Inductance Screening and Inductance Matrix Sparsification 1

Outline Inductance Screening Inductance Matrix Sparsification 2

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 C L is the loading at the far end of the transmission line l is the length of the line with the characteristic impedance Z 0

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 t r 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 f s = 0.34/t r 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)

Outline Inductance Screening Inductance Matrix Sparsification 9

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

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

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 but at the cost of

18 Sparsity Pattern for

19 Band Matching Method Preserve inductive couplings between neighboring wires

20 Horizontal layer 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 Shielding effect by the neighboring vertical layer is perfect

25 Intersection of Horizontal and Vertical Layer

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

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.)

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