A quasi-static tool for the EMI/EMC analysis of analog circuits: parasitic extractor tool and simulator of EMI parameters (PET+SEP)

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 2, MAY 1998 127

A Quasi-Static Tool for the EMI/EMC Analysis ofAnalog Circuits: Parasitic Extractor Tool andSimulator of EMI Parameters (PET SEP)

S. Piedra, J. E. Fernandez, J. Basterrechea,Member, IEEE, and M. F. Catedra

Abstract—A quasi-static approach for evaluating inductiveand capacitive coupling and radiated noise in the standard elec-tromagnetic compatibility (EMC) measurement conditions hasbeen developed. Multilayer structures are analyzed considering aquasi-static Green’s function formulation. Variational approacheshave been followed to compute inductive and capacitive par-asitics. Electric and magnetic multipolar expansions are usedfor a fast computation of near- and far-field radiated noise ina large broad band. The approach has been validated consid-ering different cases ranging from simple circuits to realisticswitched-mode power supplies and some of the validation casesare presented. The main advantages of this method is the shortcentral processing unit (CPU) time it requires to provide reliableresults for EMC analysis and design purposes.

Index Terms—Circuit modeling, parasitics, quasi-static ap-proach, radiated fields.

I. INTRODUCTION

T HE use of efficient simulators has become essential inthe early development stages of printed circuit boards

(PCB’s). Nowadays, the spectral content of signals is be-coming higher and layout components are confined to tighterspaces, making designs more conflictive from an EMC pointof view: radiated interferences and coupling between tracks oflayout increase and they become an important problem. Thereare a lot of good electrical simulators (e.g., SPICE, ELDO,etc.) and software packages for designing and producingPCB’s (e.g., TANGO, CADENCE, MENTOR, etc.), however,there are not so many EMC simulators, therefore, designersfind themselves forced to employ the trial-and-error designcycle. A review of the existing EMC tools, their methods, andfuture necessities is detailed in [1] and [2]. Following thesereferences, we can classify the different approaches to evaluateEMC into three main categories.

•Heuristic design rules. These may not properly estimateradiated fields.

•Quasi-TEM analyses. These approaches are easy to applyand efficient but fail as frequency increases to gigahertzlevels.

•Full-wave solutions of Maxwell’s equations. These provideexact solutions but require a lot of computer resources.

Manuscript received November 6, 1996; revised December 15, 1997. Thiswork was supported in part by the Spanish Advisory Commission for Scientificand Technological Research (CICYT) Project Ref. TIC96-653.

The authors are with the Grupo de Sistemas y Radio, Departmento deIngenierıa de Comunicaciones, Universidad de Cantabria, Santander 39005,Cantabria, Spain

Publisher Item Identifier S 0018-9375(98)04010-1.

(a) (b)

(c)

Fig. 1. Part of a circuit considered to introduce the terminology. (a)Schematic. (b) Layout. (c) Updated schematic, which includes parasiticelements.

In this paper, an approach for the EMC study of circuitsimplemented on multilayer PCB’s is described. The approachis based on a quasi-static analysis that for frequency up to afew hundred MHz is accurate and extremely efficient: broad-band results can be obtained in near real time using a PC. Theapproach was included in the framework of an ESPRIT Projectcalled POWERCAD. It involved the modeling of severalaspects in power electronic circuits and in particular, in themodeling of parasitic effects and radiated noise in PCB’s.

To introduce the methodology considered in this work, anexample of a part of a circuit is outlined in this paragraphand shown in Fig. 1. The starting point is the schematic,which is formed by branches connecting nodes correspondingto lumped elements [Fig. 1(a)]. Next, the layout is extractedand each branch in the electrical scheme becomes a track[Fig. 1(b)]. The goal is to evaluate the inductive and capacitiveparasitics associated to these tracks, to include them in theelectrical scheme together with the initial lumped elements[Fig. 1(c)], to analyze the circuit with the updated scheme,and, finally, with the results of this analysis and the geometryof the PCB, to predict the radiated interference.

Two main computer tools have been obtained from theapproach: 1) parasitic extraction tool (PET) for the evaluation

0018–9375/98$10.00 1998 IEEE

128 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 2, MAY 1998

of the inductive and capacitive parasitic coupling betweentracks of PCB’s layouts and 2) simulator of EMI parameters(SEP) for the computation of the radiated noise the circuitwould give in the standard EMC measurement conditions.

Two main features were defined for this approach and itscorresponding computer codes. First, to obtain tools that canrun in the shortest CPU time considering accurate approxi-mations for frequencies up to a few hundred megahertz. Thesecond feature is that the codes should be easily integratedin a broader framework, which includes electrical simulator(ES), coupling extractor, layout extractor, etc. In this sense,the flow of data of PET and SEP, once they have beenintegrated into the framework, is as follows: the layout toolgives the track geometrical data to PET, the user selectsthe tracks where PET should compute the parasitics, PETbackannotates the symbolic netlist to include the inductiveand/or capacitive parasitic elements, then, afterwards, the usercan run the electrical simulator, and the current waveformstogether with the layout geometrical data are passed to SEPthat finally provides the desired EMI data.

Printed circuits have been studied with rigorous methodslike, for instance [3], where the mixed-potential electric fieldintegral equation is solved by moment method (MM) to obtainthe current distribution and the radiated fields. The approachof [4] solves a couple of integral equations for equivalentelectric surface currents using MM; [5]–[11] also utilize MM.Others (as in [12]) employ the finite-element method, witha TEM approximation to evaluate current distribution, anduses a simple approach to compute radiated fields. The finite-difference time-domain method has also been used to treatPCB’s, [13]. All of these rigorous approaches start from com-puting the current distribution with methods that require muchmore time and memory storage than our proposed quasi-staticscheme, where the current and charge density are consideredconstant in each track of the circuit. Techniques similar to theone used here have been used historically to obtain equivalentcircuits (for example, the partial-element equivalent-circuit(PEEC) technique proposed by Ruehli in [14] or the ones in[15] and [16] for capacitances and inductances) and continuebeing used at this moment [17], [18]. Of course, the quasi-static approach is justified as long as the PCB tracks can beconsidered electrically small.

Moreover, the use of an approximated method is justifiedconsidering that in any circuit there are some components liketransformers, transistors, etc., for which it is difficult to obtainaccurate enough models, so it makes no sense to seek theexact computation of radiation but an estimation of radiationlevels. This could be sufficient for an EMC analysis if an errormargin is admitted. This means that when the radiated noiseoriginated by the PCB under study computed with this tool iscompared to the normative, the user should consider a securitylimit more restrictive than the normative in order to assurethe validity of the test. Furthermore, although approximatemethods for computing radiated noise imply to considering anerror margin the computing time is very short.

This paper is arranged as follows: first, in Section II, theapproach for the evaluation of inductive and capacitive cou-pling between tracks of a layout is presented. The effect of

the dielectric layers supporting the conductor tracks has beenconsidered in the Appendix. In Section III, the computationof the radiated field is described. Fields have to be computedin a wide band of frequencies in order to compare to EMCnormative; this can require a lot of CPU time if one computesthese field using the complete expressions as are deduceddirectly from Maxwell’s equations. In these expressions, theintegrands depend upon the frequency and that means thatfor each frequency sample, an integral shall be evaluated. Toreduce the CPU time three different approaches have beendeveloped in such a way that depending on the situation,the most suitable approach is chosen to compute radiatednoise. Section IV summarizes the characteristics of the toolsdeveloped and shows the flow of data and computation whena PCB is analyzed starting from the layout design, parasiticbackannotation, current waveform analysis using an electricalsimulator, and, finally, the radiated EMI analysis. In Section V,some numerical and experimental validations of the approachproposed will be presented and discussed. Finally, some con-clusions are given in Section VI.

II. I NDUCTIVE AND CAPACITIVE PARASITICS

As it has been previously stated we are considering onlyharmonic components for frequencies up to a few hundredsof MHz where quasi-static approximations are applicable. Thecircuit layout has been modeled using a set of rectangulartracks, as shown in Fig. 1. The tracks are defined by thestarting and ending points and their width. There are otheroptions for geometrical description as the modeling by meansof Bezier’s patches or nonuniform rational B splines (NURBS)[19], [20]. These are very efficient for arbitrary complexcurved surfaces, but dealing with them requires much mathe-matical computation. As the study presented here is focusedon planar tracks of layout, the geometrical description withrectangular tracks is sufficient for our purposes.

A. Inductive Coupling

The magnetic energy stored in a circuit with tracks canbe evaluated with any of the following expressions (e.g., [21]):

(1)

(2)

where is the mutual inductance between tracksand , ,and are the currents on tracksand , respectively, isthe surface of track, is the current distribution in trackand is the potential vector in track due to the current ofthe track

(3)

where is the Green dyadic which relates the value ofthe potential vector in with an infinitesimal current sourcelocated in . If there is not any ground plane and all the

PIEDRA et al.: QUASI-STATIC TOOL FOR THE EMI/EMC ANALYSIS OF ANALOG CIRCUITS 129

media have the vacuum permeability, the potential vector isthen defined by the expression

(4)

From (1), (2), and (4) we can obtain

(5)

This expression is variational, which means that we can obtaingood approximations for from rough approximations ofthe current distributions. For the purposes of this paper, thecurrent distribution has been assumed constant in transversalcuts of the tracks

(6)

where is the width of track and is a unit vector tangentto the track. Therefore, can be approximated by

(7)

The integral in (7) becomes singular when . This problemhas been solved by using the analytical integration techniquesdeveloped in [22], which are suitable for uniform sourcedistribution on simple-shape domains as is the case in thispaper (rectangular patches with constant current distribution).

It has to be noted that these self and mutual inductances willprovide the values of the coupled inductances representing thedifferent tracks in the layout. Each track (node in the schemat-ics) will be replaced by a set of inductances representing eachstraight section.

In most circuits there is a ground plane in order to reduce themutual inductance between tracks. This effect has been takeninto account in the computation of inductance by applyingimage theory [23]. The system form by the tracks and theground plane is substituted by the tracks and its images. Thenthe value of the inductance is given by

(8)

where represents the mutual inductance between loopsand when there is no ground plane and is the mutualinductance between track and the corresponding image oftrack .

B. Capacitive Coupling

By analogy with the previous case, the electrostatic energyin a system of conductors (tracks) can be expressed as(e.g., [21])

(9)

where and are the charges on tracksand , respec-tively, and represents the voltage in conductorwhenconductor has a total charge of 1 Cb and the rest of

the conductors are free of charge. From field theory, theelectrostatic energy is given by

(10)

where is the charge distribution in track andis the electrical potential in track due to the charge densityin the track. Using the Green’s function formulation, thispotential can be obtained from

(11)

where is the Green’s function for the electrostaticpotential and is the charge density in track. This lastGreen’s function depends on the permittivity of the multilayermedia as can be seen in the Appendix.

From (9) to (11) we can obtain the following variationalexpression for :

(12)

It must be noted that the charge density in this last expressionare those that make the electrical potential to be constant ineach circuit track. However, considering the variational natureof (12) the charge density can be assumed uniform in eachconductor and (12) can be approximated by

(13)

Using the coefficients the voltage vector of the tracks isrelated with the charge vector by

(14)

Equation (14) can be written in a more compact form asfollows:

(15)

By inverting this equation, we obtain

(16)

where

(17)

To finally obtain the values of capacitive coupling betweentracks, the following expressions are considered:

for

(18)

being the coefficients of self capacitance and thecoefficients of induction.


This allows us to write

...

(19)

where is the capacitance between the ground andconductor and is the capacitance between conductor

and conductor . These capacitances areintroduced into the netlist (back annotation) along with theinductances described in previous section, to perform signalintegrity analyses.

III. RADIATED EMISSIONS

Currently, there is a trend toward a desired speedup in thealgorithms involved in EMC [1]. This trend arises from theneed to simulate (in real time) circuits that are becoming moreand more complex. On the other hand, taking into account thata large number of factors (some of them of difficult prediction)are responsible for radiated noise from PCB’s, it makes senseto reach a compromise between the accuracy in the solutionand the computer time expended. An attempt to reach this goalis made herein in such a manner that the radiated emissionswill be computed over a broad bandwidth as is demandedby EMC regulations by solving a few integrals that do nothave frequency dependence. This approximation will be validwhenever the PCB tracks are electrically small. In other words,this new formulation will perform well up to several hundredsof megahertz for tracks of some centimeters in size.

In the next part, two approximations for both near andfar electric and magnetic fields will be derived, starting fromthe electric field integral equations (EFIE) and magnetic fieldintegral equations (MFIE), respectively.

Since the tracks will be considered electrically short, a uni-form distribution along them is a good enough representationfor the current [3]. Once the current distribution is known, theradiated emissions will be computed by using the approachesfor the EFIE and the MFIE.

A. First Approach: Near Field

The radiated fields by an arbitrary current distributioninfree-space are given by

(20)

(21)

where a harmonic time dependence has been assumed andis the magnetic vector potential

(22)

By simple inspection of (22), the need for computing one inte-gral for each frequency seems evident due to the wavenumber

in the Green’s function. In order to circumvent this

inconvenience and compute just one integral in a large broad-band, one might expand the Green’s function exponential ina Taylor series

(23)

Combining (20)–(23) and, after a long but straightforwardmanipulation, the resultant equations for the fields are

(24)

(25)

Equations (24) and (25) will be useful from a computationalpoint of view if one can neglect most of the terms of thesummation and work with a few of them. An optimumperformance of this formulation is reached [according to (23)]whenever

(26)

in which case only one term from (24) and (25) is necessary,leading to

(27)

(28)

In this work, (27) and (28) have been used to estimate theradiated field at points near the PCB and are being studied.Under our approach, the current distribution comes given by(6) and, hence, a unitary vector can replace the current densityin the integrands in (27) and (28) with the constantextracted from the integrals.

1) Contribution of the Different Terms from (24) and (25):In order to see the specific contribution of the terms in theexpansion of powers, some structures were analyzed. As anexample, this formulation will be applied to a simple 1-cmradius circular loop. The loop is carrying a sinusoidal current(a pure harmonic) at 300 MHz and 2-A peak-to-peak. Fig. 2shows the first terms of the component of along the line

(a cylindrical coordinates system is assumed). Due tothe symmetry, was chosen without loss of generality.From Fig. 2 it can be drawn that the first term of the series


Fig. 2. Contribution of the different terms of anW power expansion for thez component of the magnetic field along the line� = z.

is dominant over the rest if the distance from source point toobservation point is less than 0.1, which is coherent with (26).

B. Second Approach: Far Field

If electrically small tracks are assumed and the referencesystem is centered in the PCB, the following conditions aresatisfied for far fields:

(29)

where (30)

Equation (29) means that the distance from the origin of thereference system to the observation point is much greater thanthe distance from the same origin to the source point. On theother hand, (30) is related to the electrically small size of thePCB and tracks. Assuming (29) and (30), Green’s function forfree-space can be approached as

(31)

Using (31) in (22), the magnetic vector potential is ex-pressed as

(32)

According to (29), it is a reasonable approach to neglectthe width of the tracks, leading to a modified geometricalmodel where a track segment of length and carrying acurrent density is replaced by a segment of the samelength but carrying a total current . With this lastconsideration in mind, the electric and magnetic moments fora set of track segments can be defined, respectively, as

(33)

(34)

The first integral in (32) is related to, whereas the secondone is to as can be seen after applying some basic vectorialproperties for divergenceless currents in (32). The result canbe summarized in the following approached expression:

(35)

Substituting (35) into (20) and (21) and after some manipula-tions, the new expressions for the fields result in

(36)

(37)

where

(38)

(39)

(40)

(41)

(42)

(43)

(44)

The pursued objective in this section, i.e., the extraction of thefrequency dependence from the integrands, has been reachedin (36) and (37). Furthermore, the integrands do not dependon the observation point, as one can see in (33) and (34). Inthis paper, (36)–(44) will be used to compute the radiated fieldin far-field conditions. For usual PCB dimensions, the termswhich include are dominant—this approach being valid forfrequencies ranging up to some hundreds of megahertz.

2) A First Validation of the Formulation:The first attemptof validation for the last approaches lies in comparing theresults obtained with this formulation to the ones providedby standard formulas for a small circular loop in the far-fieldregion [24]. The geometry considered was a circular loop alongwhich a time periodic trapezoidal current flowsmA, ns, ns, period s . Theloop lies on the plane, its area is 25 square centimeters,and the field is computed at 10 m along theaxis. The currentspectrum was computed using the fast Fourier transform (FFT)and then (36) and (37) were applied to each harmonic. Fig. 3shows the comparison between this technique, the analyticalapproach for a magnetic dipole in the far-field region, and theasymptotic expression. A remarkable agreement between bothmethods is achieved.


Fig. 3. Comparison between moments and analytical formulations. Thegeometry consisted of a circular loop carrying a 10-mA trapezoidal currentat 10 MHz.

IV. EMI/EMC TOOL

In previous sections, different theoretical aspects in the com-putation of parasitics and radiated fields have been outlined. Ithas been shown those approaches were valid for frequenciesup to a few hundred megahertz and, therefore, can be usedto obtain correct estimates of couplings between tracks aswell as values of radiated field according to measurementsetups marked by the different normatives on EMC concerningradiated noise. To reach this last objective some additionaleffort has been carried out. It is a common practice to usesemi-anechoic chambers (Fig. 4), instead of open area testsites to perform EMC measurements. They are characterizedby the presence of absorbers in the walls and of a metallicground plane in the floor. In order to properly model thismeasurement situation, reflections in the ground plane haveto be accounted for apart from the direct radiation from thePCB to the point where the antenna is placed. Basically, themeasurement setup consists of a radiation receiving system(antenna and spectrum analyzer) and a rotary wooden table inwhich the device under test (DUT) is placed as sketched inFig. 4. When measurement distance is great enough (normallymeasurements are performed at 3 m), conditions specified in(29) and (30) are satisfied and the formulation of the previoussection can be used if a proper account is given of reflectionsin the ground plane. These can be taken into account easily byusing images theory and defining new sources of field given bythe images through the ground plane of the currents flowingin the layout.

A software package for the evaluation of inductive andcapacitive coupling between tracks of layout and radiated noisedue to them has been developed considering all the modelspreviously described. This software along with an electricalsimulator and a PCB layout designer, constitute a full designtool. The simulation flow is shown in Fig. 5 where PETstands for parasitic extractor tool and SEP for simulator ofelectromagnetic interference (EMI) parameters.

The whole process is as follows.

—A PCB designer generates the geometrical information ofthe layout and the circuit netlist.

—PET computes the inductive and capacitive parasiticsbetween tracks of the layout. These parasitics are usedto update the netlist, therefore, the user can see howwaveforms have been modified.

—With the electrical simulator, the currents in the para-sitic elements are evaluated, then SEP, using both thesecurrents and the geometrical information, evaluates theradiated noise including the effect of the images throughthe ground plane.

V. RESULTS

Some results were already shown in previous sections inorder to validate some of the models there introduced. In thispart, another set of results will be presented and discussed.The cases analyzed deal with the integration of the two toolsdeveloped (i.e., first the signal integrity is studied by meansof the parasitics computation as was explained in Section II)once the inductive and capacitive coupling is added to thecorresponding lumped circuit netlist, the radiated emissionsare estimated by using suitable expressions from Section III.

A. Differential and Common Mode Currents

In this example, taken from Paul [25], two tracks of 6 inin length and 25 mils in width and separated 355 mils, wereconsidered. The circuit is shown in Fig. 6. The tracks wereetched on 62 mils glass-epoxy board and matchedwith a load of 330 , then driven by a 5-V trapezoidal voltageat 10 MHz with ns, ns, and ns.In order to reduce the unwanted effects of a power supply,the circuit was powered by a battery. Also, the whole devicewas placed in a symmetric and compact manner in order toreduce the common-mode current. In spite of everything, thiswas the dominant radiation mode as described by Paul. Themeasurements were carried out in a semi-anechoic chamber, asin Fig. 4, with the circuit placed at 1-m height and the currentthrough the tracks had direction.

The process followed in this paper to simulate the radiatedemissions was first to compute the parasitics due both to theprinted circuit tracks in the presence of the dielectric layerand the semi-anechoic chamber ground planeand to the external body of the battery in the presence of thechamber ground plane . All of them were computedusing the technique described in Section II-B., , and

were determined using the three-layer- - Green’sfunction for the PCB placed 1 m from the ground plane, asdescribed in [25]. was computed considering an openparallelpiped standard battery dimensions placed 1 m fromthe ground plane. Other parasitic effects like capacitancesbetween PCB tracks and battery external plate or the onesassociated with the connecting cables were neglected. As aresult, the equivalent circuit shown in Fig. 6 was obtained.Thereafter, the equivalent circuit was run using ELDO as anelectrical simulator [26] to obtain the time domain currents(differential and common modes) extracted from the ES. Forthe differential mode, the currents through the inductancesand with the capacitances removed from the circuit wereconsidered. The current in common mode was computed like


Fig. 4. Schematic representation of a semi-anechoic chamber.

Fig. 5. Simulation flow of a complete analysis process.

the one through and or equivalently through .Later, the spectrum of the current was obtained by using theFFT and, finally, the radiated emission was estimated for eachcurrent component in the frequency domain by applying themore convenient approach from Section III along with imagetheory to simulate the effect of the ground plane of the semi-anechoic chamber. Simulation results and measurements in[25] are compared in Table I.

B. Radiation from Complex PCB’s

Although the structure analyzed in previous section was asimple PCB, this type of analysis helps us to get an insightinto the radiation phenomenon. Besides, the tools have beenchecked with several switched-mode power supplies (SMPS)providing good results. As an example, some results for the

Fig. 6. Simple case considered for the validation of the tool. Configurationand measurement results were taken from [25]. The equivalent circuit includ-ing parasitic elements to ground is also shown. The capacitance between thetwo tracks has been neglected due to its negligible effect in the frequencyband being studied.Cbatt represents the parasitic capacitance between thebattery and the ground plane.


TABLE ICOMPARISON BETWEEN MEASUREMENTS IN [25] AND SIMULATIONS OF THE

y COMPONENT OF THEELECTRIC FIELD RADIATED BY THE STRUCTURE OFFIG. 6

dc–dc converter whose schematic is shown in Fig. 7(a) andlayout in Fig. 7(b) are presented here. This layout is nota typical one by far. It was designed to stress couplingand radiation problems (long and close tracks, large loopareas, etc.) and to allow its reconfiguration by using differentpositions for the jumpers between the points marked with 1 to7 and 1 to 7 in Fig. 7(b). A comparison between measuredand simulated near electric field at 5 cm from the centerof the board is shown in Fig. 8(a) and (b), respectively, fora configuration of the jumpers which maximizes the looplengths. Those measurements were carried out with a 3.5-cmelectric field ball probe and an spectrum analyzer at AlcutelStandard Electrica S.A. (ASESA) EMC facilities in Madrid.A reasonable agreement between both can be observed. It hasto be noted that only the line connecting the peaks is shownin the simulation to avoid confusion.

A comparison between measured and simulated results ofthe electric field in a semi-anechoic chamber is shown inFig. 9(a) and (b). In the measurements the probe, a biconicalantenna, has been placed at 1 m from the floor and at 3 m awayfrom the SMPS, which was placed on a wooden table of 0.8-mheight It has to be noticed that no input filter was used so theinput cable (1.2 m long) powering the circuit was includedin the model. From these two figures one can remark thatthe estimated fields remain reasonably close to the measuredradiation levels, especially if one thinks about the number ofunknowns involved in this problem. In all the cases presentedhere, the simulations have been made in a HP9000 Model

715-50 MHz computer. For the simulation of the circuit inFig. 8(a), the CPU time and the size of executables for PETare 9 s and 116 Kb, while for SEP, they are 47 s and 456 Kb.

VI. CONCLUSION

An approach to analyze parasitics and radiated noise inanalog PCB’s has been presented. The tool is helpful in thedesign stages to minimize parasitics effects by making signalintegrity analyses as well as to estimate the degree of EMCcompliance of PCB’s. Thanks to the quasi-static algorithmsconsidered for the parasitics analysis and the expansions interms of multipoles the tool is reliable and very efficient;results can be obtained in nearly real time using modestcomputer resources (e.g., a Pentium-based PC).

The approach has been implemented in a set of computerstools, which have been integrated with other tools in a CADenvironment (as shown in Fig. 5) where our tools interactwith layout generators (the geometrical input data of thetrack is imported), with circuits simulators to obtain thecorrected currents, or voltage waveforms (the circuit netlist isimported and exported after a back annotation which includesthe parasitics). The EMC analysis is made considering thecorrected waveforms.

On the other hand, their most important handicap is thefrequential limitation as they are valid up to a few hundredsof MHz. Improvements of the tools need to consider additionaleffects like models for magnetic components, models forboxes, and discontinuities. Its extension to higher frequencyranges needs the connection with a rigorous method like,for example, the method of moments. It will also be quiteinteresting to consider the susceptibility problem.

APPENDIX: GREEN’S FUNCTION COMPUTATION

Obtaining the Green’s function in the spatial domain for amultilayered structure is usually a difficult task and in mostpractical situations it has no analytical solution, however, it ispossible to achieve an analytical expression in the transformedspatial domain. The evaluation of Green’s functions for multi-layered structures in the transformed domain has been widelystudied for both quasi-static [27]–[30] and dynamic [30]–[36]cases. The solution process is quite similar in both analyses,the main difference between them being the fact that in adynamic analysis, the starting point is the Helmholtz equationwhereas in a quasi-static one, the Laplace equation must beconsidered. After applying the Fourier transform, the solutionof these equations along with appropriate boundary conditionsfor the configuration considered provide the desired Green’sfunction transform.

A simple method to obtain the Green’s function transformmakes use of the eigenfunctions for linear systems (complexexponentials) to obtain the eigenvalues (Green’s function) forthe system by taking into account the following property ofthis type of systems:

(A.1)


(a)

(b)

Fig. 7. (a) Schematic representation of a dc–dc converter. (b) Reconfigurable layout for the converter. Points marked with 1–7, 10–70 can be switchedby jumpers to provide different routings to the signal.

With this property, if a complex exponential is used to excitethe linear system it will provide as output the transform of theimpulse response (Green’s function) multiplied by the sameexponential.

For the application considered in this paper, a two-dimensional charge distribution placed in the interfacebetween two dielectric layers (tracks of circuits) is consideredand the potential at any point comes given by

(A.2)

where indicates convolution.If the charge distribution is taken as

(A.3)

(A.2) will become

(A.4)

being and the transformed spatial domain variables.Hence, can be evaluated directly from the po-tential if a charge as given by (A.3) is considered for theinterface of interest. Keeping this last result in mind, theLaplace equation is going to be solved in the following.

From (A.4), Laplace equation can be written as follows:

(A.5)

where . This last equation along with theboundary conditions for each interface [shown in (A.6)] andthe cancellation of potential at the reference surfaces (groundplanes) will provide the solution for the Green’s functiontransform.


(a)

(b)

Fig. 8. (a) Measurement results of near electric field For the dc–dc converterof Fig. 7. (b) Simulation results (only the line connecting the peaks is shown).

Continuity of at the interfaces:

(A.6a)

Jumps in D at the interfaces:

(A.6b)

and being the position along of the interface consideredand of the planar charge distribution, respectively.

Taking into account that the general form of Laplace equa-tion (including the charges) comes given by

(A.7)

and remembering the expression for the voltage along atransmission line (TL)

(A.8)

as well as the boundary conditions (A.6) and the ones forchanges in characteristic impedance (or admittance) of TLcircuit given by

(A.9a)

(a)

(b)

Fig. 9. (a) Measurement results of electric field at 3 m from the dc–dcconverter of Fig. 7. (b) Simulated results.

(A.9b)

an analogy can be stated between them if the followingrelations are established:

(A.10a)

(A.10b)

(A.10c)

(A.10d)

This analogy avoids the analytical resolution of the system ofequations which appears when using the boundary conditions.Basically the idea is that a semi-infinite medium is identifiedby an admittance , a finite medium is replaced bya section of TL with characteristic admittance , andthe sources (charges) are substituted by equivalent unit currentsources, as shown in Fig. 10(b). The problem of obtaining theGreen’s function then reduces to analytically obtaining theadmittances seen to the left and the right of the current sourcein the TL circuit equivalent and obtaining the total impedanceas the inverse of the shunt of admittances.


(a)

(b)

(c)

Fig. 10 (a) Example of three-layered medium. (b) Equivalent TL circuitwhen the charge is placed inz = 0. (c) Simplified equivalent circuit toobtain the Green’s function transform.

As an example, Fig. 10(a) shows the case of a three dielec-tric medium, where the charge distribution is placed at .The total admittances at the right and left of the current source[Fig. 10(c)] are

(A.11)

Therefore, the Green’s function transform for this configura-tion is

(A.12)

Once its transform is computed, the Green’s function in thespatial domain can be obtained from it by using the inverseHankel transform (cylindrical symmetry) as follows:

where (A.13)

being the Bessel function of first kind and zero order.To compute the integral in (A.13) and due to the oscilla-

tory nature of the integrand, the Gauss quadrature [37] hasbeen used between each couple of consecutive zeroes of theBessel function giving rise to slowly convergent series. To

speed up the convergence of the series, the nonlineal Sank’stransformation [38] has been considered.

In order to reduce CPU time evaluating the Green’s func-tion a preprocessing is performed. Before starting parasiticscomputation, the Green’s function is evaluated for a set ofobservation points and a polynomial fitting of these data usingthe least-squares approach [37] is carried out. This polynomialexpression is later used instead of evaluating the integrandin (A.13). This technique has proved more efficient thanusing other alternatives like interpolation or extrapolation offunctions, which require much more CPU time.

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S. Piedra was born in Limpias, Spain, in 1968.She received the M.Sc. degree in physics from theUniversity of Cantabria, Spain, in 1991. She iscurrently working toward the Ph.D. degree at thesame university.

Since 1992, she has been with the Departamentode Ingenierıa de Comunicaciones, University ofCantabria, as a Research Assistant. Her researchinterests include electromagnetic compatibility, nu-merical methods in electromagnetics, and antennas.

J. E. Fernandez was born in Santona, Cantabria,Spain, on December 28, 1965. He received the B.S.(physics/electronics), M.S., and Ph.D. (electrical en-gineering) degrees from the University of Cantabria,Spain, in 1992, 1994, and 1997, respectively.

From 1994 to 1995, he was a visiting scholarin the Department of Electrical and Computer En-gineering at Syracuse University, Syracuse, NY,doing research for his Ph.D. dissertation. He joinedRYMSA, Madrid, Spain, in 1997, where he ismainly involved in the design of antennas for satel-lites.

J. Basterrechea (S’92–M’93) was born in San-tander, Spain, in 1963. He received the M.S. andPh.D. degrees in (physics/electronics), both from theUniversity of Cantabria, Spain, in 1987 and 1992,respectively.

He joined the Communications Engineering De-partment (former Electronics Department), Univer-sity of Cantabria, as a Research Assistant in 1987.In 1992 he became an Assistant Professor in thesame department and earned an Associate Professorposition in 1995. His research interests include

electromagnetic compatibility analysis and modeling techniques applied toradio and telecommunications equipment and systems and numerical methodsapplied to the design of antennas and microwave passive components. Hehas coauthored several journal articles and a book on conjugate gradient fastfourier trasform method.

M. F. Catedra received the M.S. and Ph.D. de-grees in telecommunications engineering from thePolytechnic University of Madrid (UPM), in 1977and 1982, respectively.

From 1976 to 1989, he was with the Radio-communication and Signal Processing Department,UPM, teaching and doing research. He became aFull Professor at Cantabria University, Spain, in1989. He has worked on about 25 research projectssolving problems of electromagnetic compatibilityin radio and telecommunication equipment, anten-

nas, microwave components, and radar cross section and mobile communica-tions. He has directed about ten Ph.D. dissertations and has published about30 papers, two books, about ten chapters in different books.

Documents

A quasi-static tool for the EMI/EMC analysis of analog circuits: parasitic extractor tool and simulator of EMI parameters (PET+SEP)