Skin Effects In System On A Chip Interconnects

1
Efficient Model Order Reduction Including Skin
Effect
Shizhong Mei, Chirayu Amin, and Yehea I.
Ismail High Performance VLSI Design and Analysis
Lab ECE Department, Northwestern University June
3, 2003
2
Outline

• Importance of modeling frequency dependent
  effects
• Model for skin effect impedance
• Modifying model order reduction techniques to
  handle skin effect impedance
• AWE
• PVL
• Results
• Conclusions

3
Outline

• Importance of modeling frequency dependent
  effects
• Model for skin effect impedance
• Modifying model order reduction techniques to
  handle skin effect impedance
• AWE
• PVL
• Results
• Conclusions

4
Interconnect Delay Dominates

• Interconnect delay dominates the overall delay

5
Interconnect Wires
6
Skin Effect

• Due to skin effect, the effective conducting area
  decreases as frequency increases, causing
• an increase in resistance
• a drop in inductance

7
Extra Delay Caused by Skin Effect
Response of an RLC tree
8
Outline

• Importance of modeling frequency dependent
  effects
• Model for skin effect impedance
• Modifying model order reduction techniques to
  handle skin effect impedance
• AWE
• PVL
• Results
• Conclusions

9
Approximate Skin Effect Impedance

• Approximate R(f) and L(f) with a simple yet
  accurate expression, e.g.,

10
Accuracy of the Approximation

• Analytical skin effect impedance for conductors
  of circular cross-section

11
Exact Skin Effect Impedance vs. Approximate Skin
Effect Impedance of Round Wires

• The largest discrepancy occurs at low frequencies

12
Validity of
Response of an RLC tree
13
Outline

• Importance of modeling frequency dependent
  effects
• Model for skin effect impedance
• Modifying model order reduction techniques to
  handle skin effect impedance
• AWE
• PVL
• Results
• Conclusions

14
Transfer Function of Constant RLC Circuits

• Transfer function is uniquelydetermine
  d by the circuit itself
• The Taylor expansion about s 0 is
• High order terms in the expansion can be
  neglected at low frequencies, e.g.,

15
Reduced Transfer Function

• At low frequencies, a reduced transfer function
  can approximate the original transfer function
• The coefficients in the Taylor expansion of the
  transfer function, mis are called moments
• Theoretically, arbitrary accuracy can be achieved
  by matching a higher number of moments

16
Square Root Moments When Skin Effect is
Considered

• At low frequencies, a reduced transfer function
  can approximate the original transfer function
• The coefficients in the Taylor expansion of the
  transfer function, mis are called square root
  moments
• Theoretically, arbitrary accuracy can be achieved
  by matching a higher number of square root
  moments

17
Node Voltages in RLC Trees With Skin Effect
Impedance
18
Square Root Moments for RLC Trees
With , the original transfer
function for voltage at node i is
where
,
,
,
and
, for
19
Square Root Moments for General RLC Circuits
for n ? 2
moments
20
Reduced Transfer Function

• The reduced order transfer function is

Or
21
Responses in the Time Domain

• Since the impedance contains the term , The
  node voltage in time domain takes the form

• The output is stable only under the condition
  that the real parts of or are
  non-positive
• Unstable poles are simply discarded and s in
  the remaining terms are rescaled such that the
  stable output takes the expected DC value

22
Numerical Instability

• In finite-precision arithmetic, moments are
  inherently numerically unstable
• Vectorsquickly converge to an eigenvector
  corresponding to the dominant eigenvalue of
• A maximum of 40 square root moments can be used
  in comparison to 20 traditional moments

, for
.
23
Padé Via Lanczos Method

• In some cases, accurate simulation of RLC
  interconnect circuits may require higher order
  approximation
• PVL avoids direct calculation of moments and is
  numerically more stable than AWE
• PVL needs to be modified to include skin effect
• New state space is defined

24
Outline

• Importance of modeling frequency dependent
  effects
• Model for skin effect impedance
• Modifying model order reduction techniques to
  handle skin effect impedance
• AWE
• PVL
• Results
• Conclusions

25
Response of an RLC Tree
26
Response of a General RLC Circuit
PVL with constant R PVL with skin effect AWE with
skin effect
27
Conclusions

• The square root moments are proposed to handle
  skin effect in the simulation of RLC interconnect
  circuits
• Square root moments can be easily calculated
• In practice, the model order reduction approach
  based on the square root moments can use almost
  twice the number of moments that AWE can achieve
  for constant elements using traditional moments
• The PVL method is modified to implicitly match
  the square root moments