1) The document summarizes a presentation given at the 2007 ACES Conference on using the Partial Element Equivalent Circuit (PEEC) technique to model frequency dependent phenomena like skin effect. 2) It describes using volume filaments and macro-basis functions to model skin effect in conductors with thickness between the skin depth. Broadband macromodels are generated from a frequency domain PEEC solver. 3) The presentation outlines modeling skin effect using an analytic solution of Maxwell's equations to derive impedance terms dependent on conductor thickness and frequency, and examines the asymptotic behavior of the model in the low- and high-frequency limits.

1. 2007 ACES Conference
Combined Loss Mechanism and Stability
Model for the Partial Element Equivalent
Circuit Technique
Giulio Antonini
UAq EMC Laboratory, Department of Electrical Engineering
University of L’Aquila, AQ 67040, ITALY
Albert Ruehli
IBM T. J. Watson Research Center
Yorktown Heights, NY 10598, USA
Verona, March 22, 2007 Slide 1 of 33

2. Outline
q PEEC modeling of skin-effect
• volume filament approach
• 1D macro-basis functions
• higher order basis functions
q Diffusion equation
q Broadband combined loss and stability models
q PEEC equivalent circuit
q Numerical results
q Conclusions and future work
Verona, March 22, 2007 Slide 2 of 33

3. Frequency dependent phenomena modeling
• Frequency dependent phenomena (skin-effect, dielectric losses)
– easier to be modeled in the frequency domain.
• Stability and passivity issues:
– easier to be handled at a macromodel level.
• Macromodels can be generated from either time-domain or
frequency domain solvers.
• Fast techniques (FMM, MLFMM, QR) are well established for
frequency domain solvers.
• A viable solution is: use frequency domain analysis to generate
samples to be used for macromodeling (Adaptive Frequency
Sampling can be useful).
Verona, March 22, 2007 Slide 3 of 33

4. Advanced frequency domain PEEC solver
Broadband
macromodel
PEEC frequency
domain solver
Frequency dependent phenomena
(skin-effect, dispersive
and lossy dielectrics)
Fitting techniques Acceleration techniques
powered by FMM, MLFMM, QR
AFS algorithm
Verona, March 22, 2007 Slide 4 of 33

5. Overview of skin effect modelling
Key parameters
2
• skin-depth δ = µσω
• layer thickness t
• if t δ the field diffusion can be neglected and the layer can be
approximated as a PEC surface
• if t δ the tangential electric field is virtually constant through the
layer: the sheet can be modeled by a surface resistance 1/σt;
• if t ∼ δ the spatial resolution inside the conductor has to be very fine,
=
much shorter than the skin depth ⇒ memory and time consuming
Target
• a symmetric, two-way macromodel which is valid for arbitrary t/δ
ratios
Verona, March 22, 2007 Slide 5 of 33

6. PEEC modeling of skin-effect
Skin-effect modelling
• Volume filament approach: large number of unknowns
• Non-uniform discretization helps in reducing the number of unknowns
• Macro-basis functions represent another possibility
Verona, March 22, 2007 Slide 6 of 33

7. PEEC modeling of skin-effect
z I1
d
I d y
2
I1
x
I2
W
J (r, s)
i
E (r, s) = + sA(r, s) + φ(r, s)
σ
V1 (s) = Z11 I1 + Z12 I2 + Lp11 sI1 + Lp1j sIj
j
V2 (s) = Z21 I1 + Z22 I2 + Lp22 sI2 + Lp2j sIj
j
Verona, March 22, 2007 Slide 7 of 33

8. PEEC modeling of skin-effect
    
V1 Z11 Z12 I1
 =  
V2 Z21 Z22 I2
The 2D and 3D cases are a combination of the 1D case.
l
z
y
x
Iy w
Ix
Hypothesis: transverse electric field distribution
Verona, March 22, 2007 Slide 8 of 33

9. PEEC modeling of skin-effect
Ex (z) = Ae−γz + Beγz Jx (z) = σ Ae−γz + Beγz
Since we associate a current with each cell, we can relate the current I1
to the the electric field as
d d
I1
e−γz dz + B eγz dz
=A
σW 0 0
0 0
I2
e−γz dz + B eγz dz
=A
σW −d −d
After integration
I1 γ
= −A(e−γd − 1) + B(eγd − 1)
σW
I2 γ
= A(eγd − 1) − B(e−γd − 1)
σW
Verona, March 22, 2007 Slide 9 of 33

10. PEEC modeling of skin-effect
Jx (s, z) = σEx (s, z) = σ Ae−γz + Beγz
γ 1
eγz − e−γz+γd I1
=
W (e−2γd − e−γd + eγd − 1)
e−γz − e−γd+γz I2
+
= f1 (s, z)I1 + f2 (s, z)I2
For a conductor translated along the z axis the two basis functions de-
scribing current density profile in the same direction are:
γ 1
eγ(z−zmax2 ) − e−γ(z−zmin2 )
f1 (s, z) =
W (e−2γd − e−γd + eγd − 1)
γ 1
e−γ(z−zmin1 ) − eγ(z−zmax1 )
f2 (s, z) =
W (e−2γd − e−γd + eγd − 1)
Verona, March 22, 2007 Slide 10 of 33

11. PEEC modeling of skin-effect
The voltage across the top cell is given by integrating the electric field as
L
V1 (s) = Ex (s, d)dx = LEx (d)
0
which results in
eγd − e−2γd e−γd − 1
γL
V1 (s) = I1 + I2
σW den den
where den = e−2γd − e−γd + eγd − 1.
Finally, the impedances are identified as:
eγd − e−2γd
γL
Z11 (s) = Z22 (s) =
σW (e−2γd − e−γd − 1 + eγd )
e−γd − 1
γL
Z12 (s) = Z21 (s) =
σW (e−2γd − e−γd − 1 + eγd )
Verona, March 22, 2007 Slide 11 of 33

12. Asymptotic behavior
Limit as γ → 0
3L
lim Z11 (s) = lim Z22 (s) =
2 σW d
γ→0 γ→0
1L
lim Z12 (s) = lim Z21 (s) = −
2 σW d
γ→0 γ→0
At DC currents are equal, I1 = I2 = I:
L
RDC = lim [Z11 (s) + Z12 (s)] =
σW d
γ→0
Verona, March 22, 2007 Slide 12 of 33

13. Asymptotic behavior
Limit as γ → ∞
lim Z12 (s) = lim Z21 (s) = 0
γ→∞ γ→∞
⇒ currents I1 and I2 are decoupled.
γL
lim Z11 (s) = lim Z22 (s) = lim
γ→∞ σW
γ→∞ γ→∞
√
√
Considering that for good conductors γ = jωµσ = (1 + j) πf µσ =
(1 + j) /δ, the standard surface impedance is obtained.
Verona, March 22, 2007 Slide 13 of 33

14. Thin and thick objects
Thin object Thick object
1-d macro PEEC model
2-d macro PEEC model
Standard PEEC model
• Middle region: standard PEEC model adopting brick basis function
• side regions: 1D macro PEEC model adopting 1D macro-basis func-
tions
• corner regions: 2D macro PEEC model adopting 2D macro-basis
functions
Verona, March 22, 2007 Slide 14 of 33

15. Macro-basis functions
The basis functions can be used as testing functions within a Galerkin’s
procedure.
Let F (r, s) represent a generic vector term in the electric field integral
equation:
J (r, s)
i
E (r, s) = + sA(r, s) + φ(r, s)
σ
The inner product to test this equation is defined as:
f ∗ (s, z) · F (r, s)dr j
F (r, s), f i (s, z) = i
Vj
where Vj is the j − th elementary volume.
Verona, March 22, 2007 Slide 15 of 33

16. Macro-basis functions
J (r) = f j (s, r)Ij
j
1
f ∗ (s, r) · f j (s, r) dr i dr j
Rij = i
σ Vj Vi
e−jk0 |r−r |
µ
f ∗ (s, r)
Lij = · f j (s, r) dr i dr j
i
4π |r − r |
Vj Vi
Verona, March 22, 2007 Slide 16 of 33

17. Diffusion equations
Inside a good conductor (e.g. σ ωε) the displacement current can be
neglected and the fields can be described in terms of diffusion equations.
∂Hy (z, s)
= −σEx (z, s)
∂z
∂Ex (z, s)
= −sµHy (z, s)
∂z
sµ
Zc (s) =
σ
√
γ (s) = sµσ
The diffusion of magnetic and electric fields in the conductor can be
interpreted as diffusion of voltage and current in a LG transmission line.
Verona, March 22, 2007 Slide 17 of 33

18. Diffusion equations
Two-way macromodel
    
E Z (s) Z12 (s) H
 x,1  =  11   y,1 
Ex,2 Z21 (s) Z22 (s) Hy,2
  
sµ(s)
coth γ (s) t − sµ(s) csch γ (s) t H
  y,1 
 γ(s) γ(s)
=
− sµ(s) csch γ (s) t sµ(s) coth γ (s) t Hy,2
γ(s) γ(s)
where Ex,1 , Ex,2 are proportional to V1 , V2 and Hy,1 , Hy,2 are proportional
to the currents on side 1 and 2.
Verona, March 22, 2007 Slide 18 of 33

19. Stability
If the diffusion of magnetic and electric fields in the conductor are inter-
preted as diffusion of voltage and current in a LCG, (R = 0) transmission
line, poles can be analytically computed, by using a modal expansion, as:
nπ 2
2
1 G 1 G t
pn,12 =− ± −
2 C 4 C LC
• all the poles have a negative real part ⇒ stability is ensured
• for large G (good conductors)
∼ 0, pn,2 ∼ − G , ∀n
pn,1 = =
C
⇒ the poles are multiple (numerically)
Verona, March 22, 2007 Slide 19 of 33

20. Broadband stable PEEC models for skin-effect
and magnetic-electric field couplings
Frequency dependent partial inductances ⇒ damping in the partial
elements (Kochetov, Wollenberg, 2005)
Distributed approach-suitable for time domain implementation
e−s|r m −r n |/c0
µ
Lp,mn (s) = dvm dvn
4πam an |r m − r n |
vm vn
nL c
Np
p
Resc ∗
Resr Resc
ZL,mn (s) = sLp,mn (s) = dL + k k k
+ +
mn
s − pc ∗
s − pr s − pc
k k k
k=1 k=1
• Stability can be enforced during the fitting.
• Passivity is enforced as a-posteriori step.
• A large number of lumped elements is needed to synthesize
ZL,mn (s).
Verona, March 22, 2007 Slide 20 of 33

21. Broadband stable PEEC models for skin-effect
and magnetic-electric field couplings
ZI Z V Lp
12 2 11 L 11
I1
+ +
V1 V
2
− −
 
np R11 R12
p
d11 + d12 +
k=1 s−pk k=1 s−pk
Z(s) =  
np R21 R22
p
d21 + d22 +
k=1 s−pk k=1 s−pk
Adaptive Frequency Sampling (AFS) can be useful to generate broadband
macromodels keeping the order np and the number of samples as low as
possible.
Verona, March 22, 2007 Slide 21 of 33

22. Global approach-frequency domain
implementation+fitting
• the macromodel allows to reduce the number of unknowns
• the macromodel is incorporated into a frequency domain solver
• acceleration techniques are used to speed-up the solution at each
frequency
• frequency samples are used to generate a macromodel of the
overall system via fitting techniques
Verona, March 22, 2007 Slide 22 of 33

23. Single conductor modeling with symmetrical boundary conditions
t L
W
• length L = 1 mm
• width W = 150 µm
• thickness t = 2d = 30 µm
• electrical conductivity σ = 5.8 · 107 S/m
Frequency range 0-50 GHz
Symmetrical boundary conditions ⇒ currents I1 = I2 = I = 1 mA.
Surface model: sµ/σL/W .
Verona, March 22, 2007 Slide 23 of 33

24. Single conductor modeling with symmetrical boundary conditions
V (s) = Z11 (s) I1 (s) + Z12 (s) I2 (s) = [Z11 (s) + Z12 (s)] I (s)
˜
Zse (s) = Z11 (s) + Z12 (s) Zse (s) = RDC + Zs (s)
−1
10
Z
s
RDC
Z
11
−2
10
−3
10 −3 −2 −1 0
10 10 10 10
Frequency [GHz]
Comparison of impedance Zs (s), the DC resistance RDC and Z11 (s)
Verona, March 22, 2007 Slide 24 of 33

25. Single conductor modeling with symmetrical boundary conditions
Surface model Zs (s) = sµ/σL/W : well established skin-effect
˜
Zse (s) = Z11 (s) + Z12 (s) Zse (s) = RDC + Zs (s)
1
10
Z +Z
11 12
Zs+RDC
0
10
−1
10
−2
10
−3
10 −5 −4 −3 −2 −1 0 1 2 3
10 10 10 10 10 10 10 10 10
Frequency [GHz]
Comparison of the exact impedance Zse (s) = Z11 (s) + Z12 (s) with the
˜
approximated one Zse (s) = RDC + Zs (s)
Verona, March 22, 2007 Slide 25 of 33

26. Single conductor modeling with symmetrical boundary conditions
Transfer impedance Z12 = Z21 , f = 0 − 50 GHz
0
10
−5
10
−10
10
−15
10
−20
10
−25
10 −5 −4 −3 −2 −1 0 1 2
10 10 10 10 10 10 10 10
Frequency [GHz]
With the increasing of the frequency the coupling between the two half-
cells rapidly decreases.
Verona, March 22, 2007 Slide 26 of 33

27. Single conductor modeling with symmetrical boundary conditions
I1=I2=1 mA
6
x 10
2.5
2
1.5
J [A/m2]
x
1
0.5
0
2
1.5
1
1
0.5
0
−4
0
x 10
−0.5 −5
−1 x 10
−1
−1.5
−2 −2
x [m]
z [m]
Left: electric field profile as a function of z coordinate and frequency;
right: current density distribution over the cross section.
Verona, March 22, 2007 Slide 27 of 33

28. Single conductor modeling with symmetrical boundary conditions
Frequency dependent basis functions
10 10
x 10 x 10
3.5 3.5
3 3
2.5 2.5
2 2
1.5 1.5
1 1
0.5 0.5
0 0
−2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
z [m] z [m]
−5 −5
x 10 x 10
f = 30 GHz
Verona, March 22, 2007 Slide 28 of 33

29. Single conductor modeling with symmetrical boundary conditions
Frequency dependent basis functions
Verona, March 22, 2007 Slide 29 of 33

30. Resistance and internal inductance
−1
10
Rs
RDC
Rhf
−2
10
−3
10 −3 −2 −1 0
10 10 10 10
Frequency [GHz]
Rs = Re(Z11 + Z12 )
Verona, March 22, 2007 Slide 30 of 33

31. Resistance and internal inductance
−8
10
L
i
Li,hf
−9
10
−10
10
−11
10
−12
10
−13
10 −6 −4 −2 0 2 4
10 10 10 10 10 10
Frequency [GHz]
Li = Im(Z11 + Z12 )/jω
Verona, March 22, 2007 Slide 31 of 33

32. Comparisons of per unit length resistance and internal inductance
t L
W
f = 100 MHz
W ×t Ri [Ω/m] (MoM) Ri [Ω/m] (Macro)
1.4 mil × 15 mil 4.8480 4.8504
W ×t ωLi [Ω/m] (MoM) ωLi [Ω/m] (Macro)
1.4 mil × 15 mil 4.0287 4.0232
Reference: G. Antonini, A. Orlandi, C. R. Paul, Internal Impedance of
Conductors of Rectangular Cross Section, IEEE Tran. on Microwave and
Techniques, vol. 47, n. 7, July 1999.
Verona, March 22, 2007 Slide 32 of 33

33. Conclusions and future works
• Macro-model for conductor modeling in the PEEC frame-
work
– New macro-basis functions have been computed;
– a two-way symmetric macro-model has been developed
The proposed method:
• can be easily combined with frequency-dependent models of par-
tial elements (partial inductances and coefficients of potential);
• allows a reduction of the number of unknowns;
• models the two-way coupling (broadband modeling);
• can be used in a framework different from PEEC (TLs, power-
bus structures).
Verona, March 22, 2007 Slide 33 of 33