Workshop on PEEC Modeling - LTU

Workshop on PEEC Modeling

PEEC Modeling of Magnetic Materials

and Dispersive-Lossy Dielectrics

Giulio Antonini

UAq EMC LaboratoryDepartment of Electrical Engineering

University of L’Aquila67040 AQ, Italy

[email protected]

Lulea, November 13, 2007 Slide 1 of 59

Outline 1st part

● PEEC modeling of magnetic materialsX magnetization currents and related effects

● Efficient computation of integralsX volume to surface integrals

● PEEC equivalent circuit

● Numerical results

• Integrals computation

• Canonical problem

• 3 column transformer

• Trace over a magnetic ground plane


PEEC modeling of magnetic materials

The vector potential due to electrical and magnetization currents is

A(r, t) =µ0

4π

∫

V ′

J(r′, t)| r − r′ | dV +

+µ0

4π

[∫

V ′

∇×M(r′, t)| r − r′ | dV ′ +

∫

S′

M(r′, t)× n′

| r − r′ | dS′]

The magnetic flux density B is related to A by:

B = ∇×A(r, t) Divergencyless is thus enforced



Hp: magnetic polarization uniform in volume V ′

A(r, t) =µ0

4π

∫

V ′

J(r′, t)| r − r′ | dV ′ +

+µ0

4π

∫

S′

M (r′, t)× n′

| r − r′ | dS ′

EFIE at a point r as in the standard PEEC method

Ei(r, t) =J(r, t)

γ+

∂A(r, t)∂t

+∇Φ(r, t)

where

Φ(r, t) =1

4πε0

∫

S′

σ(r′, t)| r − r′ | dS ′



Density currents, magnetization currents and density chargeare expanded as:

J (r, t) =NV∑j=1

Jj (t) fj (rj)

M (r, t) =NV∑j=1

Mj (t) bj (rj)

σ (r, t) =NS∑

m=1

qm (t) pm (rm)



The discretization process allow to re-write the EFIE as:

Ei(r, t) =J(r, t)

γ+

µ0

4π

NV∑j=1

∫

Vj

∂Jj(t)∂t

f j(rj)

| r − rj | dVj

+µ0

4π

NV∑j=1

∫

Sj

∂Mj(t)∂t

bj(rj)× nj

| r − rj | dSj +

+∇

4πε

NS∑m=1

∫

Sm

σ(rm, t)| r − rm | dSm

NV additional unknowns (magnetization currents have been added→ NV additional equations are required


PEEC modeling of magnetic materialsGalerkin’s testing procedure is applied to generate a discrete(circuit) problem

1ak

∫

Vk

F (rk) · fk(rk) dVk

−vOk = RkIk +NV∑j=1

Lp,kj∂Ij

∂t+

NV∑j=1

Lm,kj∂Mj

∂t

+NS∑

m=1

Qm(tm)(p+k,m − p−k,m)

In a matrix form yields:

−AΦ = V S + RI + LpdI(t)

dt+ Lm

dM(t)dt



The total magnetic field in the material is given by:

Hk = H ik + Hmk + Hsk =Bk

µk

where

• H ik is the magnetic field due to the electrical current(establishes the coupling with the EFIE)

• Hmk is the magnetic field due to the magnetization

• Hsk is the magnetic field due to the source current

Bik =NV∑j=1

λkjJj Bmk =NS∑j=1

αkjMj Bsk =N∑

j=1

βkjIsj



where vectors λkj and αkj are:

λkj =µ0

4π∇×

∫

Vj

f j (rj)

| rk − rj | dVj

and

αkj =µ0

4π∇×

∫

Sj

bj (rj)× nj

| rk − rj | dSj

where rj is the source point, rk is the observation point where Hk

is evaluated.

βkj depends on the source and must satisfy ∇ ·B = 0



The constitutive equation becomes:

AM = BI + CIs

where

A =

[α

µ0− α

µ− I

]

and

B =

[λ

µ− λ

µ0

]

C =

[1µ− 1

µ0

]U

where U is the unitary matrix.


Models for magnetic materials

The impact of magnetica phenomena over the EFIE is modeled bymeans of the time derivative of the magnetic vector potential as:

∂A(rk, t)∂t

=µ0

4π

∫

V ′

∂J(r′, t)∂t

1| rk − r′ | dV ′ +

+µ0

4π

∫

S′

∂[M(r′, t)× n′]

∂t

1| rk − r′ | dS ′

• Galerkin’s weighting procedure is applied (weighting over vol-umes+ projection) leading to the following definition of induc-tance Lm implementing the effects of magnetization currents.

Lm,kj =µ0

4π

1Sk

∫

Vk

∫

Sj

[bj(rj)× nj] · fk(rk)| rk − rj | dSjdVk



Such inductance is placed in series with the one describing the effectof electrical currents ⇒ the resulting equivalent circuit is topologi-cally identical to the standard one (non magnetic materials).The elementary PEEC cell becomes:

ji+ -

1-

iiP 1-

jjP

k kPL,

c iic ji

+ - + -

R k

kLi ,

å¹=

nN

nn

cn

ii

in iP

P

11

å= ¶

¶VN

j

j

kjpt

IL

1

, å= ¶

¶VN

j

j

kjmt

ML

1

,

å¹=

nN

jnn

cn

jj

jni

P

P

1



Considering the equivalent circuit, enforcing

• Kirchhoff current law (KCL) at each node

• Kirchhoff voltage law (KV L) at each loop

• Constitutive relation (CR)

the following system is finally obtained:

dΦ(t)dt − PAT I (t) = PIS (t) KCL

−AΦ (t) = V S (t) + RI (t) + LpdI(t)

dt + LmdM (t)

dt KV L

AM (t) = BI (t) + CIS (t) CR



Magnetization currents can be removed

M (t) = A−1 (BI (t) + CIS (t))

and KVL can be recast as:

−AΦ (t) = V S (t) + RI (t) +(Lp + LmA−1B) dI (t)

dt+ LmA−1CdIS (t)

dt

Thus, the PEEC method is re-formulated as the standard one pro-vided that additional partial inductances and current controlledvoltage sources are introduced.



Two different types of kernels are to be considered:

Type Ie−jk|r−r′|

|r − r′| Type II∂

∂h(e−jk|r−r′|

|r − r′| )

with h = x, y, z where r′ = [x′, y′, z′] is the source point and r = [x, y, z]

is the observation point.

• The computation of integrals by means of Gauss quadrature schemescan be extremely expensive for electrically large systems.

• This observation motivates the development of a set of acceleratedintegration schemes applicable to inner volume integration, whichsignificantly enhances the efficiency of the overall double folded vol-ume integrations.



The volume integration of the type I can be transformed into a surfaceintegration:∫

V ′

e−jkR

RdV ′ =

∑

j

hj

∫

∂jV ′

[1−jk

e−jkR

R2 − 1(−jk)2 (

e−jkR

R3 − 1R3 )

]dS′

where R is a vector from r′ to r, and R = ‖r − r′‖ = ‖R‖.



The volume integration of the type II can be transformed into a surfaceintegration:

∫

V ′

∂

∂h

e−jk|r−r′|

|r − r′| dV ′ = −∑

j

njh

∫

Sj

e−jkR

RdSj

where njhis either the x, y or z component of nj in h direction.


Volume2surface: type I kernel

Kernel Type I:e−jk0|r′−r|

|r′ − r|∫

V ′

e−jk0R

RdV ′

Order of gaussian integration: 10

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6x 10

−11

Frequency [GHz]

Typ

e I K

ern

el

VolumeSurface

Unphisical effect due to numerical inaccuracies

The surface integration allows to filter high frequency inaccuracieswhich may cause time domain instabilities


Volume2surface: type II kernel

Kernel Type II:∂

∂h

e−jk0|r′−r|

|r′ − r|∫

V ′

∂

∂h

e−jk0|r−r′|

|r − r′| dV ′

Order of gaussian integration: 10

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4x 10

−8

Frequency [GHz]

Volume

Surface

Unphysical effect due to numerical inaccuracies

Again, the surface integration allows to filter high frequency inac-curacies which may cause time domain instabilities


Volume vs surface: λ full-wave computation

λjk =µ0

4π∇×

∫

Vj

e−jk0|rj−rk|f j (rj)

| rj − rk | dVj

Current flowing along x, order of gaussian integration: 6

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5x 10

−6

Frequency [GHz]

B [T

]

λvol

λsurf

λstatic

• surface integration more accurate at high frequencies


λ full-wave computationCurrent flowing along z; order of gaussian integration: 7


Problems classification

• Magnetic problems

– input Bs → M s

– output M → B = Bm + Bs (Bi ≡ 0)

• Electric field integral equation

– direct coupling through the voltage induced by time varyingsource currents (−∂As(t)/∂t)

– indirect coupling through magnetization currents M

→ B = Bm + Bi + Bs

• Permanent magnets:

– M t = M + Mp

– B = Bm + Bi + Bs + Bp


Canonical problem

µ = 105 · µ0, I = 1 A

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

B distribution at 1 Hz

x [m]


3 columns transformer

µ = 105 · µ0, 40 windings, I = 1 A

0 0.05 0.1 0.15 0.2 0.250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2B distribution at 50 Hz

x [m]

z [m

]


Trace over a magnetic ground plane

µ = 103 · µ0

1.4 m

2 m

17.5 cm

17.5 cm

1 V

h

Short-circuit

(Courtesy of Peugeot)


Trace over a magnetic ground plane

Current distribution:left-top: f = 10 Hz, right-top: f = 10 kHz,

left-bottom: f = 10 MHz, right-bottom: f = 100 MHz.


Outline 2nd part

PEEC modeling of dispersive dielectrics

• Debye, Lorentz dielectric models• Generally dispersive media• Recursive convolution approach• Equivalent circuits approach• MNA stamps• PEEC solver including dispersive and lossy di-

electrics• Numerical results


PEEC Model Including Finite Dielectric Blocks

Equation for Total Electric Field

● KVL: v =∫

E · dl

Ei(r, t) =J(r, t)

σ+µ

∫

v′G(r, r′)∂J(r′, td)

∂tdv′+∇

ε0

∫

v′G(r, r′)q(r′, td)dv′

● KVL: Voltage = R I + s Lp I + Q/C + Vc

Vc is Excess capacitance volume term for dielectric

Ec(r, t) = εo(εr − 1)µ∫

v′G(r, r′)∂

2E(r′, td)∂t2

dv′


Basic PEEC Circuit Cell for Dielectrics

Coupled Loop For Two Basic PEEC Cells of aNon-Dispersive Dielectric Bar[7]

��

��

cc ii 21

p p11 22

1 1i i1 2

1 2

ic3

p1

33i 3

3Lp11 22LpC

1C

2

1 2 3

Excess capacitance Ck = ε0(εr−1)Sk

lk


Lossy Dielectric Model Issues

●● Lossy finite dielectric equivalent circuit models forPEEC

● Important: Models are Hilbert consistent.Means stable and passive equivalent circuits are gen-erated.

● Can include Debye narrow-band or wide-band mate-rials as required.

● Circuit solution means easy incorporation of MNAstamps into circuit solvers; uncoupled elements im-ply fast compute times.


Dispersive Dielectrics Model for PEEC

Frequency-domain permittivity function for a single pole Debyemedium

ε = ε0ε∞ + ε0εS − ε∞1 + sτ

The real and imaginary parts are Hilbert consistent (causality is guar-anteed)

The corresponding PEEC excess capacitance

Ce (s) = (ε− ε0)Sm

dm

= ε0Sm

dm

((ε∞ − 1) +

εS − ε∞s + 1/τ

)


Recursive convolution scheme

Time domain model via inverse Laplace transform and convolution

Ce (t) = ε0Sm

dm

(ε∞ − 1) δ (t) + ε0Sm

dm

(εS − ε∞)τ

e−t/τ

The charge qc (t) for excess capacitance due to voltage vc (t) needsconvolution of Ce (t) with vc (t)

qc (t) = Ce (t) ∗ vc (t) =∫ t

0Ce (t′) vc (t− t′) dt′

qc (t) = ε0Sm

dm

(ε∞ − 1) vc (t)+ε0Sm

dm

(εS − ε∞)τ

∫ t

0e−t′/τvc (t− t′) dt′


Recursive convolution scheme

By assuming t = n∆t and t′ = m∆t, its discrete counterpart reads

qc (n) = ε0Sm

dm

(ε∞ − 1) vc (n)+ε0Sm

dm

(εS − ε∞)τ

n−1∑m=0

e−m∆t/τ vc (n−m) ∆t

and after some manipulations

qc (n) = ε0Sm

dm

(ε∞ − 1) vc (n) + ε0Sm

dm

(εS − ε∞) vc (n)(1− e−∆t/τ

)

+ ε0Sm

dm

(εS − ε∞)τ

n−1∑m=1

vc (n−m) κ (m)


Recursive convolution schemewhere

κ (m) =∫ (m+1)∆t

m∆t

e−t′/τdt′ = τ(1− e−∆t/τ

)e−m∆/τ

which has the simplified form:

κ (m) = aemα

The recursive (at each time step) evaluation of Ψ for κ (m) is

Ψn =n−1∑m=1

aemαvc (n−m)

Ψn = aieαvc (n− 1) + eαΨn−1

Importance of rational kernels (≡ Debye-Lorentz models)


Equivalent Circuit Models

• Incorporation of lossy dielectrics in PEEC solver• Circuits are a convenient way• Which way is faster, convolution or circuit ?• Model circuit elements are local, works for both time

and frequency analysis• Make circuit for excess capacitances• More generality, χ (s)

Ce (s) = (ε (s)− ε0)Sm

dm

= ε0Sm

dm

[(ε∞ − 1) + χ (s)]



Debye medium

Ce (s) = (ε (s)− ε0)Sm

dm

= ε0Sm

dm

[(ε∞ − 1) +

(εS − ε∞)1 + sτ

]

sCe (s) = (ε (s)− ε0)Sm

dm

= ε0Sm

dm

[s (ε∞ − 1) +

s (εS − ε∞)1 + sτ

]

sCe (s) = sC∞ (s) + YRC (s)

The RC circuit parameters are:

CD = ε0Sm/dm (εS − ε∞)

RD = τ/ (ε0Sm/dm (εS − ε∞))



Debye medium equivalent circuit

RD

CD

CDe`

Static excess capacitance CeS = CD + CD∞ = ε0Sm/dm (εS − 1)

V consistent with static excess capacitance.


Volume Model for Dispersive Dielectrics

Debye Medium Equivalent Circuit

for PEEC Loss Model

e oo

Lp

RD C

DC

D



Lorentz medium

Ce (s) = (ε (s)− ε0)Sm

dm

= ε0Sm

dm

[(ε∞ − 1) +

(εS − ε∞) ω20

s2 + 2sδ + ω20

]

sCe (s) = s (ε (s)− ε0)Sm

dm

= ε0Sm

dm

[s (ε∞ − 1) +

s (εS − ε∞) ω20

s2 + 2sδ + ω20

]

sCe (s) = sC∞ (s) + YRLC (s)

CL∞ = ε0Sm

dm(ε∞ − 1) CL = ε0

Sm

dm(εS − ε∞)

RL = 2δdm

ε0Sm(εS−ε∞)ω20

LL = dm

ε0Sm(εS−ε∞)ω20



Lorentz medium equivalent circuitR

LL

LC

L

CL

vC

1

2 3

4

iCL

Static excess capacitance CeS = CL + CL∞ = ε0Sm/dm (εS − 1)


General Dispersive Model

General formula, frequency domain permittivity

ε (s) = ε0ε∞

+ ε0

ND∑m=1

(εDS (m)− εD∞ (m))1 + sτ (m)

+ ε0

NL∑m=1

(εLS (m)− εL∞ (m)) ω0 (m)2

s2 + 2sδ (m) + ω0 (m)2

ε∞ =ND∑m=1

εD∞ (m) +NL∑

m=1

εL∞ (m)

Static permittivity εS =∑ND

m=1 εDS (m) +∑NL

m=1 εLS (m)


Equivalent circuit for general dispersive medium

Excess equivalent admittance

sCe (s) = sC∞ (s) +ND∑m=1

YD (s) +NL∑

m=1

YL (s)

RL

,1L

L,1

CL

,1

CL

e`

,1

RL

,NL

LL

,NL

CL

,NL

CL

e`

,NL

RD

,1C

D,1

CD

e`

,1

RD

,ND

CD

,ND

CD

e`

,ND


MNA Stamps

RD

CD

1

2

3

iCD

vCD

CD

Cv

i

Geq

1 3i

vC

iSeq

Equivalent circuit for the excess capacitance of Debye medium andcorresponding time discrete equivalent circuit

Geq is function of circuit parameters and iSeq depends on circuitparameters and past values of voltages vc and vCD.


MNA Stamps

Table 1: MNA stamp for dispersive dielectrics

v1 v3

Geq −Geq

−Geq Geq

The right hand side (RHS) is characterized by the following stamp:

Table 2: RHS stamp for dispersive dielectrics

v1 v3

iSeq −iSeq


Numerical tests

• Debye medium:

1. Test D1: applied voltage - finite step;

2. Test D2: applied voltage - finite pulse;

Parameters:

Sm = 5 102m; dm = 10−3m; ε0 = 8.85 10−12 Fm

; ε∞ = 4.5; τ = 30ps; rise time =

10ps; fall time = 10ps; width = 0.1ns; dt = 0.4ps

• Lorentz medium:

1. Test L1: applied voltage - finite step;

2. Test L2: applied voltage - finite pulse;

Parameters:

Sm = 5 102m; dm = 10−3m; ε0 = 8.85 10−12 Fm

; ε∞ = 4.096; εS = 4.301; f0 = 39.5 109Hz; δ =

1.257 1012Hz; rise time = 10ps; fall time = 10ps; width = 0.1ns; dt = 0.4ps


Numerical tests

Legend

Std− Conv Standard Convolution

Rec− Conv Recursive Convolution

Static Static solution

BE Backward Euler

lsim Matlab function

stamp MNA stamp


Numerical tests. Debye medium: Test D1

0 1 2 3 4 5

x 10−10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−9

Time [s]

q c [C]

Std−convRec−convStaticBElsimstamp

0 1 2 3 4 5

x 10−10

0

1

2

3

4

5

6

7

Time [s]ic

d [A

]

Std−convRec−convBElsimstamp

Total charge RC branch current


Numerical tests. Debye medium: Test D2

0 1 2 3 4 5

x 10−10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−9

Time [s]

q c [C]

Std−convRec−convStaticBElsimstamp

0 1 2 3 4 5

x 10−10

−6

−4

−2

0

2

4

6

8

Time [s]ic

d [A

]

Std−convRec−convBElsimstamp

Total charge RC branch current


PEEC solver

ji+ -

å¹=

nN

inn

cn

ii

in iP

P

1

iv1-

iiP

å¹=

nN

jnn

cn

jj

jni

P

P

1jv

1-

jjP

å¹=

bN

mnn

nL

nmPdt

diL

1

,

,,

LiCg

+mmPL

,,c iic ji

ji+ -

å¹=

nN

inn

cn

ii

in iP

P

1

iv1-

iiP

å¹=

nN

jnn

cn

jj

jni

P

P

1jv

1-

jjP

å¹=

bN

mnn

nL

nmPdt

diL

1

,

,,

LimmPL

,,c iic ji

¥

RD CD

CD

PEEC elementary cell. Left: non dispersive dielectric; right:dispersive dielectric (Debye medium)


Full PEEC example

Figure 1: Microstrip geometry.


Full PEEC example

εS,k ε∞,k τk[ns]

pole 1 4.7 4.55 1.59

pole 2 4.55 4.40 0.159

pole 3 4.40 4.25 0.0159

pole 4 4.25 4.10 0.00159

Table 3: FR-4 Debye model parameters.


Full PEEC example: Debye medium

106

107

108

109

1010

1011

0

0.005

0.01

0.015

0.02

0.025

Frequency [Hz]

Loss

tangent

ND

=1N

D=2

ND

=3N

D=4

106

107

108

109

1010

1011

3.7

3.8

3.9

4

4.1

4.2

4.3x 10

−11

Frequency [Hz]

Mag

nitu

de(ε

) [F

/m]

ND

=1N

D=2

ND

=3N

D=4

FR-4 loss tangent and permittivity for increasing number of poles


Full PEEC example

Microstrip line: impact of dielectric losses

0 1 2 3 4 5 6

x 10−9

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Vol

tage

[V]

Non DispDisp−1 pole MNADisp−1 pole Rec−convDisp−3 poles MNA

0 1 2 3 4 5 6

x 10−9

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Vol

tage

[V]

Non DispDisp−1 pole MNADisp−1 pole Rec−convDisp−3 poles MNA

Port voltages. Left: input port voltage; right: output port voltage.


Impact of dielectric losses: pulse propagation

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time [ns]

Non dispDisp−4 poles

1 1.1 1.2 1.3 1.4 1.50.66

0.68

0.7

0.72

0.74

0.76

0.78

Time [ns]

Non dispDisp−4 poles

Left: the output voltage obtained with a dispersive dielectric antici-pates that of the non-dispersive case. Right: the output voltage witha dispersive dielectric shows larger losses than in the non-dispersivecase.


Lossy Dielectric Example

Meandering Board Line over a lossy substrate


Example Waveforms

Input and Output Waveforms for Meander TypeProblem

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(ns)

Volta

ge(V

)

Input Debye4Output Debye4


Example Output Waveforms

Comparison for Lossless, Debye 2 Pole and 4 Pole

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(ns)

Outpu

t Volt

age(V

)

losslessDebye 2Debye 4


Workshop on PEEC modeling

References[1] A. E. Ruehli, P. A. Brennan. Efficient capacitance calculations for three-dimensional multiconductor systems. IEEE

Transactions on Microwave Theory and Techniques, 21(2):76–82, February 1973.

[2] A. E. Ruehli. Equivalent circuit models for three dimensional multiconductor systems. IEEE Transactions on MicrowaveTheory and Techniques, MTT-22(3):216–221, March 1974.

[3] G. Antonini, M. Sabatini and G. Miscione. PEEC Modeling of Linear Magnetic Materials. In Proc. of the IEEE Int.Symp. on Electromagnetic Compatibility, Porland, OR, USA, August 2006.

[4] G. Antonini. PEEC modelling of Debye dispersive dielectrics. In Electrical Engineering and Electromagnetics, pages126–133. WIT Press, C. A. Brebbia, D. Polyak Editors, 2003.

[5] G. Antonini, A. E. Ruehli, A. Haridass. PEEC equivalent circuits for dispersive dielectrics. In Proceedings of Piers-Progressin Electromagnetics Research Symposium, pages 767–770, Pisa, Italy, March 2004.

[6] G. Antonini, A. E. Ruehli, A. Haridass. Including dispersive dielectrics in PEEC models. In Digest of Electr. Perf.Electronic Packaging, pages 349 – 352, Princeton, NJ, USA, October 2003.

[7] A. E. Ruehli and H. Heeb. Circuit models for three-dimensional geometries including dielectrics. IEEE Transactions onMicrowave Theory and Techniques, 40(7):1507–1516, July 1992.


Workshop on PEEC Modeling - LTU

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