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
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
Lulea, November 13, 2007 Slide 2 of 59
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
Lulea, November 13, 2007 Slide 3 of 59
PEEC modeling of magnetic materials
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 ′
Lulea, November 13, 2007 Slide 4 of 59
PEEC modeling of magnetic materials
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)
Lulea, November 13, 2007 Slide 5 of 59
PEEC modeling of magnetic materials
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
Lulea, November 13, 2007 Slide 6 of 59
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
Lulea, November 13, 2007 Slide 7 of 59
PEEC modeling of magnetic materials
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
Lulea, November 13, 2007 Slide 8 of 59
PEEC modeling of magnetic materials
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
Lulea, November 13, 2007 Slide 9 of 59
PEEC modeling of magnetic materials
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.
Lulea, November 13, 2007 Slide 10 of 59
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
Lulea, November 13, 2007 Slide 11 of 59
Models for magnetic materials
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
Lulea, November 13, 2007 Slide 12 of 59
Models for magnetic materials
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
Lulea, November 13, 2007 Slide 13 of 59
Models for magnetic materials
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.
Lulea, November 13, 2007 Slide 14 of 59
Efficient computation of integrals
The evaluation of parameters α, λ and Lm calls for volume andsurface integration:
λkj =µ0
4π∇×
∫
Vj
f j (rj)
| rk − rj | dVj
αkj =µ0
4π∇×
∫
Sj
bj (rj)× nj
| rk − rj | dSj
Lm,kj =µ0
4π
1Sk
∫
Vk
∫
Sj
[bj(rj)× nj] · fk(rk)| rk − rj | dSjdVk
Lulea, November 13, 2007 Slide 15 of 59
Efficient computation of integrals
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.
Lulea, November 13, 2007 Slide 16 of 59
Efficient computation of integrals
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‖.
Lulea, November 13, 2007 Slide 17 of 59
Efficient computation of integrals
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.
Lulea, November 13, 2007 Slide 18 of 59
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
Lulea, November 13, 2007 Slide 19 of 59
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
Lulea, November 13, 2007 Slide 20 of 59
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
Lulea, November 13, 2007 Slide 21 of 59
λ full-wave computationCurrent flowing along z; order of gaussian integration: 7
Lulea, November 13, 2007 Slide 22 of 59
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
Lulea, November 13, 2007 Slide 23 of 59
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]
Lulea, November 13, 2007 Slide 24 of 59
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
]
Lulea, November 13, 2007 Slide 25 of 59
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)
Lulea, November 13, 2007 Slide 26 of 59
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.
Lulea, November 13, 2007 Slide 27 of 59
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
Lulea, November 13, 2007 Slide 28 of 59
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′
Lulea, November 13, 2007 Slide 29 of 59
Basic PEEC Circuit Cell for Dielectrics
Coupled Loop For Two Basic PEEC Cells of aNon-Dispersive Dielectric Bar[7]
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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
Lulea, November 13, 2007 Slide 30 of 59
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.
Lulea, November 13, 2007 Slide 31 of 59
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
)
Lulea, November 13, 2007 Slide 32 of 59
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′
Lulea, November 13, 2007 Slide 33 of 59
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)
Lulea, November 13, 2007 Slide 34 of 59
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)
Lulea, November 13, 2007 Slide 35 of 59
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)]
Lulea, November 13, 2007 Slide 36 of 59
Equivalent Circuit Models
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∞))
Lulea, November 13, 2007 Slide 37 of 59
Equivalent Circuit Models
Debye medium equivalent circuit
RD
CD
CDe`
Static excess capacitance CeS = CD + CD∞ = ε0Sm/dm (εS − 1)
V consistent with static excess capacitance.
Lulea, November 13, 2007 Slide 38 of 59
Volume Model for Dispersive Dielectrics
Debye Medium Equivalent Circuit
for PEEC Loss Model
e oo
Lp
RD C
DC
D
Lulea, November 13, 2007 Slide 39 of 59
Equivalent Circuit Models
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
Lulea, November 13, 2007 Slide 40 of 59
Equivalent Circuit Models
Lorentz medium equivalent circuitR
LL
LC
L
CL
vC
1
2 3
4
iCL
Static excess capacitance CeS = CL + CL∞ = ε0Sm/dm (εS − 1)
Lulea, November 13, 2007 Slide 41 of 59
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)
Lulea, November 13, 2007 Slide 42 of 59
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
Lulea, November 13, 2007 Slide 43 of 59
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.
Lulea, November 13, 2007 Slide 44 of 59
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
Lulea, November 13, 2007 Slide 45 of 59
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
Lulea, November 13, 2007 Slide 46 of 59
Numerical tests
Legend
Std− Conv Standard Convolution
Rec− Conv Recursive Convolution
Static Static solution
BE Backward Euler
lsim Matlab function
stamp MNA stamp
Lulea, November 13, 2007 Slide 47 of 59
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
Lulea, November 13, 2007 Slide 48 of 59
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
Lulea, November 13, 2007 Slide 49 of 59
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)
Lulea, November 13, 2007 Slide 50 of 59
Full PEEC example
Figure 1: Microstrip geometry.
Lulea, November 13, 2007 Slide 51 of 59
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.
Lulea, November 13, 2007 Slide 52 of 59
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
Lulea, November 13, 2007 Slide 53 of 59
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.
Lulea, November 13, 2007 Slide 54 of 59
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.
Lulea, November 13, 2007 Slide 55 of 59
Lossy Dielectric Example
Meandering Board Line over a lossy substrate
Lulea, November 13, 2007 Slide 56 of 59
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
Lulea, November 13, 2007 Slide 57 of 59
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
Lulea, November 13, 2007 Slide 58 of 59
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.
Lulea, November 13, 2007 Slide 59 of 59