Electromagnetic computer modelling for EMC (integral methods)

R.F. Milsom

Indexing terms: Electromagnetic compatibility; Computer-aided design

Abstract: The paper reviews methods for com- puter modelling of electromagnetic fields with par- ticular application to EMC. The interconnection system of a piece of electronic equipment, which comprises PCB, cables, packages etc. is described as an additional component, which acts as both a source and receiver of stray electromagnetic fields. This component, which in general has extremely complex shape, is modelled by solving Maxwell’s equations. Emphasis is given to integral methods of generating the appropriate model. Integrating this model with simulators used in electronic design, thus allowing combined electronic/ electromagnetic modelling is also discussed. Cur- rently available methods of numerical analysis and computer hardware restrict application of many tools to small subsets of full EMC prob- lems. Various approximations which allow realis- tic structures to be modelled are therefore examined. An application of the Philips’ PCB layout simulator FASTERIX is given.

1 Introduction

All electronic equipment, when in operation, generates electromagnetic energy. Similarly, all such equipment could receive energy via either a conduction or radiation path from a source. This mutual interaction may be either wanted, as in the case of transmission and recep- tion of radio signals, or unwanted in which case it is called EM1 (electromagnetic interference).

Equipment is said to have EMC (electromagnetic compatibility) [l] if the unwanted energy it emits and its susceptibility to incident energy are below prescribed levels. The introduction of the EC Directive on EMC [Z] and its associated standards, together with increased use of electronics and the consequent emission levels and crowding of the electromagnetic spectrum, have made it imperative to treat EMC as central to the product design process.

In recent years, CAD (computer aided design) tools have been used in most areas of design, e.g. mechanical, PCB and IC layout, analogue circuit analysis, logic simu- lation and synthesis, and mixed mode simulation. By contrast, an empirical approach has been used to tackle EMC, with ‘fixes’ based on experience being applied at a late stage of development. However, the complexity and importance of the subject is now at a level that such an

Q LEE, 1994 Paper 1186A (EZ), received 22nd October 1993 The author is with Philips Research Laboratories, Redhill, Surrey RHI SHA, United Kingdom

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approach is no longer satisfactory. To improve design for EMC it is necessary to predict electromagnetic field effects within or near equipment, whether such fields be self generated or incident from the environment. This requires new CAD tools based on 3D electromagnetic field analysis, and integration with other analysis tech- niques and tools used in electronic simulation.

Commercially available software has limited capability in relation to the requirement. This can be understood by considering the boundary conditions imposed by the extraordinarily complex geometrical configurations involved. A piece of electronic equipment can be con- sidered as a set of discrete and integrated conventional components, and an additional component which makes up the interconnections and encapsulations, referred to as the interconnection system. In general this extra com- ponent comprises PCBs, connectors, cables, screens and packages, forming a set of complex shaped conductors and insulators.

Currents are induced in the conductors of the inter- connection system by electrical sources within the equipment and also by externally generated incident fields. These currents and the electromagnetic field excited by them may cause EMI. In order to perform computer simulations of this interference it is necessary first to derive a mathematical model of the fields associ- ated with the interconnections, and second, to determine a method of interfacing this to models of the other com- ponents.

The required model is based on a solution of Maxwell’s equations, subject to the boundary conditions imposed by geometry and material properties. Because of the complexity involved, numerical methods must be used to find an approximate solution. However, the geometries concerned are so complex that currently available computer hardware and numerical techniques limit solutions to relatively small subsets of the total problem. The derivation of accurate approximations is therefore crucial.

A wide variety of numerical methods has been devel- oped for electromagnetic field analysis. Most of this work predates concerns about EMC, and was used for applica- tions ranging from electric motors at a few Hz to radar scattering problems at tens of GHz. Numerical methods

The author would like to acknowledge the support of the UK Department of Trade and Industry under the EUREKA programme EU127, JESSI Subprogram ACS, and help from Dr. K.J. Scott and A.T. Yule of Philips Research Labor- atories UK, and Ir. G.P.J.F.M. Maas, Dr. A.J.H. Wachters and Dr. R. du Cloux of Philips Research Laboratories Eindhoven.

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for 3D field analysis can be broadly divided into two types; differential and integral. All methods discretise the continuum problem and hence transform the field equa- tions into a system of linear equations. In general, differ- ential methods (e.g. finite element method [3]) require the generation of a mesh over the volume domain of the problem, while integral methods (e.g. method of moments [4]) require a mesh over a much smaller domain, nor- mally just conductor surfaces.

The number of linear equations which must be solved at each frequency is proportional to the number of volume or surface elements in the mesh. With the complex geometrical shapes that occur an accurate solu- tion requires a fine mesh and therefore a very large system of equations. Fig. 1 illustrates this complexity

Fig. 1 PCB metallisation pattern and discretisation

with a 2D mesh generated for one metallisation layer of a small PCB. With differential methods several hundreds of thousands of unknowns would be required for the problem shown, whereas for integral methods the number is a few thousand. By contrast, the left hand side matrix is sparse in the former case but, in principle, full in the latter. One very significant advantage of the surface integral approach is that the domain which must be meshed is finite. In a differential approach this domain is infinite, since fields are radiated throughout space, so in practice an artificial boundary is introduced, and this must be at a large distance from the region of real inter- est to minimise artifacts. This increases the number of elements and therefore the number of unknowns.

Integral methods are of prime concern here. Despite their advantages, conventional integral techniques still require massive computing resources. Many ways of speeding up the computations have been studied. In the following sections, first, the strategy for modelling the total problem, which requires integration of different models, is considered. Second, the integral method is described in general terms, and third, some of the approximation techniques are discussed.

2 Integration of models

Since the objective of EMC simulation is to predict the behaviour of electronic equipment in the presence of elec-

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tromagnetic fields, it is necessary to develop a model of the interconnection system which can be coupled to models of other components. When EMC is not a con- sideration circuit analysis tools are used to model ana- logue parts and logic simulators to model digital parts. Mixed mode simulators couple these two approaches using special interfacing techniques. From an EMC point of view digital circuits must be treated as analogue. Circuit models are therefore used to characterise the logic drivers and inputs to digital blocks. The model derived from field analysis to characterise the interconnection system must then be in a form consistent with analogue circuit analysis, whether the application is digital, ana- logue or a mixture of both.

Let the interconnection system be connected to other components, analogue or digital, at N points or nodes (typically pads on a PCB). Then the model for this inter- connection component must provide a relationship, valid over a range of time or frequency, between the vector of N currents I and the vector of N node voltages V . Here voltage is defined with respect to some reference, typic- ally an ideal ground plane or point on a non-ideal ground plane. The combination of node and reference form a port.

Assume, for the moment, that there is no field due to external sources, but there are one or more electrical sources within the equipment. Then, since the intercon- nection component may also be assumed linear, a given frequency component of current vector is related to the correspondng component of voltage vector by

I ( 0 ) = Y(w)V(w) (1)

where o is angular frequency, and Y is the order N complex admittance matrix which characterises the inter- connections. Harmonic time dependence exp (jolt), where f is time, is common to both I and V and all field vari- ables and is therefore dropped from all relevant equa- tions. Clearly, from eqn. l, the jth column of Y may be obtained by applying the port boundary conditions

to the interconnections in the absence of other com- ponents, then solving for the electromagnetic fields and extracting the set of node currents from this solution.

In principle, all ports are electromagnetically coupled, therefore Y is a full matrix. However, in practice, subsets may be suficiently decoupled to allow a partitioning of the problem, or at least introduce some sparsity into the matrix.

If the circuit simulator used allows N-port admittance matrix models, and only frequency domain analysis is required, then it is suficient to determine Y ( 0 ) from the field analysis and download it to the simulator. If an external field is incident on the structure then it can be shown that this can be represented in the model by a set of equivalent current sources connected at the A' ports.

In many EMC problems time domain analysis is required. This is because spurious signals and radiation are often the result of intermodulation or harmonic dis- tortion, caused by the presence of non-linear com- ponents. Although the interconnection system itself may be considered linear, its model must be usable for analysis involving other components which are non- linear. Two approaches to treating this problem are pos- sible. First, Fourier transformation and the convolution

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theorem may be applied to eqn. I , or other equations derived from it, giving time domain relationships of the form

i(t) = y ( t ~ t ‘ )u( t ’ ) dt‘ I:= (3)

where y is an impulse response matrix which is related to the Fourier transform of Y , and i and U are the Fourier transforms of I and V . Alternatively, a lumped equivalent circuit having admittance matrix approximately equal to Y over the frequency range of interest may be derived as discussed below. Several special purpose simulators based on the convolution approach have been developed. The lumped model has the advantage that it can be used in most general purpose analogue simulators, since it is just a network of standard frequency-independent passive components: resistors, capacitors, self and mutual induc- tors. Both time and frequency domain simulation are then possible.

These methods give solutions for the effect of both external fields and internal coupling on electrical per- formance. However, with EMC, the spatial distribution or some spatial integration of (one or more frequency components of) the emitted field may also be required. Since the admittance matrix is determined by extracting node currents from the set of N field solutions defined above, it follows that the field at any point in space or time may be found by appropriate superposition of these solutions weighted by the node voltages determined from the circuit analysis.

Field solutions may be fully dynamic [4] or quasi- static [SI. In the former case, the far field radiation is implicit in the solution. In the latter case only the near field is included. At frequencies of concern in many EMC problems (typically below I GHz) this quasi-static approximation is sufficiently accurate to model the current distribution. The radiated field is then determined to good accuracy as a second order correction.

3 integral equation analysis

The integral method is now considered in general terms. Under certain circumstances Maxwell’s partial differen- tial equations of electromagnetism may be transformed to one of a variety of integral formulations. These include the IFIE (electric field integral equation), MFIE (magnetic field integral equation) and CFlE (combined field integral equation) [SI. Only the EFIE is considered as this has been shown to be well suited to EMC prob- lems. Again omitting harmonic time dependence, the EFIE has the general form,

c c , .

(4)

where r and r’ are position vectors, E is electric field, J is current density, C is the domain of the conductors and G is a Dyadic Green’s function [7] appropriate to the geo- metrical configuration concerned, typically the PCB layer structure. From eqn. 4 it can be seen that the Green’s function is equal to the electric field due to a unit Hertz- ian dipole embedded in this structure. It will be recalled from the discussion above that solutions for J(r). and hence port currents, are required for particular boundary conditions on the port voltages. In order to introduce these boundary conditions it is necessary to define a

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relationship between the circuit variable voltage and field variable electric field E . Thus, for example, the voltage at port i is defined by the line integral,

E(r) dl I: v = -

over some path between the reference and the ith node. Strictly, this voltage depends on the selected path so care is needed over the definition. To minimise uncertainty, the path should be electrically short. To obtain the required field solution a localised incident source field Ej”(r) is introduced at port j , and defined to satisfy the relationship,

- i I f E $ ) . dl = 1

This source field is also defined to be zero outside the source region, but via conduction it excites the intercon- nection system giving rise to a current distribution J,{r). From eqn. 4 this can be seen to generate a scattered field E:s’(r) given by,

Ey’(r) = J^ 1 G(r I r’)J,{r’) dr’3 (7)

To satisfy continuity at the source the scattered field just outside the source region must match the source field just inside this region over port j . In order to obtain the required short circuit port currents, ideal conductors are introduced across each port. Thus the full set of conduc- tors includes both the interconnections and these short circuits. The solution for current density in conductors is expanded over a set of M basis functions h,(rXm = 1 . . M ) , each typically associated with a particular element. Thus

where the ajm are complex constants to be determined. Substituting eqn. 8 into 7 the scattered field is then

where the integral is over the domain of the mth basis function. Since the interconnection conductors are non- ideal, i.e. have finite local conductivity U(*), and the port conductors have infinite local conductivity, then applying the field matching condition and Ohm’s law

to the conducting regions

m = : I a,, j jm G(r I J)k, (r ’ ) dr‘3

= Ey’(r) over port j

= 0 over port i (i # j )

over interconnect conductors (1 1 )

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To find the ujm a set of M test functions e,(r) (n = 1, . . . , M ) is introduced and the method of moments then gives

"

or = 0

(n = 1, ..., M ) (12)

where the choice of right hand side depends on whether the particular e, is defined over mesh element(s) at port j, port i (i # j ) or the interconnection conductors, respect- ively. In one embodiment of MOM, known as the Galer- kin Method, the sets of test and basis functions are identical. This is usually more accurate than other imple- mentations but not necessarily more efficient. Eqn. 12 forms a linear system which is solved for the ajm ( j = 1, ..., N , rn = 1, ..., M). From the solution for the net port currents the jth column of Y is given by.

Yij = j k j r ) dr* (i = 1, . . . , N ) (13)

where the integral is over a surface normal to the ideal conductor connected across port i. This process is re- peated with excitation at each port in turn, thus provid- ing all columns of Y(w). Solutions are obtained for all required w. The frequency dependence of Y derives from the frequency dependence of G.

When used in a circuit simulator the combination of admittance matrix of the inter-connection system, models of other components and source models yields a solution for branch currents and node voltages, and in particular the voltages 6 ( j = I , . . . , N ) across the ports of the inter- connection system. Applying the superposition principle then gives the net scattered field,

N M r r r

where G may be a different approximation to the Green's function, e.g. it may include radiated fields as well as evanescent fields, whereas that used in the solution for currents may only include the latter.

When an external field E")(r) is incident on the inter- connection system, its effect can be solved for in a manner similar to that described above to determine Y except that it is the net field [E")(r) + E@'(r)] which must satisfy eqn. 10. The vector of equivalent current sources I, which will model this incident field is given by the solu- tion obtained for the short circuit currents at the N ports of the interconnection system with no other components connected. These current sources, which appear across the ports in the circuit simulation, are of course different for each incident field. Their values are downloaded to the circuit simulator along with the solution for Y . The

solution for the net field is then the superposition of E"'(r) with eqn. 14.

4 Approximation techniques

Although suitable for some EMC simulations direct application of the method of moments is too computer intensive for many real applications, since the solution for a very large number of unknowns is required at each frequency. References 8, 9 and 10 are examples of pack- ages that use variants of the method for planar struc- tures. However, these were developed as MMIC (monolithic microwave integrated circuit) design tools rather than EMC tools, and would typically be limited to around loo0 geometric elements by constraints on com- puter memory. Because the maximum size of problem that can be addressed is small, a number of special approximation techniques are used in EMC tools to reduce computational demands. Two of these are con- sidered here.

4.1 Transmission line model The most widely used approach treats cables and PCB tracks only rather than the full interconnection system. Separate admittance matrices (or their equivalents) are obtained either for each single straight conductor section and its associated ground plane return path, or for groups of parallel conductors. These simplified geomet- ries are treated as multiconductor transmission lines [ I l l . The analysis effectively reduces to a 2D field problem since mode propagation is assumed parallel to the conductors. Separate models derived from 3D field analysis for each discontinuity are sometimes used in conjunction with the transmission line models. Coupling between models and the behaviour of arbitrarily shaped polygonal partial ground planes cannot be modelled.

The transmission line model is numerically efficient but has limited application. It is used primarily on digital designs to model signal integrity and crosstalk. Refer- ences 12, 13 and 14 are examples of packages that use this approach. As discussed above, radiation can be included directly in the analysis, or treated as a second order effect which involves a correction to the Green's function. However, radiation models derived from trans- mission line analysis are likely to underestimate true radiation. The reason is that common mode signals, which are the primary source of radiation, are ignored in the model. By definition, every component of current in a straight transmission line has an equal and opposite com- ponent in a nearby conductor which almost cancels the radiation field, whereas in the more general config- urations found in real applications, i.e. where return paths are much less well defined, no such automatic can- cellation occurs.

4.2 Lumped equivalent circuit model It is well known that a transmission line may be charac- terised by sections comprising lumped components con- nected in cascade, as illustrated in Fig. 2. This model is accurate at frequencies for which each such section is

electrically short, say less than a tenth of a wavelength. As stated above, such a model has the advantage that it may be used in a general purpose analogue simulator for either time or frequency domain analysis. This concept is not, in fact, restricted to transmission lines, and has been generalised to arbitrary 3D structures [5, 15-17].

Such an approach leads initially to a lumped circuit where the number of nodes is related directly to the (typically very large) number of geometrical elements. However the Philips EMC simulator FASTERIX [SI provides a method of dramatically reducing the size of equivalent circuit [18, 191. This is based on analogy with the lumped model of the transmission line, for which sec- tions are uniformly spaced along the line at electrically short intervals. In the much more general 3D problem the physical locations associated with equivalent circuit nodes are distributed over the conductor surfaces. This set of nodes includes the external nodes of the intercon- nection system. However, additional nodes are intro- duced because an accurate lumped model requires a conducting path of small electrical length (say less than a tenth of a wavelength at the highest frequency of interest) between each node and at least one other node. Fig. 1 shows the location of such nodes (dotted) for one applica- tion, determined automatically (together with the element mesh) by the FASTERIX geometry preprocessor.

The imaginary part of the admittance matrix defined by such a set of nodes is dominated by the sum of two terms, one proportional to w and the other inversely pro- portional to w. There is also a typically much smaller real part which characterises loss. An equivalent circuit of the form shown in Fig. 3, for an example with just four nodes, has an approximately equal admittance matrix.

Fig

c 2 3

2 3

:,?12 nL33 R33

grid . 3 Lumped equivalent circuit model for arbitrary structure

This quasi-static method has been shown to give a solution which is almost as accurate as the full wave solu- tion at frequencies used in EMC tests (typically up to I GHz for consumer electronic products), and consider- ably higher if required. However, the solution is orders of magnitude faster, an advantage that is due mainly to two factors. First, the computation of the passive component

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values is a once only field analysis, and employs static Green’s functions [20] which are much simpler, and more efficiently computed, than the fully dynamic Green’s function. Second, the time or frequency doman analysis of the full EMC problem is handled by the circuit simulator modelling in terms of scalar rather than vector quantities, and using a much smaller number of unknowns than the field analysis. Full wave solutions on the other hand require field analysis at each frequency of interest.

Although the solution for the current distribution is quasi-static, the radiated field is computed using the dynamic Green’s function. Any component of electric or magnetic field may be displayed in any plane. Fig. 4 shows an example of such a radiation plot.

I

Fig. 4 Magnitude of radiated magneticfield above a PCE

5 References

I WILLIAMS, T.: ‘EMC for product designers’ (Butterworth-Heine- mann, Oxford 1992)

2 Directives 89/336/EEC and 92/3I/EEC, Oficial J. of the European Communities, European Information Centres, No. L139, pp. 19-26, 1989, and No. L126,1992, p. 11

3 KHAN, R.L., and COSTACHE, G.I.: ‘Finite element method applied to modelling crosstalk problems on printed circuit boards’, IEEE Trans., 1989, EMC-31, ( I ) , pp. 5-15

4 CERRI, G., DE LEO, R., and MARIANI-PRIMIANI, V.: ‘A rigor- ous model for radiated emission prediction in PCB circuits’, IEEE Trans., 1993, EMC-35, ( I ) , pp. 102-109

5 DU CLOUX, R., MAAS, G.P.J.F.M., WACHTERS, A.J.H., MILSOM, R.F., and SCOTT, K.J.: ‘FASTERIX, an environment for PCB simulation’, Proc. 10th Internat. Symp. EMC, 1993, pp. 213-218

6 MEDGYESI-MITSCHANG, L.N., and PUTNAM, J.M.: ‘Integral equation formulations for imperfect conducting scatterers’, IEEE Trans., 1985, AP-33, (2). pp. 206-214

7 COLLIN, R.E.: ‘The Dyadic Green’s function as an inverse oper- ator’, Radio Science, 1986, 21, (6). pp. 883-890

8 ‘em’, Sonnet Software Inc. 135 Old Cove Road, Suite 203, Liverpool, NY 13090-3746. USA, and EEsof Inc., 5601 Lindero Canvon Road. Westlake Village, CA 91362, USA

CA 91362, USA 9 ‘EMSim’, EEsof Inc., 5601 Lindero Canyon Road, Westlake Village,

IO ’HP Momentum’. Hewlett-Packard Co., P.O. Box 58059, MSSILSG. Santa Clara, CA 95051-8059, USA

I 1 WEI, C., HARRINGTON, R.F., MAUTZ, J.R., and SARKAR, T.P.: ‘Multiconductor transmission lines in multilayered dielectric media’, IEEE Trans., 1984, MIT-32, (4). pp. 439-450

12 ‘TLC‘, ‘XTK’ and other tools. Quad Design Technology Inc., 1385 Del Norte Road. Camanlo, CA 93010, USA

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13 ’Greenfield‘ family of tools, Quantic Laboratories Inc., Suite 200, 281 McDermot Avenue, Winnepes Manitoba, Canada, R3B 059

14 JOHN, W., HOENER, J., and RETHMEIER, 0.: ‘FREACS - a fast reflection and crosstalk simulator’. CompEuro 1991, pp. 526- 530

15 MILSOM, R.F., SCOlT, KJ., CLARK, G., McENTEGART, J.C., AHMED, S., and SOPER, F.N.: ‘FACET - a CAE system for RF simulation includng layout’, 26th Design Automation Conf. (DAC), 1989, Las Vegas, USA, pp. 622-625

16 HEEB, H., RUEHLI, A., JANEK, J., and DAIJAVED, S.: ‘Simulat- ing electromagnetic radiation of printed circuit boards’, I E E E lnt.

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ConJ on Computer Aided Design, ICCAD-90, Santa Clara, pp. 392-395

17 PAUL, C.R.: ‘Modelling electromagnetic interference properlies of printed circuit boards’, IBM J . Res. Dewlop., 1989,33, (1). pp. 33-50

18 MILSOM, R.F., SCOTT, K.J.. and SIMONS, P.R.: ‘Reduced equiv- alent circuit model for PCB, Philips J . Res., 1993.44, (1)

19 DU CLOUX, R., MAAS, G.P.J.F.M., and WACHTERS, A.J.H.: ‘Quasi-static boundary element method for electromagnetic simula- tion of PCB’, ibid, 1993

20 S c o r n , K.J.: ‘Efficient image theory for electromagnetic field mod- elling in PCB’, ibid. 1993

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