Albert Ruehli. IBM Research Division, T. J. Watson Research Center. Yorktown Heights, NY 10598. Abstract. The PEE'C approach is a full wave electromagnetic.
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
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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
