1. Inductance Screening and Inductance Matrix Sparsification

2. Outline
Inductance Screening
Inductance Matrix Sparsification

3. Inductance Screening Accurate modeling the inductance is expensive
Only include inductance effect when necessary
How to identify?

4. Off-chip Inductance screening
The error in prediction between RC and RLC representation will exceed 15% for a transmission line if
CL is the loading at the far end of the transmission line
l is the length of the line with the characteristic impedance Z0

5. Conditions to Include Inductance
Based on the transmission line analysis, the condition for an interconnect of length l to consider inductance is
R, C, L are the per-unit-length resistance, capacitance and inductance values, respectively
tr is the rise time of the signal at the input of the circuit driving the interconnect

6. On-chip Inductance Screening
Difference between on-chip inductance and off-chip inductance
We need to consider the internal inductance for on-chip wires
Due to the lack of ground planes or meshes on-chip, the mutual couplings between wires cover very long ranges and decrease very slowly with the increase of spacing.
The inductance of on-chip wires is not scalable with length.

7. Self Inductance Screening Rules
The delay and cross-talk errors without considering inductance might exceed 25% if
where fs = 0.34/tr is called the significant frequency

8. Mutual Inductance Screening Rules
SPICE simulation results indicates that most of the high-frequency components of an inductive signal wire will return via its two quiet neighboring wires (which may be signal or ground) of at least equal width running in parallel
The potential victim wires of an inductive aggressor (or a group of simultaneously switching aggressors) are those nearest neighboring wires with their total width equal to or less than twice the width of the aggressor (or the total width of the aggressors)

9. Outline
Inductance Screening
Inductance Matrix Sparsification

10. C Matrix Sparsification
Capacitance is a local effect
Directly truncate off-diagonal small elements produces a sparse matrix.
Guaranteed stability.

11. L Matrix Sparsification
Inductance is not a local effect
L matrix is not diagonal dominant
Directly truncating off-diagonal elements cannot guarantee stability

12. Direct Truncation of 1

13. Direct Truncation of
next

14. Direct Truncation Resulting inductance matrix quite different
Large matrix inversion.
No stability guarantees.

15. Window-based Methods

16. Window-based Methods
Since the inverse of the original inductance matrix is not exactly sparse, the resulting approximation is asymmetric.

17. Window-based Methods Avoid large matrix inversion.
No stability guarantees.
Advanced methods exist to guarantee the stability

18. Sparsity Pattern for 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

19. Band Matching Method
Preserve inductive couplings between neighboring wires

20. Horizontal layer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Shielding effect by the neighboring horizontal layer is perfect.
Inverse of Inductance matrix is block tridiagonal.

21. Block Tridiagonal Matching
If L has a block tridiagonal inverse,
L can be compactly represented by

22. Block Tridiagonal Matching
Sequences and are calculated only from tridiagonal blocks.
Tridiagonal blocks match those in the original inductance matrix.
Inverse is a block tridiagonal matrix.

23. Properties
The resulting approximation minimizes the Kullback-Leibler distance to the original inductance matrix.
The resulting approximation is positive definite.

24. Vertical Layer 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Shielding effect by the neighboring vertical layer is perfect.

25. Intersection of Horizontal and Vertical Layer1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

26. Multi-band matching method
Horizontal Block Tridiagonal band matching
Vertical Block Tridiagonal band matching
Converge to an unique solution.

27. Intersection of Horizontal and Vertical Layer

28. has the minimum distance
Optimality
In every step, the distance to another space is minimized.
(Final solution is optimal.)
has the minimum distance

29. Stability
In every step, the resulting matrix is positive definite. Final solution is stable.