1. Chapter 2 Interconnect Analysis
Prof. Lei He
Electrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.edu

2. Organization Chapter 2a First/Second Order Analysis
ECE902 VLSI Interconnect
Organization
Chapter 2a First/Second Order Analysis
Chapter 2b Moment calculation and AWE
General Moment Matching Models
Moments Computation
AWE
Chapter 2c Projection based model order reduction
Prepared by Lei He

3. 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

4. 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.
(ii)

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

6. 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

7. Components of Moment Matching Methods
2-pole method (S2P introduced previously)
Asymptotic Waveform Evaluation (AWE) [Pillage- Rohrer, TCAD’90]
Critical Component
Moment computation

8. Organization General Moment Matching Models Moments Computation AWE
ECE902 VLSI Interconnect
Organization
General Moment Matching Models
Moments Computation
AWE
Prepared by Lei He

9. Basis of Moment Computation by DC Analysis
Applicable to general RLC networks
Used in Asymptotic Waveform Evaluation (AWE)
Represent a lumped, linear, time-invariant circuit by a system of first-order differential equations:
where x represents circuit variables (currents and voltages)
G represents memoryless elements (resistors)
C represents memory elements (capacitors and inductors)
bu represents excitations from independent sources
y is the output of interest

10. Transfer Function Assume zero initial conditions and perform Laplace
transform:
where X, U, Y denote Laplace transform of x, u, y, respectively
Transfer function:
Let s = s0 + s, where s0 is an arbitrary, but fixed expansion
point such that G+s0C is non-singular

11. Taylor Expansion and Moments
Expansion of H(s) about s = 0:
Recursive moment computation:

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

13. Interpretation of Moment Computation
Compute:
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

14. Interpretation of Moment Computation (Cont’d)
Compute:
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

15. Interpretation of Moment Computation (Cont’d)
Compute:
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

16. 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

17. 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 an eigenvalue of A with largest absolute value

18. Organization General Moment Matching Models Moments Computation AWE
ECE902 VLSI Interconnect
Organization
General Moment Matching Models
Moments Computation
AWE
Prepared by Lei He

19. Moment Matching by AWE [Pillage-Rohrer, TCAD’90]
Recall the transfer function obtained from a linear
circuit
Let s = s0 + s, where s0 is an arbitrary, but fixed expansion point such that G+s0C is non-singular
When matrix A is diagonalizable

20. 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

21. Asymptotic Waveform Evaluation
Recall EQ1:
Let

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

23. 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):

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

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

26. 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

27. 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

28. Homework
[1] Given the circuit as shown below and a unit step voltage source at the input node s, use SPICE to simulate the circuit and obtain the accurate 50% delay at node n. Also analytically calculate the delay using Elmore method and S2P method. How do they compare with the result obtained by SPICE? R1 C1 s R2 C2 R4 C4 C3 R3 C5 R5 n
R1 = 1mΩ
R2 = 2mΩ
R3 = 2mΩ
R4 = 1mΩ
R5 = 4mΩ
C1 = 1nF
C2 = 1nF
C3 = 4nF
C4 = 4nF
C5 = 2nF 1v

29. Homework
[2] Give the circuit as shown below and a unit step voltage source at node s, can we still use the “shared-path” formula to calculate the Elmore delay? Explain why or why not. Use DC analysis method via MATLAB or SPICE to get the 0th -3rd moments of C3 and C5. R1 C1 s R2 C2 R4 C4 C3 R3 C5 C6 n
R1 = 1mΩ
R2 = 2mΩ
R3 = 2mΩ
R4 = 1mΩ
C1 = 1nF
C2 = 1nF
C3 = 4nF
C4 = 4nF
C5 = 2nF
C6 = 1nF 1v

30. Steps for Problem 1
1. Write the SPICE netlist of the circuit and probe the voltage response at node n.
2. Record the time when the voltage at node n reaches 0.5V. That time is the 50% delay.
3. Use the Elmore delay formula to calculate the Elmore delay.
(find the shared path between each node and node n).
4. Write down the transfer function and driving point admittance of the circuit with input s and output n.
5. Expand the transfer function to get the moments m1* and m2*. Expand the driving point admittance to get m1, m2, m3, and m4.

31. Steps for Problem 1
6. Follow the S2P algorithm to get k1, k2, p1 and p2.
7. Use the frequency domain expression (h(s)) to derive the time domain expression (h(t)).
8. Plot the obtained time domain waveform to get the 50% delay for the S2P model.
9. Compare the results.

32. Steps for Problem 2
1. Follow the DC analysis method to reconstruct the circuit (e.g. replace C with zero current source for 0th moment calculation, etc).
2. Stamp the G and C matrices for MATLAB analysis or write the corresponding netlist for SPICE analysis.
3. Get the voltage across the capacitance as the moment.
4. The above should be done repeatedly until all the desired moments are acquired.