Recent progress and applications in partial element equivalent circuit modeling

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RECENT PROGRESS AND APPLICATIONS IN PARTIAL ELEMENT EQUIVALENT CIRCUIT

MODELING

Jonas Ekman

Email: Jonas.Ekman@csee.ltu.se Lule˚ a University of Technology

SE-971 87 Lule˚ a Sweden

This paper presents a number of recent developments and applications within electromagnetic modeling using the PEEC method. The recent extensions of the classical PEEC formulation to handle nonorthogonal geometrical ob- jects and frequency dependent dielectric losses are discussed together with numerical examples. Further, the time complexity for current PEEC based EM solvers, both quasi-static, full-wave, time- and frequency- domain, are detailed. We show that the bottleneck for future PEEC based EM solvers are the partial element calculation procedure. Last, applications of the PEEC method to new areas including modeling of embedded passives and antenna analysis is discussed.

1 Introduction

The partial element equivalent circuit (PEEC) [1]-[3] method was developed at IBM by A. E. Ruehli to facilitate VLSI inductance calculations [1] in the early 1970s and is nu- merically equivalent to a full-wave method of moments solution with Galerkin matching.

In the method, the electromagnetic problem is transformed into the circuit domain and solved using circuit analysis techniques. This enables a combined solution of electronic functionality and electromagnetic effects. The method is often used for electrical inter- connect and package analysis but has evolved over the years with new application areas continuously reported.

2 Nonorthogonal PEEC Formulation

This section gives an outline of the geometrical nonorthogonal extension to the PEEC method. The objects can be quadrilateral and/or hexahedral conductors or dielectrics which can be both orthogonal and/or nonorthogonal.

The starting point for the derivation is the total electric field at the material which is E ~

ⁱ

( ~ r

g

, t) = J( ~ ~ r

g

, t)

σ + ∂ ~ A( ~ r

g

, t)

∂t + ∇φ( ~ r

g

, t) (1)

where ~ E

ⁱ

is the incident electric field, ~ J is the current density in a conductor and ~ A and φ

are vector and scalar potentials, respectively. As indicated above, the dielectric areas are

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taken into account as an excess current rather than a capacitance with the scalar potential using the volume equivalence theorem. This is accomplished by adding and subtracting

²

⁰^{∂ ~}_∂t^E

in the Maxwell equation for ~ H, or

∇ × ~ H( ~ r

g

, t) = ~ J( ~ r

g

, t) + ²

⁰

(²

r

− 1) ∂ ~ E( ~ r

g

, t)

∂t + ²

⁰

∂ ~ E( ~ r

g

, t)

∂t . (2)

In (2), the current is written as a total current

J( ~ ~ r

g

, t) = ~ J

C

( ~ r

g

, t) + ²

0

(²

r

− 1) ∂ ~ E( ~ r

g

, t)

∂t (3)

where ~ J

C

is the conductor current and the remainder of the equations is the equivalent polarization current due to the dielectric.

By using the vector potential ~ A and the scalar potential φ using the free space retardation time we can formulate an integral equation for the electric field at a point ~ r

g

which is to be located either inside a conductor or inside a dielectric region. Starting from (1) with the externally applied electric field set to zero, and substituting for ~ A and φ. The final integral equation to be solved is

~ˆn × ~ E

ⁱ

( ~ r

g

, t) = ~ˆ n ×





J( ~ ~ r

g

, t) σ





+ ~ˆ n ×





µ

Z

v0

G( ~ r

g

, ~ r

g

0) ∂ ~ J( ~ r

g

0, t

d

)

∂t dv0





+ ~ˆ n ×





²

0

(²

r

− 1)µ

Z

v0

G( ~ r

g

, ~ r

g

0) ∂

²

E( ~ ~ r

g

0, t

d

)

∂t

²





(4)

+ ~ˆ n ×

·

∇

²

0

Z

v0

G( ~ r

g

, ~ r

g

0)q( ~ r

g

0, t

d

)dv0

¸

where ~ˆ n is the surface normal to the body surfaces.

2.1 Interpretation as circuit equivalent

To represent the current flow in orthogonal cells, we use the convenient weighting function J = I/(W T ) where I, J are the current and current density and W, T are the cell width and thickness, respectively. The generalization of the current distribution for nonorthog- onal hexahedral cell shapes is given by

J

a

= h

a

¯

∂ ~rg

∂a

· (

^{∂ ~}_∂b^r^g

×

^{∂ ~}_∂c^r^g

)

^¯^¯_¯

I

a

(5)

where J

b

and J

c

can easily be found by permuting the indices. We call the quotient in (5)

the weight w

a

which simplifies (5) to J

a

= w

a

I

a

. We should note that all the above

quantities are a function of the local position coordinates a, b, c. Next, we use an integral

or inner product operator V

a

=

^R_a^R_b^R_c

w

a

~ˆa · ~ E

^¯^¯_¯^{∂ ~}_∂a^r^g

· (

^{∂ ~}_∂b^r^g

×

^{∂ ~}_∂c^r^g

)

^¯^¯_¯

da db dc to integrate the

terms of (4) where ~ E(a, b, c) is the ~ E field term to be integrated. We need to apply the

inner product to each term in (4) to transform each term to a voltage drop across a circuit

element in the KVL equation. Next, we integrate the right hand terms of the integral

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equation (4). After applying the inner product, the first element on the right hand side leads to the series resistance term in the form

R

a

= 1 σ

Z

a

Z

b

Z

c

h

a 2

¯

∂ ~rg

∂a

· (

^{∂ ~}_∂b^r^g

×

^{∂ ~}_∂c^r^g

)

^¯^¯_¯

da db dc (6) The second right hand side term of (4) gives a generalization of the partial inductance concept for nonorthogonal problems, or

Lp

aa⁰

= µ

Z

a

Z

b

Z

c

Z

a⁰

Z

b⁰

Z

c⁰

(~ˆ a · ~ ˆ a

⁰

) h

a⁰

G[ ~ r

g

(a, b, c); ~ r

g

(a

⁰

, b

⁰

, c

⁰

)] da

⁰

db

⁰

dc

⁰

h

a

da db dc (7) From the third right hand side term of (4), the nonorthogonal PEEC excess capacitance expression is extracted, see further [4]. Finally, the fourth right hand side term of (4) we get for the normalized coefficients of potential

P n

aa⁰

= 1

²

Z

a

Z

b

Z

a⁰

Z

b⁰

G[ ~ r

g

(a, b, c); ~ r

g

0(a

⁰

, b

⁰

, c

⁰

)] da0 db0 da db (8) Here a, b, a

⁰

, b

⁰

are interpreted in a general sense for the appropriate cell orientations.

P_ii

1 I_Pⁱ P_jj

1 I_P^j

Lpmm Rmm

I_Cⁱ I_C^j

a I b

Vmm^L

Figure 1: Basic PEEC cell.

In the PEEC circuit solution, terminal or nodal variables are associated with each of the re- sultant circuit elements and are collected in an overall circuit solver vector of unknowns.

The solution vector variables are quantities like the potentials ~ Φ, and other conventional circuit variables like i, v, q where~i is the currents, ~v the ~ voltage, and ~q some charges etc. If no other cir- cuit elements are included, the solution vector used is (~ Φ,~i)

^T

as the only unknowns since this

vector directly yields the most useful EM circuit output variables. Here we point out that in part, the inclusion of both ~ Φ and ~i is, responsible for the full spectrum property of the solution methodology such that the correct dc solution is obtained.

2.2 Example

To demonstrate the nonorthogonal PEEC formulation we consider the non-uniform trans-

mission line (TL) depicted in Fig. 2 (a). The TL is 100 mm. long (L), the width (W)

varies from 5 to 9 mm., the thickness (T) of the conductors are 0.1 mm., and the two

conductors are spaced 10 x T. The TL is differentially excited, at the near-end, using a 1

ns., 1 mA current pulse and terminated, at the far-end, in a 50 Ω resistor. The structure

is studied using three different PEEC models. First, the structure is modelled using the

original PEEC formulation with orthogonal surface and volume cells (giving a stair-case

approximation of the geometry) entitled Orthogonal, coarse mesh in Fig. 2 (b). Second,

the first model was refined using a much larger number of surface and volume cells (to

make a better representation of the geometry), this test is entitled Orthogonal, fine mesh

in Fig. 2 (b). Finally, the TL is modelled using the nonorthogonal formulation using

the same number of cells as for the first test, see Table 1 for details on cell count. The

nonorthogonal test is entitled Nonorthogonal, coarse mesh in Fig. 2 (b). The results in

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Table 1: Detail for non-uniform TL test.

PEEC model PEEC model

Coarse mesh Staircase approximation Ind. cells

self 10 mutual 45

Ind. cells

self 490

mutual 119 805 Cap. cells

self 20

mutual 190

Cap. cells

self 300

mutual 44 850

Fig. 2 (b) shows clearly that the response from the nonorthogonal PEEC formulation is more similar to the orthogonal formulation using a large number of cells compared to the first test using a coarse discretization.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

0 0.01 0.02 0.03 0.04 0.05 0.06

Time [/ns]

[/V]

Near−end voltage

Orthogonal, coarse mesh Orthogonal, fine mesh Nonorthogonal, coarse mesh

(a) (b)

Figure 2: Nonorthogonal TL geometry (a) and comparison for near-end voltages using orthogonal (staircase approximation) and nonorthogonal PEEC formulation (b).

3 Frequency dependent lossy dielectric formulation

The inclusion of finite, lossy dielectrics in the PEEC method is detailed in [5] by the use of excess capacitance of a dielectric cell. Since the frequency dependent dielectric losses are becoming more important, the PEEC method was recently extended [6] to deal with dielectrics described in terms of a frequency dependent permittivity, ε(s), which can be approximated by a sum of Lorentz and Debye terms. Assuming N

D

to be the number of Debye terms and N

L

to be the number of Lorentz terms, the frequency dependent permittivity get the following representation in the Laplace domain

ε(s) = ε





ε

∞

+

N_D

X

m=1

(ε

DS

(m) − ε

D∞

(m)) 1 + sτ (m) +

N_L

X

m=1

(ε

LS

(m) − ε

L∞

(m))ω(m)

²

1 + sδ(m) + ω

⁰

(m)

²





(9)

where ε

∞

=

^P^N_m=1^D

ε

D∞

(m) +

^P^N_m=1^L

ε

L∞

(m), ε

DS

(m), and ε

LS

(m) are the static dielectric

constants for the m-th Debye and Lorentz medium, ω

⁰

(m) is the resonance frequency of

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R

_D

C

_D

C

D 8

b a

R

L

C

L

C

L 8

L

a b

(a) (b)

Figure 3: Debye (a) and Lorentz (b) media equivalent circuits.

the m-th Lorentz medium.The loss constant of the m-th Debye term is τ (m). The static permittivity of such a medium can be defined as ε

S

=

^P^Nm=1^D

ε

DS

(m) +

^P^Nm=1^L

ε

LS

(m). The admittance corresponding to the PEEC excess capacitance can be represented as

sC

e

(s) = sC

∞

(s) +

ND

X

m=1

Y

D

(s) +

NL

X

m=1

Y

L

(s) (10)

which can be synthesized to an equivalent circuit and easily included in the PEEC solver.

By using equivalent circuits for the Debye and Lorentz media, as detailed in Fig. 3, the MNA stamp is easily realized by the use of an equivalent admittance, G

eq

, and a current source, i

Seq

. This gives the following MNA stamp for a dielectric cell connected between node α and β

α . β . .

α . β .







G

eq

−G

eq

−G

eq

G

eq













φ

¹

φ

α

. φ

β

~i

1

.

~i

nv







=







. i

Seq

.

−i

Seq

v

S₁

. v

Sn







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where φ

n

is the solution of node potentials, ~i

n

is the solution of volume cell currents, and v

Sn

are external voltage sources (excitation).

4 Time Complexity for PEEC Solvers

It is relevant to estimate the order of complexity for each part in a PEEC, full-wave solver.

For this purpose, Table 2 details the serial time complexity for the serial, full-wave, time- and frequency- domain, MNA based PEEC formulation outlined in [7]. In the table, the variable n = (n

C

+ n

L

) is the number of unknowns (node voltages and volume cell currents) hence the number of columns and rows of the MNA system matrix. However, for a general PEEC model, n

²

is not the maximum number of partial elements, since the specific problem at hand affects the sparsity of the partial inductance matrix. For example, the number of partial elements for a 2D (L

p

, R, P )PEEC model of a plate with same number of partition in the x- and y- directions would be

(n

L

− 1)

²

+ 8n

L

4 + (n

C

− 1)

²

+ 2n

C

2 , (12)

(6)

equal to O(m)

²

complexity (m = max(n

L

, n

C

)). From (12) it is evident that the number of nodes (coefficients of potential) contributes substantially to the total number of partial elements for a PEEC model.

Table 2: Serial time complexity for full-wave PEEC solvers.

Moment FD PEEC TD PEEC

Time complexity Time complexity Meshing

↓

Element calculation

↓

System build-up†

↓

System solution

O (n) O (n

²

)

^?

O (n

²

)

^¦

O (n

^iter

n

²

)

O (n) O (n

²

) O (n

¹^.5

) O (n

^iter

n

¹^.5

)

?

Complex elements in the FD updated at each frequency of interest.

¦

System updated at each frequency of interest.

†

Creating MNA coefficient sub-matrices.

Traditionally, the time complexity of solving PEEC based EM models, in both time- and frequency- domain, has been dominated by the solution of the resulting linear system of the form

Ax = b (13)

where the coefficient matrix A is (1) a large scale, dense, and complex valued matrix for frequency domain modeling and (2) a large scale, sparse, and real valued matrix for time domain modeling. The progress within linear equation solving has made it possible to solve the FD PEEC systems on the order of O(n

^iter

n

²

) if the convergence is achieved in n

^iter

iterations by using a Krylov subspace method such as the BiConjugate Gradient method. The required computations of the matrix-vector products, in some Krylov sub- space methods, at each iteration account for the major computational cost of this class of methods. However, by using the fast multipole method (FMM) [8] the complexity of the matrix-vector products can be reduced from O(n

²

) to O(n

¹^.5

). With the multilevel fast multipole algorithm (MLFMA) [9] the complexity is further reduced to O(n log n). To further improve the solution of the FD PEEC systems, the convergence rate of the iter- ative, Krylov subspace methods, can be improved by using a preconditioning technique.

However, many of the preconditioning techniques have been developed in the context of

iteratively solving linear equations involving large sparse matrices. In the electromag-

netic scattering simulation field, the diagonal and block diagonal preconditioners have

been considered. To note is that many preconditioning techniques rely on a fixed spar-

sity pattern, obtained from the sparsified coefficient matrix by dropping small magnitude

entries [10]. This is not to be preferred when using the PEEC method due to possible

stability problems, especially for full-wave PEEC modeling. For the solution of the sparse

TD PEEC systems iterative methods can be used with a time complexity on the order of

O(

^iter

n

¹^.5

).

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5 Applications in PEEC Modeling

There are several new application areas within PEEC modeling. For example the mod- eling of parasitic effects in the generic 64-lead quad-flat-pack (QFP) [11], optimization design of LTCC filters [12], and the analysis of arbitrary conductor-magnet structures [13]. However, there are some application areas that are specially interesting including modeling of embedded passives, railway structures and antennas as detailed below.

One of the challenges in radio frequency integrated circuits (RFIC) is the fabrication of high-Q spiral inductors. Due to geometrical complexity of planar or quasi- planar inductors, accurate computation of the inductance is usually difficult and time consuming.

In [14], regular (L

p

, R)PEEC models are used in the calculations showing reasonably good agreement compared to measurements. In [15], evaluation of the inductance of spiral inductors is performed by using results previously obtained from the method of moments.

The calculated values are compared against measurements and other techniques with good agreement.

The PEEC method was used in [16] to study simple EM effects in local railway structures with some comparison to measurements. In that paper, the ground was discretized using dielectric cells. However, for the analysis of railway, or power-like, structures the PEEC method could be used with layered media Green’s functions to further speed up the modeling.

Due to the equivalent circuit formulation utilized in the PEEC method, analyzing anten- nas in which lumped components are used can be especially suitable. For example, planar inverted F antennas that uses LC resonators to interconnect the antenna elements and to tune the antenna resonance frequencies. Further, the nonorthogonal PEEC formulation gives new possibilities for the analysis of conformal antennas [17] in a straightforward manner.

6 Summary

This paper details recent progress within PEEC based EM modeling including the nonorthogonal formulation and the inclusion of frequency dependent lossy dielectric by the use of equivalent circuits. The nonorthogonal PEEC model is suitable for modeling of complex geometries by the use of quadrilateral and hexahedral elements in the meshing.

However, the formulation requires numerical (space) integration in the calculation of the partial elements and this limits the applicability for large problems, as shown in Sec.

4. The recent applications of PEEC modeling includes several new areas within RFIC and power system EM analysis. The possibility to use PEEC with layered media Greens functions opens up new possible application areas for future PEEC based EM modeling.

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