1 Dipanjan Gope Solving Low-Frequency EM-Ckt Problems Using the PEEC Method Dipanjan Gope* Electrical Engineering University of Washington Vikram Jandhyala Circuit Technology CAD INTEL Corporation Albert Ruehli System Level Design T.J. Watson IBM Research
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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?
• 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?
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