passive model reduction including dielectrics

1
A New Surface Integral Formulation For Wideband
Impedance Extraction of 3-D Structures
Ben Song, Zhenhai Zhu, John D. Rockway and Jacob
White RLE VLSI-CAD group, MIT
2
Outline

• Motivations and Background
• A New Surface Integral Formulation
• Low Frequency Problem and Its Remedy
• Conclusions

3
Parasitic Extraction at High Frequency

• Higher Frequencies
• Need distributed RLC or full-wave analysis
• Dense couplings
• Simultaneously analyze more complicated
  geometries
• Lossy substrate
• Multilayer structures !!

RF/Mixed-Signal
Courtesy of Harris semiconductor
4
Fast Integral Equation Approach

• Solves the discretized integral equations
  Iteratively
• Requires only matrix-vector products
• Computes Matrix-Vector products approximately
• Can often be performed in order n or nlogn ops
  (FMM, Hierarchical SVD, pFFT, Wavelet, etc.)

5
State-of-the-Art Fast Impedance Solvers

• FastImp
• Analyzes structures with millions unknowns over
  night on a desktop PC
• Available at http//rleweb.mit.edu/vlsi/codes.htm
  
• Other University programs
• University of Illinois (AIM methods)
• University of Washington (Direct Inversion)
• Several Commercial products
• Hierarchical SVD
• Fast Multipole

6
Why Surface Formulations?

• Handles Skin Effects Easily

No interior mesh needed

• Does not generate interior modes
• Interior modes confuse model order
• reduction

7
Surface Formulation Alternatives

• Many sets of variables to choose
• Scalar and Vector Potentials
• Electric and Magnetic Fields
• Charge and Current densities

• Many Equations to choose
• Maxwells Equations
• Constitutive Equations
• Potential Equations
• Conservation Law

• Established Formulations
• FastImp (proved success for impedance
  extraction)
• Surface PEEC (with effective surface impedance)
• PMCHW (widely used in scattering problems)

8
Properties Desired for A Surface Formulation

• Well conditioned at both low and high
  frequencies

• fastImp formulation

• Ease of use of layered media Greens function

• fastImp formulation

• Reasonable accuracy for coarse discretization

• fastImp formulation

• Unified and convenient method for computing
  
• impedances

• fastImp formulation

9
Outline

• Motivations and Background
• A New Surface Integral Formulation
• Low Frequency Problem and Its Remedy
• Conclusions and future work

10
Stratton-Chu Integral Representation for Magnetic
Field
Vector Greens Second Identity
11
Stratton-Chu Integral Representation for Electric
Field
Vector Greens Second Identity
12
The Properties of Stratton-Chu Integral
Representations
13
Derivation of Governing Equations
eq 1
Et, Ht, Hn
Gcond
0
eq 2
Ht, Et, En
Gair
0
J
eq 3
Gair
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Summary
eq 1
Gcond
Gcond
eq 2
Gair
Gair
Gair
eq 3
Gair
Gair
Gair
15
Advantage IEase of Use of Multilayer Greens
Function
16
Advantage IEase of Use of Multilayer Greens
Function
17
Advantage IIConsistent Port Current Calculation
One formula works for all the frequencies
18
Outline

• Motivations and Background
• A New Surface Integral Formulation
• Low Frequency Problem and Its Remedy
• Conclusions and future work

19
Unknowns and Basis Functions
RWG Edge based, Linear varying, Vector Basis
functions has supports on the two panels sharing
one edge Non-divergence Freee on each supporting
panel.
20
Low Frequency Problem

• Matrix ? Infinity at D.C.

• Large errors in low frequency results

Sometimes L lt 0
21
Low Frequency Problem The Reason
Two parts scale with frequency differently
Separates it and scales the N.D.F. part
explicitly
22
Loop Basis and Star Basis
Span RWG basis Span Loop Span Star
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Low Frequency Problem The Remedy
??
Loop Basis
Star Basis
New system of equations can give correct
results down to D.C.
24
Wire Example Resistance
Straight wire 1?1?4 (mm)
FastHenry 128 ? FastHenry 692 ? FastHenry
2048 Surface 132
Resistance (Ohm)
Frequency (Hz)
25
Wire Example Inductance
Straight wire 1?1?4 (mm)
Inductance (nH)
FastHenry 128 ? FastHenry 692 ? FastHenry
2048 Surface 132
Frequency (Hz)
26
A Ring Example Resistance
FastHenry 960 ? FastHenry 3840 ? FastHenry
15360 Surface 272 ? Surface 496
Resistance (Ohm)
Frequency(Hz)
27
A Ring Example Inductance
Inductance (nH)
FastHenry 960 ? FastHenry 3840 ? FastHenry
15360 Surface 272 ? Surface 496
Frequency(Hz)
28
Conclusions

• A new surface integral formulation has been
• developed
• use as
  unknowns
• use RWG basis and pulse basis
• use voltage sources as excitation

• The advantages over fastImp formulation

compatible with multilayer dyadic Greens
function
consistent in port current calculation
Thanks Prof. Luca Daniel for help in presentation