1. Chapter 2 Interconnect Analysis Model Order ReductionQ D C - +
Prof. Lei He
Electrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.edu

2. Math Representation of RLC Circuits
Any RLC circuit can be represented by a first order differential equation
G x(t) C = B u(t)
(G+ sC)x(s) = Bu(s) (Laplace(s) domain)
u is an mx1 vector for the inputs of the circuit (e.g. current sources)
x is an Nx1 vector for the response of the circuit (e.g. node voltage)
G and C are NxN sparse matrices corresponding to the R, L, C element values and their connections
B is an Nxm matrix indicating the locations of the current sources
dx(t) dt

3. The Curse of Complexity
Number of nodes in an RLC Circuit: N
Need to play with N x N matrices
Ω(N2) Floating Point Operations (FLOP)
Number of possible input patterns m
Ω(N2m) FLOP
E.g. One block in Intel Pentium μP [ICCAD’04]
N=349,706, m=36,129
4,418,370,274,646,244 FLOP
4,418,370 seconds (1 FLOP/ns)
51 days …
By FLOP I mean the addition, subtraction, multiplication and division of floating numbers.
At first look, this complexity does not seem to be high. But let’s just look at an example.
Unfortunately, things are still complicated …

4. Can we save our computers?
The circuit may need to be repeatedly solved for many different inputs u(t)
=> Model Order Reduction
x(t) can be obtained in two ways:
x(t) = h(t) conv u(t), x(s) = H(s)*u(s) => reduce h(t)
Directly solve differential equation => reduce G, C sizes
transfer function

5. Outline Moment calculation AWE PRIMA ECE902 VLSI Interconnect
Prepared by Lei He

6. Transfer Function of RLC Circuits
The system equation in s domain
(G+sC)x(s)=Bu(s)
For simplicity, we will consider a single-input-single-output (SISO) circuit, then
u is a scalar for the input current source
x is an Nx1 vector for the node voltage of the circuit
B is an Nx1 vector indicating the location of the current source, e.g.
B=( … 0)T indicates the current source at node 3
x(s) can be solved
x(s) = (G+sC)-1Bu(s)
The output voltage at one node can be expressed as
y(s)=LT(G+sC)-1Bu(s)
L is an Nx1 vector selecting the output node location, e.g.
L=( … 0)T selects the voltage at node 4 as output
Transfer function H(s) = LT(G+sC)-1B

7. Moments of H(s)
Moments of H(s) are coefficients of the Taylor’s Expansion of H(s) about s=0

8. Some Calculations
H(s) = LT(G + sC)-1B = LT[G(I + sG-1C)]-1B = LT(I + sG-1C)-1G-1B (AB)-1 = B-1A-1 = LT(I - sG-1C + s2(G-1C)2 - …)G-1B (1+x)-1=1-x+x2-… = LT G-1B - s LT (G-1C)(G-1B) + s2 LT (G-1C)2(G-1B) - … 0th order moment m(0): LT G-1B 1st order moment m(1): -LT (G-1C)(G-1B) 2nd order moment m(2): LT (G-1C)2(G-1B) …… kth order moment m(k): (-1)kLT (G-1C)k(G-1B)

9. Expansion at Arbitrary Frequency
m(k) = (-1)kLT (G-1C)k(G-1B) implies that G is invertible.
If not, we can do expansion at some frequency different from s=0
Let s = s0 + σ, where s0 is an arbitrary, but fixed expansion
point such that G+s0C is non-singular, then
H(s) = LT(G + sC)-1B
=> H(σ) = LT(G + s0C + σ C)-1B
= LT[I-σ(G+ s0C)-1C]-1(G+s0C)-1B
Denote A= -(G+ s0C)-1C, R= (G+s0C)-1B, then
H(σ) = LT(I-σ A)-1R

10. Taylor Expansion and Moments
Expansion of H(σ) about s = 0
 Expansion of H(s) about s=s0
H(σ) = LT(I-σ A)-1R
Recursive moment computation:

11. Taylor Expansion and Moments (Cont’d)
Expansion of H(s) around
Recursive moment computation:

12. Interpretation of Moment Computation
Compute:
System equation: (G+sC)x(s)=Bu(s)
When s0 = 0, equivalent to DC analysis:
setting shorting inductors (0V) and opening capacitors (0A)
compute currents through inductors and voltages across capacitors as moments
Convert:
Inductor
Voltage source
Capacitor
Current source

13. Interpretation of Moment Computation (Cont’d)
Compute:
System equation: (G+sC)x(s)=Bu(s)
When s0 = 0, equivalent to DC analysis:
setting voltage sources of inductor L= LmL, current sources of capacitor C = CmC
external excitations = 0
compute currents through inductors and voltages across capacitors as moments
Convert:
Inductor
Voltage source
Capacitor
Current source

14. Interpretation of Moment Computation (Cont’d)
Compute:
System equation: (G+sC)x(s)=Bu(s)
When s0 = 0, equivalent to DC analysis:
setting moments as currents through inductors and voltages across capacitors
external excitations = 0
compute voltage sources of inductors and current sources of capacitors
Convert:
Inductor
Voltage source
Capacitor
Current source

15. Moment Computation by DC Analysis
Perform DC analysis to compute the (i+1)-th order moments
voltage across Cj => the (i+1)-th order moment of Cj
current across Lj => the (i+1)-th order moment of Lj
DC analysis:
modified nodal analysis (used in original AWE )
sparse-tableau ……
Time complexity to compute moments up to the p-th order: p  time complexity of DC analysis

16. Advantage and Disadvantage of Moment Computation by DC Analysis
Recursive computation of vectors uk is efficient since the matrix (G+s0C) is LU-factored exactly once
Computation of uk corresponds to vector iteration with matrix A ( )
Converges to an eigenvector corresponding to the eigenvalue of A with largest absolute value

17. Numerical Problems for Matrix Power
Assume λ1, λ2, … λN are the eigenvalues of matrix A, with λ1 the largest in absolute value, then

18. Outline Moment calculation AWE PRIMA ECE902 VLSI Interconnect
Prepared by Lei He

19. Pade Approximation where q < p << N
H(s) can be modeled by Pade approximation of type (p/q):
where q < p << N
Or modeled by q-th Pade approximation (q << N):
Formulate 2q constraints by matching 2q moments to
compute ki & pi

20. General Moment Matching Technique
Basic idea: match the moments m-(2q-r), …, m-1, m0, m1, …, mr-1
When r = 2q-1:
(i) initial condition matches, i.e.
Final value theorem
(ii)

21. Compute Residues & Poles
EQ1
match first 2q-1 moments

22. Basic Steps for Moment Matching
Step 1: Compute 2q moments m-1, m0, m1, …, m(2q-2) of H(s)
Step 2: Solve 2q non-linear equations of EQ1 to get
Step 3: Get approximate waveform
Step 4: Increase q and repeat 1-4, if necessary,
for better accuracy

23. Moment Matching by AWE [Pillage-Rohrer, TCAD’90]
Recall the transfer function obtained from a linear
circuit
When matrix A is diagonalizable

24. q-th Pade Approximation
Pade approximation of type (p/q):
q-th Pade approximation (q << N):
Equivalent to finding a reduced-order matrix AR such that
eigenvalues lj of AR are reciprocals of the approximating poles
pj for the original system

25. Asymptotic Waveform Evaluation
Recall EQ1:
Let

26. Asymptotic Waveform Evaluation (Cont’d)
Rewrite EQ1:
where
Solving for k:
Let
Need to compute all the poles first

27. Structure of Matrix AR Matrix: has characteristic equation:
Therefore, AR could be a matrix of the above structure
Note that:
Characteristic equation becomes the denominator of Hq(s):

28. Solving for Matrix AR Consider multiplications of AR on ml produces

29. Solving for Matrix AR (Cont’d)
After q multiplications of AR on ml
produces
Equating m’ with m:

30. Summary of AWE
Step 1: Compute 2q moments, choice of q depends on accuracy requirement; in practice, q  5 is frequently used
Step 2: Solve a system of linear equations by Gaussian elimination to get aj
Step 3: Solve the characteristic equation of AR to determine the approximate poles pj
Step 4: Solve for residues kj

31. Numerical Limitations of AWE
Due to recursive computation of moments
Converges to an eigenvector corresponding to an eigenvalue of matrix A with largest absolute value
Moment matrix used in AWE becomes rapidly ill-conditioned
Increasing number of poles does not improve accuracy
Unable to estimate the accuracy of the approximating model
Remedial techniques are sometimes heuristic, hard to apply automatically, and may be computationally expensive

32. Outline Moment calculation AWE PRIMA ECE902 VLSI Interconnect
Prepared by Lei He

33. (G+ sC)x(s) = Bu(s) (Laplace(s) domain)
General Idea
Any RLC circuit can be represented by a first order differential equation
G x(t) C = B u(t)
(G+ sC)x(s) = Bu(s) (Laplace(s) domain)
Can we reduce the equation size?
Reduce the number of variables (column # of G and C)
Reduce the number of equations (row # of G and C)
dx(t) dt

34. Reduce the Number of Variables
Guess x can be represented by linear combination of some vectors u1, …, uq
reduced
Note: q << N
original

35. Illustration dx(t) G x(t) + C = B u(t) dt + = B u(t) 1 N m q m q 1 m q

36. Reduce the Number of Equations
= B u(t) N 1 q m m q q N
Left multiply q q 1 q m 1 1
= u(t) m q q q

37. Projection Framework
qxn G
qxq
nxn
nxq

38. Approaches for picking Vq and Uq
Use Eigenvectors
Use Time Series Data
Compute
Use the SVD to pick q < k important vectors
Use Frequency Domain Data
Use Krylov Subspace Vectors?

39. Aside on Krylov Subspaces - Definition
The order k Krylov subspace generated from a matrix E and a vector b is defined as

40. Intuitive view of Krylov subspace choice for Uq
Taylor series expansion: A=-G-1C, R=G-1B
change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point

41. Combine point and moment matching: multipoint moment matching
Multiple expansion points give larger band
Moment (derivates) matching gives more accurate
behavior in between expansion points

42. Need for Orthonormalization of Uq
cannot be computed directly
Vectors will line up with dominant eigenspace!

43. Need for Orthonormalization of Uq (cont.)
In "change of base matrix" U transforming to the new reduced state space, we can use ANY columns that span the reduced state space
In particular we can ORTHONORMALIZE the Krylov subspace vectors

44. Orthonormalization of Uq:The Arnoldi Algorithm
For i = 1 to q-1
Generates k+1 vectors!
Orthogonalize new vector:
Remove the projection on other normalized vectors
For j = 1 to i
end
Normalize new vector
end

45. We know how to select Uq now…
but how about Vq?

46. Interconnected Systems
In reality, reduced models are only useful when connected together with other models and circuit elements in a composite simulation
Consider a state-space model connected to external circuitry (possibly with feedback!)
ROM
Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the reduced model?

47. Passivity
Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provide energy that is not in its storage elements.
If the reduced model is not passive it can generate energy from nothingness and the simulation will explode

48. Interconnecting Passive Systems
The interconnection of stable models is not necessarily stable
BUT the interconnection of passive models is a passive model: Q D C - + Q D C - + Q D C - + Q D C - +

49. Sufficient conditions for passivity
Note that these are NOT necessary conditions (common misconception)

50. Congruence Transformations Preserve Positive Semidefinitness
Def. congruence transformation
Property: a congruence transformation preserves the positive semidefiniteness of the matrix
Proof. Just rename
same matrix

51. PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98)
Select Vq=Uq with Arnoldi Krylov Projection Framework:
Use Arnoldi: Numerically very stable

52. Moment Matching Theorem
PRIMA preserves the moments of the transfer function up to the q-th order, i.e.,

53. Summary: Conventional Design Flow
Funct. Spec
Logic Synth.
Gate-level Net.
RTL
Layout
Floorplanning
Place & Route
Front-end
Back-end
Behav. Simul.
Gate-Lev. Sim.
Stat. Wire Model
Parasitic Extrac.

54. Summary: Parasitic Extraction
thousands of wires
e.g. critical path
e.g. gnd/vdd grid
Parasitic
Extraction
identify some ports
produce equivalent circuit that models response of wires at those ports
tens of circuit
elements for
gate level spice
simulation

55. Summary: Model Order Reduction
Electromagnetic
Analysis
thin volume filaments
with constant current
small surface panels
with constant charge
million of elements
Model Order
Reduction
tens of elements

56. Summary We have presented how to calculate moments of RLC circuits
We have discussed about AWE and PRIMA
Both are based on the moment matching
AWE has numerical problems and can only match 3-4 moments
PRIMA is inherently stable and can match high order moments
PRIMA can preserve passivity

57. References
L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Computer-Aided Design, vol. 9, pp. 352–366, Apr
Altan Odabasioglu, Mustafa Celik, and Lawrence T. Pileggi, “PRIMA: Passive Reduced-Order Interconnect Macromodeling Algorithm”, IEEE Trans. Computer-Aided Design, Vol. 17, pp , Aug