1

Dipanjan Gope

Solving Low-Frequency EM-Ckt ProblemsUsing the PEEC Method

Dipanjan Gope*Electrical Engineering

University of Washington

Vikram Jandhyala

Circuit Technology CADINTEL Corporation

Albert RuehliSystem Level Design

T.J. Watson IBM Research

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Dipanjan Gope

Outline

• Numerical Problems for Low-Frequency EFIE- Low-Frequency in Circuits: Why Should We Bother?- What are the Detrimental Numerical Effects?

• Existing Solution Methods

• PEEC Low-Frequency Solution- Basic PEEC Cell- Low-Frequency Strategies

• EFIE Low-Frequency Solution

• Results and Conclusions

3

Dipanjan Gope

“Low Frequency” in Circuits

On-Chip

1. Electrically Small Structure

BoardsPackages

1mm

UW VCO Structure

2. Local Refined Mesh

f = 40Ghzλ = 7.5mm

Courtesy: Ansoft Corporation

Electrostatic Magnetostatic

11 12 13

21 22 23

31 32 33

C C CC C CC C C

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

11 12 13

21 22 23

31 32 33

L L LL L LL L L

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

ElectroMagnetic

11 12 13

21 22 23

31 32 33

S S SS S SS S S

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Numerical Problems

1dimensionfrequency

<<

Electrically Small

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Dipanjan Gope

“Low Frequency” Problems

• “Low Frequency” Fast Solver Problem- Traditional FMM: Singularity in Hankel function

• “Low Frequency” Mixed Potential Problem- Affects solver convergence: larger number of iterations- Affects accuracy even for direct solution

Level 0 Level 1 Level 2

Cube Length < λ/5

- Solution 1: Low Frequency FMM, Chew etal.- Solution 2: QR-PEEC fast iterative solver, Ruehli etal. EPEP’04

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Dipanjan Gope

EFIE Formulation

Frequency domain, Method of Moments

Electric Field Integral Equationstantan ( ) i

sZ+ =E J E J

Scattered Electric Field( )s jω=− −∇ΦE J A

Electric Vector Potential| | ( )( )

4 | |

jk

S

e dsμπ

′−

∫′

′=′−

r -r J rA rr r

Electric Scalar Potential| |1 ( ')( )

4 | |

jk

S

e d sρπε

′−

∫ ′Φ =′−

r -r rrr r

Continuity Equation( ) ( ) 0s jωρ∇ ⋅ + =J r r

| | | |( ) 1 ( ')( )4 | | 4 | |

r - r r - rJ r rE Jr r r r

j k j ks

S S

e ej d s d sμ ρωπ π ε

′ ′− −

∫ ∫′

′ ′= − − ∇′ ′− −

Vector and Scalar potential from EM Currents

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Dipanjan Gope

Mixed Potential Problems

= Z

eN

jωLeN

Rank=Ne

P1e pN Njω ×+ ×A p e

TN N××A

pN

Rank=Np

1. Fast Solver Convergence Suffers 2. Direct Solver Result Suffers

Effects

For a closed object: Ne=1.5Np

Beyond Machine Precision

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Dipanjan Gope

Outline

• Numerical Problems for Low-Frequency EFIE- Low-Frequency in Circuits: Why Should We Bother?- What are the Detrimental Numerical Effects?

• Existing Solution Methods

• PEEC Low-Frequency Solution- Basic PEEC Cell- Low-Frequency Strategies

• EFIE Low-Frequency Solution

• Results and Conclusions

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Dipanjan Gope

Loop-Star Decomposition

Loop basis Star basis

Chew etal.

• Loop basis for solenoidal current (Magneto-static)• Star basis for curl-free current (Electrostatic)• Frequency scaling for improved spectral property• Number of iterations does not scale with frequency

Frequency vs Iteration

050

100150200250300

9.00E

+10

9.00E

+08

9.00E

+06

9.00E

+04

frequency (Hz)

Num

ber

of it

erat

ion

Loop-StarBasis Rearrangement

Courtesy: Slide by Swagato Chakraborty

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Dipanjan Gope

Loop-Star Challenges

• Loop Detection is Challenging for:- Open structures- Structure with holes- Structure with handles- Structures with junction

Open Structure

Hole

Junction

Handle/Loop

• Where to Apply Loop-Star Basis Functions- Detection of mesh where loop-star should be applied- Detrimental effects if applied wrongly

More Significant

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Dipanjan Gope

Outline

• Numerical Problems for Low-Frequency EFIE- Low-Frequency in Circuits: Why Should We Bother?- What are the Detrimental Numerical Effects?

• Existing Solution Methods

• PEEC Low-Frequency Solution- Basic PEEC Cell- Low-Frequency Strategies

• EFIE Low-Frequency Solution

• Results and Conclusions

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Dipanjan Gope

PEEC Single Matrix: EM / Circuit

Lumped Circuit Elementse.g. R; L; C; V; I

Distributed Circuit Elementse.g. Transmission Line

~r dt t

KVL; KCL Maxwell’s Equations

Elem

ents

Def

initi

onEq

uatio

nsSo

lutio

n

SPICE Port Model + SPICEPEEC

r dt t>>

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Dipanjan Gope

PEEC Methodology: SPICE Form

' '

' ' ' '( )( , ) ( , ) ( ) ( , ) ( )i

v v

J rE r j G r r J r dv G r r q r dvj

ω ωμσ ω

∇= + +∫ ∫

Circuit Model Element Identification

• KVL: Voltage = R I + sLp I + Q/sC

• RHS Term 1: Resistance

• RHS Term 2: Partial Inductance

• RHS Term 3: Coefficients of Partial Potential

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Dipanjan Gope

PEEC Stick ExampleG

eom

etry

Con

duct

or

4km

k m k m

k mp k m

k m k ma a l l

dl dlL da daa a r rμ

π⋅

=−∫ ∫ ∫ ∫

+ - + -R1 N1 N2 1.202mOhms

L1 N2 N3 5.887nH

C1 N1 0 1.702pF

K12 L1 L2 1.282nH 0.054ns

F12 N4 0 V1 0.124 0.032ns

Possible Solution Schemes: .cap, .ind, .ac, .tran

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Dipanjan Gope

PEEC DC Solution

+ - + -

• ElectroStatic: Short Inductors (.cap)

Continuity Equation NOT ValidJ jωρ∇ ⋅ = −

+ - + -

• MagnetoStatic: Open Capacitors (.ind)

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Dipanjan Gope

PEEC Low Frequency Solution

Step 1: Separate Charge and Current Basis Functions

• Charge Basis is Not Derived From Current Basis (Unlike RWG Basis)

• Enables Stamping of Scalar and Vector Potentials Differently- Vector potential is stamped in the impedance form (KVL)- Scalar potential is stamped in the admittance form (KCL)- Both cases ω is in the numerator in matrix elements

0...1:: ,L A P A

e e e p p p p e e

TN N N N N N N N i N incEFIE j t E

jω

ω× × × × =⎛ ⎞

+ < >⎜ ⎟⎝ ⎠

In Contrast RWG-EFIE is Completely KVL

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Dipanjan Gope

PEEC Low Frequency Solution

Step 1: Separate Charge and Current Basis Functions

112

11 11 1

2

21 2

11 22 1

2

11

1 1 0 0 0 0 00

0 0 0 1 00

0 0 1 1 0 0 0 000 0 0 1 0

0 0 0 0 1 0 1 00 0 0 0 0 1 1 00 1 0 1 0 0

c

c

c s

c

L

psp p V

p Vsp p I I

II

R sLp

φ

φ

−⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥− − ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ =⎢ ⎥ ⎢ ⎥− −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎣ ⎦⎣ ⎦⎢ ⎥− − −⎣ ⎦

• No Extra Memory• No Extra MatVec Product Time• No ω in Denominator• No ω in Off-Diagonal

CCCS

C C C S

Vc1 Vc2

1φ 2φ

ILKVL

KCL

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Dipanjan Gope

PEEC Low Frequency Solution

Step 2: Incorporation of Dielectric Loss

12

11 11

21

11 22

11

1 1 0 0 0 0 0

0 0 0 1 0

0 0 1 1 0 0 0

0 0 0 1 0

0 0 0 0 1 0 10 0 0 0 0 1 11 1 0 0 0 0

psGp p

psGp p

R sLp

−⎡ ⎤⎢ ⎥⎢ ⎥+ − −⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥

+ − −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥− − −⎣ ⎦

CCCS

CCCS

Vc1 Vc2

1φ 2φ

G

LinearAt Low Frequencies R and G Dominate

Seam Less Transition to DC

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Dipanjan Gope

Transmission Line Example

PEEC Solver Results

Beyond Precision

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Dipanjan Gope

Conclusions

• Separated Potential Enables Mixed KCL/KVL Form

• Loss Enables Flat Conditioning at Low Frequencies- Loss dominates over all effects at low frequencies

• Seam-Less Transition from .ac to .cap and .ind

• Does Not Harm Conditioning for Electrically Large Mesh

• No Requirement for Loop Detection

Highlight