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

CL is the loading at the far end of the transmission line

l is the length of the line with the characteristic impedance Z0

oDRV

o

nZZZRl

Cl

12

CL

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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

CL

Rl 2

LC2t r

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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.

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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 frequency4

)(2

281CL

DRVs

ZRlLlf

CL

Rl

Cl

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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

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C Matrix Sparsification

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433810047.3309.3285.3342.83310042605134634.8548.3394.16

7.33013463047126923.619.249.23834.8512693047126327.875.33448.3323.611263304813402.83394.169.2427.8713402411

10C 12

• Capacitance is a local effect• Directly truncate off-diagonal small elements produces a sparse matrix.• Guaranteed stability.

433810040002.8331004260513460094.16013463047126909.240012693047126327.87000126330481340

2.83394.169.2427.871340241110C 12

L Matrix Sparsification

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

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11= 1010.8 8.51 7.22 6.45 5.90 5.47 5.138.51 10.8 8.51 7.22 6.45 5.90 5.477.22 8.51 10.8 8.51 7.22 6.45 5.906.45 7.22 8.51 10.8 8.51 7.22 6.455.90 6.45 7.22 8.51 10.8 8.51 7.225.47 5.90 6.45 7.22 8.51 10.8 8.515.13 5.47 5.90 6.45 7.22 8.51

L

,

10.8

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Direct Truncation of

11= 10

10.8 8.51 7.22 6.45 5.90 5.47 5.138.51 10.8 8.51 7.22 6.45 5.90 5.477.22 8.51 10.8 8.51 7.22 6.45 5.906.45 7.22 8.51 10.8 8.51 7.22 6.455.90 6.45 7.22 8.51 10.8 8.51 7.225.47 5.90 6.45 7.22 8.51 10.8 8.515.13 5.47 5.90 6.45 7.22 8.51

L

,

10.8

1

1 10= 102.53 1.67 0.12 0.12 0.08 0.05 0.111.67 3.63 1.60 0.04 0.07 0.04 0.050.12 1.60 3.64 1.59 0.04 0.07 0.080.12 0.04 1.59 3.64 1.59 0.04 0.120.08 0.07 0.04 1.59 3.64 1.60 0.120.05 0.04 0.07 0.04

L

,

1.60 3.63 1.670.11 0.05 0.12 0.12 0.12 1.67 2.53

1 10= 102.53 1.671.67 3.63 1.60

1.60 3.64 1.59,1.59 3.64 1.59

1.59 3.64 1.601.60 3.63 1.67

1.67 2.53

L

1L

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Direct Truncation of

1L

next

11= 106.74 4.21 2.50 1.49 0.90 0.57 0.384.21 6.35 3.77 2.25 1.36 0.86 0.572.50 3.77 5.96 3.55 2.15 1.36 0.901.49 2.25 3.55 5.85 3.55 2.25 1.490.90 1.36 2.15 3.55 5.96 3.77 2.500.57 0.86 1.36 2.25 3.77 6.35 4.210.38 0.57 0.90 1.49 2.50 4.21

L

,

6.74

11= 1010.8 8.51 7.22 6.45 5.90 5.47 5.138.51 10.8 8.51 7.22 6.45 5.90 5.477.22 8.51 10.8 8.51 7.22 6.45 5.906.45 7.22 8.51 10.8 8.51 7.22 6.455.90 6.45 7.22 8.51 10.8 8.51 7.225.47 5.90 6.45 7.22 8.51 10.8 8.515.13 5.47 5.90 6.45 7.22 8.51

L

,

10.8

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Direct Truncation

• Resulting inductance matrix quite different• Large matrix inversion.• No stability guarantees.

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Window-based Methods

1

11

35.521.421.474.6

10

68.267.11010 1.672.53

11= 106.74 4.21 2.50 1.49 0.90 0.57 0.384.21 6.35 3.77 2.25 1.36 0.86 0.572.50 3.77 5.96 3.55 2.15 1.36 0.901.49 2.25 3.55 5.85 3.55 2.25 1.490.90 1.36 2.15 3.55 5.96 3.77 2.500.57 0.86 1.36 2.25 3.77 6.35 4.210.38 0.57 0.90 1.49 2.50 4.21

L

,

6.74

1 10= 102.53 1.671.67 3.63 1.60

1.60 3.64 1.59,1.59 3.64 1.59

1.59 3.64 1.601.60 3.63 1.67

1.67 2.53

L

1

11

96.577.350.277.335.621.450.221.474.6

10

69.260.1

67.153.21.603.631.67

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Window-based Methods11= 10

10.8 8.51 7.22 6.45 5.90 5.47 5.138.51 10.8 8.51 7.22 6.45 5.90 5.477.22 8.51 10.8 8.51 7.22 6.45 5.906.45 7.22 8.51 10.8 8.51 7.22 6.455.90 6.45 7.22 8.51 10.8 8.51 7.225.47 5.90 6.45 7.22 8.51 10.8 8.515.13 5.47 5.90 6.45 7.22 8.51

L

,

10.8

44.292.192.144.2

10)2:1,2:1( 101L

48.271.131.071.162.371.131.071.148.2

10)3:1,3:1( 101L

Since the inverse of the original inductance matrix is not exactly sparse, the resulting approximation is asymmetric.

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Window-based Methods

• Avoid large matrix inversion.• No stability guarantees.• Advanced methods exist to guarantee the stability

but at the cost of

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Sparsity Pattern for

2

8 9 10

3 4 5

7

1

6

12 13 14 1511

1L

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Band Matching Method

• Preserve inductive couplings between neighboring wires

L

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Horizontal layer

Shielding effect by the neighboring horizontal layer is perfect.Inverse of Inductance matrix is block tridiagonal.

2

8 9 10

3 4 5

7

1

6

12 13 14 1511

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Block Tridiagonal Matching

nnTn

Tn

nT

n

LLL

LLLLLL

L

21

22212

11211

1

,111

111

1 ),(

iiii

iiii

VLU

LUV

LVmmIU

If L has a block tridiagonal inverse,L can be compactly represented by

nnTT

nTT

n

nTT

n

VUUVUV

VUVUUVVUVUVU

L

21

22212

12111

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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.

iU iV

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Properties

• The resulting approximation minimizes the Kullback-Leibler distance to the original inductance matrix.

• The resulting approximation is positive definite.

1)}~det(log)~({1)~,( 11 LLLLTrN

LLd

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Vertical Layer

Shielding effect by the neighboring vertical layer is perfect.

2

8 9 10

3 4 5

7

1

6

12 13 14 1511

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Intersection of Horizontal and Vertical Layer

2

8 9 10

3 4 5

7

1

6

12 13 14 1511

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Multi-band matching method

L

Horizontal Block Tridiagonal band matching

Converge to an unique solution.L~

Vertical Block Tridiagonal band matching

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Intersection of Horizontal and Vertical Layer

v

L

h

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has the minimum distance

Optimality

• In every step, the distance to another space is minimized.

(Final solution is optimal.)

L

h

v

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Stability

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