) n iven a particular value, solve for ( j) Translate coplex voltage into agnitude and phase Probles with s as Bigger proble for L as Z L j L Y L jl n.7 Z. Feng MTU EE59

18 Nodal analysis does not like infinite conductance Z L j L Y L jl eneral solution is to treat L like a voltage source and use auxiliary equations for inductor currents.8 Z. Feng MTU EE59

19 Soeties analog designers prefer poles / zeros instead of a freq. doain ag. and phase plots We start by forulating the equations using s instead of s j x( s) x( s) B u( s).9 Z. Feng MTU EE59

20 onsider a linear interconnect ckt exaple sl s 6 5 s 5 s 8 5 in s s s s We can stap these eqns. into the for s x ( s ) x ( s ) Bu ( s ). Z. Feng MTU EE59

21 s S L i i L Z. Z. Feng Feng MTU EE59 MTU EE59..

22 5 5 IN i L S i Z. Z. Feng Feng MTU EE59 MTU EE59..

23 Isolating one response variable of interest: s x ( s ) Bu ( s ) x ( s ) s Bu ( s ) and i ( s) L T s Bu( s). Z. Feng MTU EE59

24 For exaple, if we are interested in the response at node 7: s Bu ( ) T 7 ( s ) L s T where L For a transfer fct at node 7, u( s) H ( s L s 7 ) B T. Z. Feng MTU EE59

25 i s Bu ( s ) T ( s ) L s We know fro raer s Rule that any node voltage solution will be of the for: i ( s) dett T det s s z s z s z s p s p s p n det s The roots of are the ckt poles.5 Z. Feng MTU EE59

26 det s s singular s p i Muller s root finding algorith: Search for points for which det(s+)= Select points in s-plane Interpolate with polynoial Solve for polynoial root Use root to prune search Fit new polynoial And so on.6 Z. Feng MTU EE59

27 How do we test to see if Y s LU Factor Y det Y det L det U dt det L Difficult to find all of the poles in a freq. range reliably We ll actually show a better way of doing this using odel order reduction ethods.7 Z. Feng MTU EE59

28 An alternative way of perforing frequency doain analysis is via oents We ll first show how oents can be used to represent linear syste responses Including generation of transfer functions Becae very popular for interconnect analysis probles Then we ll show how we can extend these techniques to tie-varying nonlinear systes.8 Z. Feng MTU EE59

29 Start with a linear interconnect ckt s6 5 s 5 s sl 8 5 in s s s s We know that we can stap these eqns. into the for: sx( s) x( s) Bu( s).9 Z. Feng MTU EE59

30 s S L i i L Z. Z. Feng Feng MTU EE59 MTU EE59..

31 5 5 IN i L S i Z. Z. Feng Feng MTU EE59 MTU EE59..

32 s x ( s ) Bu ( s ) Y (s) x ( s) s Bu( s) If we are interested in the response at node 7: where s Bu ( s ) T 7 ( s ) L s T L For a transfer fct at node 7, u( s) H7 ( s) L s T B. Z. Feng MTU EE59

33 enerally, we express transfer functions in a siilar for: H s a b as a b s b n s s n n It s ipractical, however, to calculate transfer functions sybolically ll for large ckts in either for Therefore, we instead start with series expansions in s. Z. Feng MTU EE59

34 By definition of the Laplace transfor: H st s h t e dt Expanding about s H s h t st s t s t dt 6 k k k! s k t k h t dt Like oents fro probability theory H s s s We call the -ters oents here. Z. Feng MTU EE59

35 We can use value for s= b a a s an s b s b s a b since can be used to represent n s s n zeros & poles But how any oents do I need to copletely specify y -pole syste?.5 Z. Feng MTU EE59

36 n a a s a s b s b s s n hapter 6 shows a th order syste exaple: a as as as bs bs bs bs s s ollect the coefficients for equations in ters of unknown coefficients Assuing we have calculated as any oents L T as necessary fro L S B.6 Z. Feng MTU EE59

37 a as as as bs bs bs bs s s a a a a b b b b b b b b b b b b b b 5 b b b b b b b b Z. Feng MTU EE59

38 The next equation would be: b b b b 5 6 But it can be shown that this is not linearly independent of original equations 7 8 In general, oents uniquely specify an order syste th linear equations in ters of unknowns when oents are known last equations can be used to calculate the b-coefficients.8 Z. Feng MTU EE59

39 b b b b 5 b b b b b b b b 6 5 b b b b b b b b b 5 b b b b b Z. Z. Feng Feng MTU EE59 MTU EE Hankel atrix ill-conditioned

40 This Hankel atrix can be LU factored to solve for denoinator coefficients: b, b, b, b The roots of the ckt poles b s bs bs bs are Once we have the b-ters, the a-ters are easily obtained fro the first four equations iven the nuerator coefficients, the roots of: a a s a s a s are the ckt zeros -- at a specific node of interest!. Z. Feng MTU EE59

41 Poles are the inverse of our tie constants Responses are sus of decaying exponentials with decay rates specified by poles pit kie i k i ' s Zeros specify the residues, -- the aount of energy at a particular frequency. Z. Feng MTU EE59

42 All of this assues that we can calculate oents easily For an ipulse response: x( s) s B x xs xs since U s L T We pre-ultiply by to get a response at one node: T s L s B s s Z. Feng MTU EE59

43 7 T s L s B L T s B set A L T sa B L T sa B L T sa B set R B sa R L T expand about s. Z. Feng MTU EE59

44 T 7 s L sa R d d s s s s s 7 ds 7 s ds 7 s d ds d ds d ds s s 7 7 s s n n s s 7 s L T sa A R L T AR sa A R L T L T A R n!! L T A n R. Z. Feng MTU EE59

45 ` learly we can solve for A and and fro the recursively calculate oents R B Then solving the Hankel atrix forulation we can obtain the coefficients for poles characteristic eqn Not a well conditioned atrix proble however - - ore on this later.55 Z. Feng MTU EE59

46 A L R B k k If and, how do we solve for the oents, K fro to A R T ust be nonsingular LU factor -- dc equivalent ckt solution Recursive application of.66 Z. Feng MTU EE59

47 dc recursive for oents v s sl v s 7 6 s 6 5 s 5 s 8 5 in i L s s s s in s s s i L L L s.77 Z. Feng MTU EE59

48 Moents are unknown, but we can express capacitor currents in ters of capacitor voltages, and inductor voltages in ters of inductor currents I s o s sl L L L o s.88 Z. Feng MTU EE59

49 dc equivalent ckt 6 I 6 5 s I 5 L sl I L i L I I I L s I s s Solve for the oents recursively.99 Z. Feng MTU EE59

50 6 k k 5 k 6 6 k c k k L Lk k k k in k L k k k k k.5 Z. Feng MTU EE59

51 oupling capacitors can be split into two current sources to aintain a tree structure for interconnect s 5 5 k c k s k I 6 6 s o s o s.5 Z. Feng MTU EE59

52 Tree-like interconnect structures can be solved efficiently via path tracing In general we won t attept to solve for oents to get an exact solution Instead we ll calculate p oents in an attept to capture the p-ost doinant poles Asyptotic Wavefor Evaluation (AWE).5 Z. Feng MTU EE59

53 Model Order Reduction Take an -th order syste alculate q oents (q<<) enerate a q-th order odel as if it were a q-th order syste Moent atching or Pade approxiation Used in any areas of science & engineering Sees straightforward, but there are any probles and issues to be dealt with.5 Z. Feng MTU EE59

54 Exaple: first order odel of -th order syste alculate first two oents at a response node of interest kˆ pˆ t s or v t ke ˆ s pˆ Single-pole ipulse response odels Moents are ˆ ˆ pt ke dt tke ˆ pt ˆ dt kˆ pˆ kˆ ˆp.5 Z. Feng MTU EE59

55 Moents are calculated fro ckt so we can use the to deterine the values for kˆ and pˆ kˆ pˆ kˆ p ˆ Two equations in ters of two unknowns.55 Z. Feng MTU EE59

56 pˆ is an approxiation of the doinant (sallest) pole (largest tie constant) v t k e p t k e p t k e p t vv t t.56 Z. Feng MTU EE59

57 Ipulse Response: n n s b b s s a s a s H If dc gain = In ters of poles & zeros: z s z s z s n p s p s p s z z z s H Z. Z. Feng Feng MTU EE59 MTU EE

58 To see the relationship to oents, we can expand H s Which is equivalent to: as ans b s b s n b s b s a s a s.58 Z. Feng MTU EE59

59 a b Fro b n a Z j P j P j If there are no low frequency zeros, is sall If p Then b is uch saller than all other poles b p p p j for j Z j,, p a.59 Z. Feng MTU EE59

60 First oent of ipulse response is a reasonable doinant pole approxiation under certain conditions Basis for use of Elore Delay as a doinant tie constant approxiation O X j s-plane X O X O XX O X But note that while varies fro node to node, the actual poles do not.6 Z. Feng MTU EE59

61 What about odels for q>? Moents are fro expansion about s= so we expect q-th order odel to capture the lower frequency poles nd order odel atching oents: ˆ ˆ pˆ t pˆ t h t ke ke After integration by parts to obtain oent forulas: kˆ kˆ kˆ pˆ ˆ p pˆ ˆ p kˆ p ˆ ˆ p ˆ pˆ p kˆ kˆ ˆ kˆ kˆ.6 Z. Feng MTU EE59

62 ould solve nonlinear equations in ters of unknown, but a better way is via Hankel atrix equations Treat this as a nd order syste: b b Solve for b-coefficients, then solve For the poles b ˆ ˆ p b p pˆ and pˆ p.6 Z. Feng MTU EE59

63 iven the poles, we can solve for the zeros and/or residues directly via a linear set of equations: kˆ kˆ pˆ ˆ p kˆ kˆ p ˆ ˆ p In atrix for we write this as k anderonde atrix Diagonal atrix of /p s First q oents *pg. 7-7 of the textbook.6 Z. Feng MTU EE59

64 Siple Exaple:. 5 5 in Z. Feng MTU EE59

65 onvergence of pole approxiations: Exact poles st order nd order rd order th order Z. Feng MTU EE59

66 R clock tree with 765 nodes v(t) input voltage rap nd, rd order AWE, and SPIE siulation st order AWE tie.66 Z. Feng MTU EE59

67 RL clock tree with 97 nodes v(t) input nd order AWE st order AWE rd order AWE SPIE siulation tie.67 Z. Feng MTU EE59

68 In theory we can apply oent atching for any order of approxiation But in practice it s not so siple: Approxiations of stable systes can be unstable Finite precision probles Inherent instability of Pade approxiations.68 Z. Feng MTU EE59