partial element equivalent circuit (peec)dipanjan/e8_202/e8202-cem_12.pdf · 4 • a. e. ruehli,...

Partial Element Equivalent Circuit (PEEC)

E8-202 Class 12

Dipanjan Gope

2

Grading Scheme

Time Frame Assignment Grading

Sep 6 HW 1 15

Sep 24 HW2

Oct 8 HW 3

Oct 11 Midterm 1 15

Nov 5 HW 4 5

Nov 20 Midterm 2 15

Dec 5 Final Exam 30

Dec 10 Final Project 20

3

• 2D vs 2.5D vs. 3D Formulations

• Electrostatic Formulation: Capacitance matrix extraction

• Magnetostatic Formulation: Inductance matrix extraction

• Electric Field Integral Equation (EFIE): S-parameter extraction

• Partial Element Equivalent Circuit (PEEC) Method

• Magnetic Field Integral Equation (MFIE) and Combined Field Integral Equation (CFIE)

• PMCHWT Formulation: Dielectric modeling

• Parallelization techniques

Module 2: Method of Moments

4

• A. E. Ruehli, “Equivalent circuit models for three-dimensional multiconductor systems” IEEE Trans. Microwave Theory and Techniques, vol. MTT22 (3), pp. 216-221, March 1974.

References

5

What does PEEC do?

Lumped Circuit Elements e.g. R; L; C; V; I

Distributed Circuit Elements e.g. Transmission Line

~r dt t

KVL; KCL Maxwell’s Equations

Elem

en

ts

Def

init

ion

Eq

uat

ion

s So

luti

on

SPICE Port Model + SPICE PEEC

r dt t

6

SPICE Review

• Sparse Tableau Analysis (STA) - IBM ASTAP simulator

• Modified Nodal Analysis (MNA) - SPICE simulators

7

Stamps Revisited

Resistor

Current Source

Voltage Source

Some excerpts from UW class notes by Prof. Richard Shi and Guoyong Shi

8

Stamps Revisited

Capacitor

N+

N-

N+ N- RHS

N+ C/h -C/h (C/h)V(t-h)

N- -C/h C/h -(C/h)V(t-h)

Inductor

N+

N-

N+ N- I-branch RHS

N+ 1

N- -1

Branch 1 -1 -L/h -(L/h)I(t-h)

9

Stamps Revisited

VCVS

CCCS

Some excerpts from UW class notes by Prof. Richard Shi and Guoyong Shi

N+ N- NC+ NC- ik

N+ 1

N- -1

NC+

NC-

ij 1 -1 E -E

10

' '

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

v v

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

j

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

PEEC: Equation

11

Basis Function

Current

Charge

12

Partial vs Loop Inductance

Loop Partial

122211 2LLLL

13

Vc2

CCVS

Vc1

Unit Cell

' '

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

v v

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

j

dldarfa

dvrfa

a lv

)(1

)(1

Testing Function

14

Vc2 CCVS

Vc1

Resistance Term

dldaJ

adv

J

aa lv

11

dldaa

I

adv

J

aa lv

11

a

lIdv

J

av

1

15

Vc2 CCVS

Vc1

Inductance Term

v va a

dvdvGaa

jGJdvdvja

'1

'1

''

'

v v

vvp dvdvGaa

L '1

''

'

16

Vc2 CCVS

Vc1

Capacitor Terms

2121111 QpQp

211

1211

11

1Q

p

pQ

p

dt

dQ

p

p

dt

dQ

dt

d

p

2

11

1211

11

1

211

121

1

11

1cc i

p

pi

dt

d

p

17

Ge

om

etry

C

on

du

cto

r

4km

k m k m

k mp k m

k m k ma a l l

dl dlL da da

a a r r

+ - + -

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

PEEC Stick Example

18

Vc2 CCVS

Vc1

Basic Dielectric Modeling

Vc2 CCVS

Vc1

vv

i dvrqrrGj

dvrJrrGjj

JrE ')'()',(')'()',(

)1(),(

'0

l

AC 0)1(

19

Integral Equation Solution to Maxwell’s Equations

Special Meshing for PEEC Cells

Transformation to PEEC Circuit Elements

SPICE Circuit Elements Other SPICE Type Models

SPICE Like Circuit Description

MNA Delay Diff. Eq. (DDE) Circuit Solver

Voltages, Currents, Electric and Magnetic Fields

PEEC Flow

20

• Orthogonal, No-retardation

• …

• Orthogonal, with-retardation

• …

• Orthogonal, with-dielectric and retardation

• Non-orthogonal, with dielectric, retardation

• ….

PEEC History

21

Volumetric formulation ideal for on-chip cases

• No thickness discretization necessary

DC to daylight solution

• Charge and current decoupled at DC

“Stamps” provide interface between techniques Multi-purpose solver (.tran / .ac / .cap / .ind)

PEEC: Advantages

22

Dense Matrix: Expensive LU (Gaussian Elimination)

• Can we use EM properties to expedite solution? Volume formulation unsuitable for “thick” structures

• Can we get similar stamps for surface-based/hybrid form?

PEEC: Drawbacks

Addressed in later research