IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998 47

Susceptibility Analysis of Arbitrarily Shaped 2-DSlotted Screens Using a Hybrid GeneralizedScattering Matrix Finite-Element Technique

Juan V. Balbastre Tejedor,Member, IEEE,Luis Nuno, Member, IEEE,and Miguel Ferrando Bataller,Member, IEEE

Abstract—The electromagnetic susceptibility (EMS) inside slot-ted screens has been studied using a hybrid technique. The screenis characterized by a generalized admittance or impedance ma-trix, computed using the finite element method (FEM), which isthen combined with a modal solution in free-space. The scatteringmatrix for the screen can then be easily computed. As a practicalapplication, the electrical performance of a slotted square en-velope has been studied. In general, it is shown that couplingto the interior of slotted screens is maximized at frequenciescorresponding to resonances of the shorted screen, provided thatthe fields do not vanish near the aperture.

Index Terms—Electromagnetic coupling, finite-element meth-ods, scattering matrices, shielding.

I. INTRODUCTION

ELECTRONIC devices are generally covered by somekind of envelope or screen. One important property of

those covering structures is the way they protect the enclosedcircuit from external electromagnetic interference. When theenvelope is slotted, environmental electromagnetic fields canoriginate an important field inside the envelope, which canaffect the electrical performance of the enclosed devices orcircuits. In this case, the knowledge of the field distributionand polarization can help the designer to choose the bestemplacement and orientation for the most sensitive devices.

The analysis of arbitrarily shaped screens is a quite difficultproblem, so it is convenient to carry it out by numericalmethods [1]–[4]. The finite element method (FEM) is a goodtool for the electromagnetic analysis of complex structures,but it cannot deal with open problems unless the Sommerfeldradiation condition is forced on the boundary. Absorbingboundary conditions (ABC’s) of different orders (based onasymptotic expansions of the Sommerfeld condition) can beused to make the boundary “transparent” to the radiatedwaves [5]. The FEM can also be combined with an integralequation to force the radiation boundary condition via theGreen function of the exterior unbounded medium [6]. Inthis paper, the study of cylindrical envelopes of arbitrarycross section has been accomplished by combining a modalanalysis in free-space with a circuital representation of thescreen obtained by the FEM.

Manuscript received December 2, 1996; revised December 2, 1997.The authors are with the Departamento de Comunicaciones, Universidad

Politecnica de Valencia, Camino de Vera s/n, Valencia, 46071 Spain.Publisher Item Identifier S 0018-9375(98)02163-2.

First, the structure is characterized by a generalized inmit-tance (admittance or impedance) matrix, as will be describedin Section II. In Section III, a generalized scattering matrix isobtained by enforcing the appropriate expansion for the fields.Then, in Section IV, the inmittance parameters are obtainedvia the FEM. In Section V, several screens have been analyzedusing the proposed method. The analysis of a general squarescreen for different frequencies shows a close relationshipbetween the resonances of the screen and the cutoff frequenciesof the corresponding waveguide, as was stated for circularstructures in [1]. Finally, in Section VI, some computationalfeatures of the method are discussed.

II. I NMITTANCE MATRIX CHARACTERIZATION

The structure to be analyzed, which is invariant in, isenclosed by two circles with radii and (as shown inFig. 1) that will be called ports 1 and 2. The analysis is carriedout in terms of the axial component of the incident wave,which is for a TM incidence and for TE waves. For

or the fields propagating toward the origin canbe expanded in terms of Bessel functions of the first kind orHankel functions of the first kind , while the fieldspropagating outwardly is written in terms of Hankel functionsof the second kind, as is stated in (1) [7]. The superscriptsand in (1) stand for waves propagating toward and from thecoordinate origin, respectively, and the limits of summationswill be chosen in order to achieve a correct representation ofthe field behavior in each region. These values may depend onthe particular problem as will be discussed later in Section VI

(1a)

(1b)

(1c)

(1d)

First of all, the structure shown in Fig. 1 will be character-ized by a generalized inmittance matrix. To do so, we enforce

0018–9375/98$10.00 1998 IEEE

48 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998

Fig. 1. General geometry of a slotted screen.

the axial component of the total field ( for TM wavesand for TE waves) at port

(2)

and compute, by using the FEM, the field (which representsthe total magnetic field for TMincidence and the total electricfield for TE incidence) in . Note that sinceand determine the extension of the computational domain,they should be chosen as close as possible to the screen.

The field is related with by Maxwell’sequations

(3)

where and are and for the TMcase, while for the TEwaves they are and ,respectively. In both cases is the free-spacewavenumber.

Once the field is computed, it can be expanded asfollows:

(4)

being

(5)

So, the following generalized inmittance matrix is obtained:

(6)

III. SCATTERING MATRIX CHARACTERIZATION

In this section, we will derive an expression for the two-port generalized scattering matrix characterizing the structurein Fig. 1

(7)

If the port is excited with the mode we have, forcingcontinuity of at both ports

(8a)

and, forcing the continuity of also at both ports

(8b)

From (8) we obtain

(9a)

(9b)

being [8]

if

if(10)

Exciting the port with the mode weighted by we have

(11a)

(11b)

where is the vector containing the weighting coefficientsand

The generalized scattering matrix relates the coefficients ofthe fields appearing in (1) in the following way:

(12)

(13)

BALBASTRE et al.: ANALYSIS OF ARBITRARILY SHAPED 2-D SLOTTED SCREENS 49

so, from (11), we obtain the expression of the scattering matrixshown in (13) at the bottom of the previous page.

IV. I NMITTANCE PARAMETERS COMPUTATION

The basic step to obtain the circuital model of the generalstructure of Fig. 1 is to compute the fields from which theinmittance parameters are obtained. To do so we mustsolve the problem

(14a)

in , subject to the boundary condition

(14b)

at , where , , , , and have been alreadydefined and is for the TM waves and for TEpolarization.

After a Galerkin procedure, the following integral equationis obtained:

(15)

where is a vector test function and the bidimensionaldivergence theorem has been applied.

Equation (15) is discretized using the FEM with edgeelements. The domain is divided into triangular elementsand inside them the following interpolation is used:

(16a)

(16b)

The interpolating functions and can be found,for example, in [9], and the subindexes stand for differen-tiation with respect to the corresponding variable. The vector

contains the tangential component of at edges.Introduction of (16) in (15) yields a system of equations of

the form

(17)

where the elements of matrix are integrals of the interpo-lating functions and its derivatives over the elements and theyhave been extensively presented in [10]. The vector isgiven by

if

if(18)

Fig. 2. Normalized bistatic radar cross section of a dielectric shell with" = 4:0 and radii 0.45� and 0.5� (TMz case).

if the edge belongs to the contour and zero otherwise.In this equation, is the length of the edge, and ,are the values of the angle(see Fig. 1) for the nodes originand end of edge .

One important feature of matrix is that it does notdepend on the excitation field, so it may be factorized onceand then (17) can be solved for the different by backsub-stitution. This procedure is very efficient, because although theFEM problem should be solved times (once pereach mode), the main computational effort, which correspondsto the LU factorization of matrix , is carried out only once.

After computing the , the inmittance parameters can bestraightforwardly obtained from (5), as shown in (19) at thebottom of the page.

V. NUMERICAL RESULTS

In this point, the method developed in the previous sectionshas been checked by computing the bistatic radar cross sectionof an empty dielectric shell for which an analytic solution isavailable. Then, the proposed method has been applied to theanalysis of a square slotted screen in two different cases: whenthe screen is empty and when the screen is covering a perfectlyconducting circular cylinder.

A. Empty Dielectric Shell

First of all, we have studied the dielectric shell shown inFig. 2, in order to test the validity of the proposed procedure.The analysis has been performed using the inner and outerradii of the shell as and , respectively.

(19)

50 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998

In this case, the fields in the region admit twopossible expressions: in terms of Hankel functions of the firstand second kind or in terms of Bessel functions of the firstkind [7]. When all the fields in are represented byHankel functions, the coefficients and must be the samesince there is nothing to reflect nor absorb the incident power.In this case, we have

(20a)

(20b)

On the other hand, since the fields inside the screen willbe stationary, they can be represented only in terms of Besselfunctions of the first kind. In this case, we have and

(21a)

(21b)

In both cases, we take , whichcorresponds to the cylindrical harmonics of a plane wavepropagating in direction .

Once the coefficients are known from (20a) or (21a),the bistatic radar cross section can be evaluated as [7]

(22)

In Fig. 2 we present the radar cross section of a dielectricshell of radii 0.45 and 0.5 m. The material from which the shellis made has a permittivity of and the incident waveis a TM one, with the electric field given by .In Fig. 2, the computed value of is also compared with itsanalytical value, showing a very good agreement that validatesthe proposed method.

B. Empty Screen

Then, we have applied it to the study of the square slottedscreen presented in Fig. 3, made of perfect conductor, using

m and m. The analysis is performed in thesame way as described in the previous example. We presentin Fig. 4 the axial electric field computed when a TMpolarized wave of the form is incident over thescreen, with m (which is the cutoff wavenumberof the TM and TM modes of the associated squarewaveguide). We can see, however, that the field distributioninside the screen is that of the TM mode of the squarewaveguide, while the TM mode is not excited, because itsaxial electric field vanishes in the plane, where theslot is placed. This behavior coincides with the results of [1]and [4] and it has been checked by computing the magneticfield inside the envelope when a TEpolarized wave of theform with m is incident over thescreen. In this case, since the frequency of the incident wavecoincides with the cutoff frequency of the TEmode of thesquare waveguide whose axial magnetic field vanishes in theplane , no significant field is obtained inside the screen.

Fig. 3. Empty slotted square envelope, withL = 1 m, e = 20 cm, andd = 5 cm.

Fig. 4. Axial electric field obtained when a TMz wave with k =p13�

m�1 is incident along they axis over the empty cavity of Fig. 2.

C. Screen Covering a Circular Cylinder

When a lossy dielectric screen covers a perfectly conductingcircular cylinder of radius (see Fig. 5) the fields in the spacebetween the screen and the cylinder may be represented byusing Hankel functions of the first and second kind. In thiscase, the cylinder can be characterized by a diagonal scatteringmatrix given by

(23a)

in the TM case and by

(23b)

in the TE case. So, the coefficients for the field expansion are

(24a)

(24b)

BALBASTRE et al.: ANALYSIS OF ARBITRARILY SHAPED 2-D SLOTTED SCREENS 51

Fig. 5. Perfectly conducting circular cylinder of radiusR = 0:25m enclosedby a lossy slotted square envelope withL = 1 m, e = 20 cm, d = 5 cm,and � = 1000 �1 m�1.

(24c)

and is the same as in the empty cavity case.The electric current in the cylinder may be computed by

(25)

obtaining

(26a)

for the TM case, and

(26b)

for the TE case.We have applied this analysis to the structure shown in

Fig. 5, where the screen dimensions and the values ofand are the same than in the previous example, butnow it is considered made of a lossy dielectric having

m . First, we have computed the current in thecircular conductor, which is presented in Fig. 6 for both theTM and TE incidences. To check the relationship betweenthe peaks arising in the graphs of current and the modes ofthe corresponding closed waveguide, we have analyzed thatwaveguide using the FEM [11]. In Tables I and II the cut-off wavenumbers for the TM and TE modes are listedand compared with the wavenumbers where the peaks appear,

. A closed relationship can be observed between the cutofffrequency of some modes and the frequencies where the peaksare located, whereas for other modes no significant current isfound at . This is due to the fact that for modes with a givensymmetry around the inner conductor, the integral definingthe current in (25) vanishes, provided that the induced fieldpresents the same symmetry than the modal pattern does. So,graphs in Fig. 6 clearly show the coupling effect between theincident wave and the modes of the bidimensional structure,described in [1] and [4]. In Fig. 7 we show, for a direct

(a)

(b)

Fig. 6. Induced current on the shielded circular cylinder of Fig. 5. (a) TMz

case. (b) TEz case.

incidence, the axial field corresponding to the first peak ofcurrent in the TM and TE case.

In Fig. 8 we present the axial magnetic field correspondingto the TE mode in the coaxial structure and the axial magneticfield produced by a TEpolarized wave of the formwith m . The correspondence between both fieldsis evident and the kind of symmetry which makes the inducedcurrent given by (25) negligible can be appreciated.

We also present (in Fig. 9) the axial electric field for theTM mode of the coaxial structure and the electric axial fieldfor a direct TM incidence with m . Since theaxial electric field corresponding to this mode vanishes alongthe plane, the mode is not excited by the incident planewave.

Finally, we have computed the axial electric field producedwhen a TM polarized wave with m is incident overthe structure. Since in that case the frequency of the incidentwave does not match the cutoff frequency of any mode of the

52 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998

TABLE ICUTOFF WAVENUMBERS FOR THE FIRST TMz MODES

IN THE COAXIAL STRUCTURE OF FIG. 5 AND FREQUENCIES

OF THE PEAKS OF CURRENT IN THE GRAPH OF FIG. 6(a)

TABLE IICUTOFF WAVENUMBERS FOR THE FIRST TEz MODES IN

THE COAXIAL STRUCTURE OF FIG. 5 AND FREQUENCIES

OF THE PEAKS OF CURRENT IN THE GRAPH OF FIG. 6(b)

coaxial structure, no significant field is produced inside thecavity.

VI. NUMERICAL CONSIDERATIONS

In (1), a finite number of cylindrical harmonics have beenconsidered in the expansion of the electric or magnetic fieldsin the regions and . So, it is necessary tofind a criterion to chose the limits in the summations. In theexterior region the criterion we have chosen is to accountonly for the propagating solutions of the wave equation, sowe have taken [12], [13]. In the interior regionthe criterion has been different for the empty envelope andfor the covered cylinder. In the former case, since the solutioninside the cavity may be represented by a convergent series ofBessel functions of the first kind, we can take more harmonics

(a)

(b)

Fig. 7. (a) Axial electric field obtained when a TMz wave withk = 10:16

m�1 is incident along they axis over the enclosed cylinder of Fig. 5. (b)Axial magnetic field obtained when a TEz wave with k = 7:59 m�1 isincident along they axis over the enclosed cylinder of Fig. 5.

than . However, a problem arises when the numberof modes is high. To compute the inmittance parameters it isnecessary to solve the wave equation with an excitation of theform . The solution to that problem is obtained via theFEM, which requires the discretization of this functions. So,if we take high-order modes, we will need a lot of nodes atthe boundary to model them correctly, leading to prohibitivelylarge matrices in (13) and (17). We have found that a goodcompromise between accuracy and memory requirements isto choose . In the case of the covered cylinder, thesolution inside the screen is stated in terms of incident andscattered waves, so the number of modes chosen in this casehas been .

With the aforementioned criteria, in the problems solvedin this section the maximum number of modes has been

. Using ten samples per period of the highermode, the equation system (17) has 7100 unknowns and the

BALBASTRE et al.: ANALYSIS OF ARBITRARILY SHAPED 2-D SLOTTED SCREENS 53

(a)

(b)

Fig. 8. (a) Axial magnetic field pattern for the fifth TE mode of the coaxialstructure shown in Fig. 5. (b) Axial magnetic field obtained when a TEz wavewith k = 6:81 m�1 is incident along they axis over the enclosed cylinderof Fig. 5.

full matrix to be inverted in (13) has 42 rows and columns.The total amount of memory used to solve that problem was15 MB and the solution took 17 s on a HP-C160 workstation.

VII. CONCLUSIONS

A very powerful tool for studying the electromagneticsusceptibility of arbitrarily shaped slotted screens has beendeveloped. The proposed method is based on a circuitalcharacterization of the structure by a generalized inmittancematrix, via the FEM, which is then combined with a modalexpansion to compute the field inside and outside the envelope.Although we have centered our analysis in square slottedstructures, the versatility of the FEM permits one to apply thismethod to any bidimensional envelope no matter how manyslots or dielectric parts it contains. The only restrictions of thishybrid method are those derived from its numerical nature.In fact, depending on the available computational resources,

(a)

(b)

Fig. 9. (a) Axial electric field pattern for the forth TM mode of the coaxialstructure shown in Fig. 5. (b) Axial magnetic field obtained when a TMz

wave with k = 10:40 m�1 is incident along they axis over the enclosedcylinder of Fig. 5.

there will be a limit for the electrical size of the objects beinganalyzed. The analysis has been carried out for both TMandTE waves, showing a close relation between the susceptibilityproblem and the modes of the bidimensional structure. Thecircuital characterization of structures also allows one to studymultiple screen designs as well as interference problems.

REFERENCES

[1] R. W. Ziolkowski and J. B. Grant, “Scattering from cavity-backedapertures: The generalized dual series solution of the concentricallyloaded E-pol slit cylinder problem,”IEEE Trans. Antennas Propagat.,vol. AP-35, pp. 504–528, May 1987.

[2] E. H. Newman and M. Kragalott, “Moment method analysis of theelectric shielding factor of a conducting TM shield at ELF,”IEEE Trans.Electromagn. Compat.,vol. 37, pp. 400–407, Aug. 1995.

[3] S. Celozzi and M. S. Sarto, “Equivalent source method for the evaluationof the electromagnetic field penetration inside enclosures,”IEEE Trans.Magn., vol. 32, pp. 1497–1500, May 1996.

[4] J. V. Balbastre and L. Nu~no, “Susceptibility analysis of arbitrarilyshaped 2-D slotted screens using a hybrid modal-finite element tech-

54 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998

nique,” in 5th Topic. Meet. Elect. Performance Electron. Packaging,Napa, CA, Oct. 1996, pp. 220–222.

[5] B. Engquist and A. Madja, “Absorbing boundary conditions for thefinite-difference approximation of the time-domain electromagneticfield equations,”IEEE Trans. Electromagn. Compat.,vol. EMC-23, pp.37–382, Nov. 1981.

[6] J. M. Jin and J. L. Volakis, “A hybrid finite element method for scatteringand radiation by microstrip patch antennas and arrays residing in acavity,” IEEE Trans. Antennas Propagat.,vol. 39, pp. 1598–1604, Nov.1991.

[7] C. A. Balanis, Advanced Engineering Electromagnetics.New York:Wiley, 1982.

[8] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions.New York: Dover, 1970.

[9] J. M. Jin,The Finite Element Method in Electromagnetics.New York:Wiley, 1993.

[10] J. V. Balbastre, “Solucion de problemas electromagneticos cerrados yabiertos mediante el metodo de los elementos finitos, incluyendo mate-riales anis´otropos e inhomog´eneos,” Ph.D. dissertation, Dept. Commun.,Univ. Politecnica, Valencia, Spain, July 1996.

[11] L. Nu~no, J. V. Balbastre, and H. Casta~ne, “Analysis of general lossy in-homogeneous and anisotropic waveguides by the finite element methodusing edge elements,”IEEE Trans. Microwave Theory Tech.,to bepublished.

[12] D. R. Wilton and R. Mittra, “A new approach to the calculationof electromagnetic scattering properties of two-dimensional bodies ofarbitrary cross section,”IEEE Trans. Antennas Propagat.,vol. AP-20,pp. 310–317, May 1972.

[13] S. K. Chang and K. K. Mei, “Application of the unimoment method toelectromagnetic scattering of dielectric cylinders,”IEEE Trans. AntennasPropagat.,vol. AP-24, pp. 35–42, Jan. 1976.

Juan V. Balbastre Tejedor (S’90–M’97) was bornin Mislata, Spain, in 1969. He received the M.S.E.and Ph.D. degrees in telecommunication engineer-ing, both from Polytechnic University of Valencia,Spain, in 1993 and 1996, respectively.

In 1993, he joined the Department of Commu-nications at the Polytechnic University of Valenciaas a Teaching Assistant, where he is currently aLecturer of electromagnetic theory since 1997. Hismain research interests lay in numerical methods,electromagnetic compatibility, and industrial mi-

crowave heating systems.

Luis Nuno (M’95) was born in Valencia, Spain, in1963. He received the M.S.E. degree in telecommu-nication engineering from the Polytechnic Univer-sity of Madrid (UPM), Spain, in 1987 and the Ph.D.degree in telecommunication engineering from thePolytechnic University of Valencia (UPV), Spain,in 1993.

Miguel Ferrando Bataller (S’81–M’83) was bornin Alcoy, Spain, in 1954. He received the M.S.E.and Ph.D. degrees in telecommunication engineer-ing both from Polytechnic University of Catalonia,Barcelona, Spain, in 1977 and 1982, respectively.

From 1977 to 1982, he was a Teaching Assistantin the Antennas, Microwave, and Radar Group atthe Polytechnic University of Catalonia and anAssociate Professor since 1982. In 1990, he joinedthe Polytechnic University of Valencia, Spain, wherehe is currently Vice-Chancellor and Professor of

Antennas and Satellite Communications in the School for TelecommunicationsEngineering. He is currently involved in research activities in numericalmethods, radar cross section, antennas, propagation, and telecommunicationssystems.