Partial Element
its Application
Equivalent
Circuit
in the F’requericy
(PEEC) Method
and
and Time Domain
Albert Ruehli
IBM Research Division, T. J. Watson Research Center
Yorktown Heights, NY 10598
Abstract
The PEE’C approach is a full wave electromagnetic
electrical modeling technique foT conductors embedded
in arbitrary dielectrics in terms of equivalent circuits.
The models can be used in both the time as well as the
frequency domain. It facilitates the solution of problems which have both an electromagnetic part as well
as a circuit part. Also, it leads to an intuitive understanding of electromagnetic problems.
1
Introduction
The solution of electromagnetic problems is formulated in terms of circuits in the Partial Element Equivalent Circuit (PEEC) approach [l] instead of the usual
field solution and field variables. The variables are
quantities like the potential ‘P, and some other circuit variables like i, v, q, where i is the current, v the
voltage, and q some charges etc.. The basic formulation is an Electric Field Integral Equation (EFIE)
full wave solution to Maxwell’s equations. The models evolved first in terms of partial inductance calculations for VLSI inductance calculations [2]. The full
PEEC models were developed from the study of partial inductances and the method of moments (MOM)
techniques [3]. Recently, the PEEC model has been
extended to accurately include models for dielectrics
[4] and also scattering or incident fields [5].
PEEC models can be employed in both the frequency as well as the time domain. So far, we have
been able to extend all the new developments of the
method to both domains without any restrictions. In
the time domain a large system of delay differential
equations (DDE) result. Many realistic problems, may
also include a nonlinear part which isbest solved in the
time domain. The time domain solution of EFIE has
traditionally
been plagued by instabilities [6] and the
O-7803-3207-5/96/$5.00
0 1996 IEEE
PEEC models are no exception. Much progress has
been made in this area and we believe that the problem has been solved [7]. The solution of the PEEC
models can be obtained in different ways. Many techniques like simplified PEEC models, macromodels and
special circuit formulations can be adapted for the efficient solution of PEEC model problems. Macromodeling is a key technique for the solution of very large
problems. These techniques are used to model complex and challenging problems as is evident from work
done in this area e.g. [S]. Attention has been given
recently to the solution of large circuit models which
result from the PEEC approach e.g. [9].
In this paper we cover several aspects of PEEC
circuit models. Section 2 describes the basic circuit
models and the derivation of the PEEC model from
Maxwell’s equations. Section 3 shows circuit models
and aspects of PEEC model. Stability and time domain issues are given in Section 4. Finally, Section 5
considers model simplification and some example results are given.
2
Circuit
Based EM
Models
Circuit type models have been in use in EM for
many years. The most widely used circuit model is
the lumped element Differential Equation (DE) model
for transmission lines [lo]. An interesting aspect is
the fact that the longitudinal direction is represented
by a DE circuit model while the transverse direction
for the coupling is represented in terms of an integral
equation(IE) model which is in fact related to a PEEC
model. The main difference is that the PEEC model
is a full wave model while the transmission line model
supports TEM modes only.
We give an abbreviated derivation of the PEEC
model which includes finite size dielectric regions besides the conductors or wires [4]. The starting point is
128
Figure 1: Two cell example
the PEEC model for conducting objects [l] and some
of the work on dielectrics for the MOM e.g. [ll]. We
specifically concentrate on approximations of the currents, charges and potentials (voltages) which are appropriate for an equivalent circuit representation. The
key idea is to treat the displacement current due to the
bound charges for to the dielectrics with er > 1 separately from the displacement currents due to the free
charges. We start from the sum of all the sources of
electric,,field [lo] at any point in space
Figure 2: Complete PEEC model for two cell example
The scalar potential
is similarly
cl,)dvr
(7)
JVI,F')q(rr,
G(i=,
where & is a potential applied electric field, 9 is the
current density in the conductors, d and 9 are vector and scalar potentials, respectively. As indicated
above, the dielectric areas are taken into account as a
current rather than a capacitance through the scalar
potential.
T-his is accomplished by adding and subtracting ~0% in the Maxwell equation for B, or
vxIT=JfeO(er-l)~+Eg~
Finally, using the above we can formulate an integral
equation for the electric field at a point F which is to be
located either inside a conductor or inside a dielectric
region. Starting from Eq. 1 with the externally applied
electric field set to zero, and substituting for A and 0
from Eq. 4 and Eq. 7 respectively
qe, t)
u
J(F, t) = &(F, t) + E&
- l)z
ai.
= /.A G(F, Ff)f(i;‘, td)dti!
J VI
where the retardation
which simply
points F and
derived here,
functions are
+
EO(G - l)p
%&,,
G(F, Fl)q(Ti,
(3)
where jc~(F,t) is the conductor current and the remainder of the equations is the equivalent polarization
current due the the dielectrics.
The vector potential d is for a single conductor at
the field point ? given by
&=,t)
G(F, i;r) “(;
p
(2)
Here, the current in Eq. 2 is written as a total current
JVI ,-,(F
;,)
d2E(er,
td)
1
d-t2
JVI
=0 (8)
JVI t&h/
+
(4
Note that Eq. 3 is used for the total current in both
the conductors and the dielectrics.
To solve the integral equation Eq. 8 numerically, we
need appropriate approximations to the current, potential, and charge variables. We do not use the usual
high frequency skineffect approximation but we use a
more accurate approximation to the internal current
flow which is detailed in [12]. This implies that we
cannot utilize the continuity equation
time is given by
is the free space travel time between the
or. It is noted that in the formulation
both the retardation and the Green’s
free space quantities where
(6)
to replace all the charge variables directly with the
current variables as is done in the moment type solutions [3]. The charges are, in all the solution approaches, assumed to be located on the conductor surfaces. In our approach, we implement the continuity
equation in the form of Kirchoff’s current law in the
circuit equation solution.
129
Three coupled integral equations (I.E.) result if we
represent the vector quantities in terms of the Cartesian coordinates. For this case, the vector quantities
are f = J,O + Jy$ + J,2 and l? = E,S + Eyfi + E,&.
Using this, the three I.E. are identical in form for all
three space directions z, y, z.
A very simple example geometry is given in Fig. 1
which is a metal strip. It should be noted that this
problem cannot be solved with the usual transmission
line circuit models. The aim is to find an equivalent
circuit where the unknowns are the potentials as well
as the partial inductance currents.
We must use pulse functions matching for a
Galerkin solution. The inner product is formed with
respect to a cell of conductor cy
-1
au
.
J
f(r)czv
=
vp
;
J
i
a,
m
da dl
1, i4”,
11
11
m
3
Tm
F
coupling to cell Q for a current 1~ in the cell /?. The
dielectrics are represented with additional circuit elements. We define the excess capacitance of a dielectric
cell as
(10)
For the second and third terms in Eq. 8 we make
use of the laminar, uniform current flow through the
cell to take
$-ail/at
integral. With some help from
the second term represents the
of the conductor cell CYIwhere
between two parallel cells a! and
JJ
hap
v.vB
Lp,p = -k-
II T 22
j;
Figure 3: Voltage source capacitance model
where er is the dielectric constant of the dielectric cell.
The equivalent circuit for the dielectric bar or cell y
is given by a partial inductance Lprr in series to a
capacitor 6’: with the current i = C+ w.
where
aa and <Ps are the potential at thi two ends of the
dielectric cell. This model is a very important addition
to the PEEC approach [4].
3
outside the
ognire that
inductance
inductance
by
m
I
;
oL
where v, is the volume of the conductor cell Q, a, is
the cross-section of the cell (z - y direction) and I,
is the length (z-direction) and f(F) is the integrand.
This corresponds to averaging the field point f over the
cell volume vcr. From the result, we can assert what
the appropriate equivalent circuit is for each term of
the IE, corresponding to Eq. 8. The first term of the
IE can easily be shown to be the resistance of the cell
R, with the dc current 1,” = v
since the current
through the cell is Iz = Jza,. The resistance is as
usual
a&
-=
ai
iji;
G(fa, $)dvadvp
(12)
[2], we recpartial self
the partial
p are given
where the first term is the partial self inductance of
the cell cz and.the second term represents the inductive
Circuit
Models
Provided that a solver exists, equivalent circuit solutions represent a very flexible approach for the solution of practical problems. We give PEEC models in
this section for both ,the time domain as well as the
frequency domain. The most efficient model for the
two cases differ in the circuit used to represent the
capacitive effects. A complete model for the example
metal conductors shown in Fig. 1 is given in Fig. 2
[5]. The model for the partial inductances is rather
straightforward following the basic implementation of
PEEC models [l]. We show in more detail how the capacitive effects are modeled. The general form of the
capacitive term is Cp= PQ where P is the coefficient
of potential matrix. The case where the retardation
cannot be ignored is very interesting in terms of the
capacitive model. Two different forms of the capacitive equivalent circuit are most appropriate depending
on whether the PEEC model is applied in the time or
frequency domain. The most efficient model for the
MNA time domain formulation is the one shown in
Fig. 2. The controlled sources are derived from the
(13)
For the Cartesian coordinates considered here, all perpendicular cells have zero mutual inductances. Hence,
the inductive terms are simply interpreted as
Equivalent
130
PEEC - Model
+
“’
Source ve+
w
Figure 4: Simple PEEC model for one cell
above coefficient of potential equation with the relationship between current and charge i = $.
The
capacitive current is for K capacitive cells
where i,, is the total capacitive current for cell m and
from Eq. 5 the retardation time is
(17)
where &,, is the distance between conductor cells m
and n and c is the speed of light. We measure the distance K&, between two points on two different cells m
and n. The second circuit model for the retarded coefficients of potential shown in Fig. 3 is more suitable for
a loop oriented formulation like the MLA in [12]. This
is the preferred formulation for the frequency domain
since it leads to the minimum number of unknowns.
This formulation without additional circuit elements
included corresponds to a Galerkin MOM method [3].
The PEEC model for this case is the same as the one
given in Fig. 2 where the capacitive part is replaced
with the one in Fig. 3. It can easily be proven that
the ThCvenin equivalent circuit has voltage controlled
voltage sources which are given by
(18)
where s@):,is the potential across the self capacitance
n, shown in Fig. 3. Both formulations use the same P
matrix and the same delays.
Figure 5: PEEC model for dipole antenna
4
Model
Iution
stability
and time
domain
so-
Stability is a key problem for time domain solution, for both the MOM and PEEC formulations e.g.
[S] and PEEC formulation.
In the time domain, the
model instability will be observed as an unstable solution which will totally mask the real solution starting
at some point in time. The confusing aspect of the
stability issue is that there may be two different contributing factors which may cause instabilities. First,
the discretized model may be unstable. However, this
can be corrected by adjusting the delay times Eq. 17
in the model shown below and by using other techniques which improve the model stability [7]. The
second issue for the stability is the time integration
methods used for the integration of the resultant Delay Differential Equation (DDE) [13]. We cannot determine from an unstable solution which one of the
sources caused the problem. However, a carefully implemented method leads to a stable time domain solution. The key issue for an efficient time domain formulation for the PEEC model is to introduce all the state
variables which are required without introducing additional unknowns. A condensed modified nodal analysis (MNA) matrix can be found by utilizing Kirchoff’s
131
DIPOLE (45cm x 9.0cm)
1
DescriDtion
*
t
Full Model
All delays small
Resistance small
High Currents
Chip power,Gnd
Low Currents
Chip, Signals
PEEC Model
(W,,W
Ignore all 7
(W,,P)
or (R,L,,C)
(Lp,P,~) or (Lp,C)
WP)
or (LPI
(R,P)
or (R,C)
4.08 _........______......
i ........._..__.....
$ _..___...............
w .............+....................
-009 ........._...........
j ................+... ..................p.....- ......t ................. .
-8dOOOO
4O;OOO
's.OdOOO
12.0bOOO
Frtlquency (GHr)
16.0bOOO
5
Figure 6: Impedance of patch dipole
current law at each node to eliminate unnecessary currents. This is possible due to the fact that the internal
nodes of the PEEC model have a very specific known
structure [5]. The values of the partial inductances Lp
for the model are found using the formulas in [2]. We
give a simple example for the complete time domain
formulation which corresponds to the one cell model
in Fig. 8. The corresponding PEEC model is shown
in Fig. 9 and the time domain equations are
where I means that its variable is evaluated at t - r.
Equation (19) can be rewritten as a Delay Differential Equation (DDE) in an MNA form similar to [9]
which is
G(t) = Py(t) + Qy(t - T) + ~(4 t - T)
(20)
where P,Q and u are evident. The model for larger
problems with multiple delays leads to the equation
for a PEEC model in the general form
Qiy(t - 7i) + c
jr = Py + c
i
zci(t - pi)
Table 1: PEEC Model Reduction
20.00000
(21)
i
For the time domain solution of the DDE, a specially
stable integration method needs to be used. The result of work in this area shows that explicit integration
methods are only marginally stable. Implicit integra
tion methods which have excellent stability properties
like the backward Euler or the Lobatto III-C have to
be used. It is clear that the solution of the implicit
system is much slower than the explicit ones.
Simplified
PEEC
models
and results
Macromodels obtained by pruning an existing
PEEC model are used quite frequently. We denote a
full model by the notation (R,L,,P,r,V,)PEEC
where
the terms mean that the quantities are included in
the PEEC part of the model. The notation which
indicates the presence of at least one element of a particular type is R for series resistances, Lp for partial
inductances, P for the coefficients of potential, C for
capacitances, T for delay and VP for potential incident
field sources [5]. Table 1 gives most of the situations
for reduced circuit PEEC models. A small delay 7 implies that for the maximum angular frequency of the
spectrum w, (UT) << 1. This may be due to a small
geometry or due to a sufficiently low frequency w. A
substantial reduction in model size may be achieved in
situations where these approximations apply. In the
(L,,R)PFEC
case we use the L, and the R matrices
only and the circuit equations degenerate to
V(s) = RI(s) + sL;l(s)
The admittances
(22)
are easily found from
I(i)
= (R + sLp)-lV(s)
(23)
This model may be further reduced if the resistance
is small compared to the partial inductances for the
frequencies of interest. Then a frequency independent
reduction results
I(s) = f(Lp)-lV(s)
(24
which leads to a simple solution. It is evident from the
extensive use of inductance calculators in the EIP and
EIP areas for noise calculations that many practical
problems can be solved with (Lp) or (Lp, R)PEEC
models. Such calculations have been used for years
132
three dimensional multiconductor systems. IEEE
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PI A.
E. Ruehli. Inductance calculations in a complex integrated circuit environment.
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0
0.2
0.4
0.6
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Ruehli and H. Heeb.
Circuit
models
for three-dimensional geometries including dielectrics. IEEE Trans. on i&Tow.
Theory and
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1
time (ns)
Figure 7: Current response for simple PEEC model
where the model complexity has been drastically reduced by, for example, including much simplified capacitances in the model. The reduction of models using systematic macromodeling techniques is an area of
intensive research activity today e.g [9]. W e conclude
the paper by giving a simple result in both the frequency and the time domain. Figure 5 gives the most
simple PEEC model for a dipole antenna. Figure 6
shows the input impedance of a patch dipole antenna
(4.5 cm wide and 9 cm long) with a comparison between the M O M solution in [ll] and a PEEC model.
The dashes curves are the imaginary parts and the
solid and dotted curves the real parts. As is evident,
the results are very close. The second example is a
time domain solution for a 0.5 cm by 2 cm patch excited by a pulse function with a PEEC model similar
to Fig. 4. The result in Fig. 7 shows the convergence
of the current waveform with the number of cells along
the length.
6
Conclusions
and Acknowledgements
The PEEC method offers several additional features above the conventional field results due to the
circuit oriented solution method. The author would
like to acknowledge many discussions with B. Rubin
as well as the frequency domain results obtained from
the collaboration with J. Garrett and C. Paul and the
time domain results obtained form the work with W .
Pine110 and A. Cangellaris.
[51 A. E. Ruehli, J. Garrett, C. R. Paul. Circuit models for 3d structures with incident fields. In Proc.
IEEE Id. Symp. on Electrom. Cornpat., pages
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PI B.
P. Rynne. Comments on a stable procedure in
calculating the transient scattering by conducting
surfaces of arbitary shape. IEEE Trans. on Ant.
and Propag., APP-41(4):517-520, April 1993.
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discretized partial element equivalent efie circuit
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by Moment
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133