Inductance Screening and Inductance Matrix Sparsification

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Inductance Screening and Inductance Matrix
Sparsification
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Outline

• Inductance Screening
• Inductance Matrix Sparsification

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Inductance Screening

• Accurate modeling the inductance is expensive
• Only include inductance effect when necessary
• How to identify?

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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

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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

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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
  frequency

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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)

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Outline

• Inductance Screening
• Inductance Matrix Sparsification

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C Matrix Sparsification

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

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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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Direct Truncation of

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Direct Truncation of

next
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Direct Truncation

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

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Window-based Methods
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Window-based Methods
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

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Sparsity Pattern for
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Band Matching Method

• Preserve inductive couplings between neighboring
  wires

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

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

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Properties

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

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Vertical Layer
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• Shielding effect by the neighboring vertical
  layer is perfect.

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Intersection of Horizontal and Vertical Layer
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Multi-band matching method
Horizontal Block Tridiagonal band matching
Vertical Block Tridiagonal band matching
Converge to an unique solution.
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Intersection of Horizontal and Vertical Layer

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Optimality

• In every step, the distance to another space is
  minimized.
• (Final solution is optimal.)

has the minimum distance
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Stability

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