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A Fast Circuit Model Description of the Shielding Effectiveness of a Box With Imperfect Gaskets or Apertures Covered by Thin Resistive Sheet Coatings Tadeusz Konefal, John F. Dawson, Member, IEEE, Andrew C. Marvin, Member, IEEE, Martin P. Robinson, and Stuart J. Porter, Member, IEEE
Abstract—This paper presents an intermediate level circuit model (ILCM) for the prediction of the shielding effectiveness (SE) of a rectangular box containing one or more rectangular gaskets of known transfer impedance or constitutive parameters. The box may also possess a rectangular aperture covered by a thin resistive sheet. The ILCM takes into account multiple waveguide modes and is thus suitable for use at high frequencies and with relatively large boxes. The gaskets may be positioned anywhere in the irradiated front face of the box, and the SE at any point within the box may be found when irradiated by a plane wave. Solution times using the ILCM technique are three orders of magnitude less than those required by traditional numerical methods such as finite difference time domain (FDTD), transmission line matrix (TLM), or method of moments (MoM), even when using a relatively slow interpreted language such as MATLAB. Accuracy, however, is not significantly compromised. Comparing the circuit model with TLM over eight data sets from 4 MHz to 3 GHz resulted in an rms difference of 3.90 dB and a mean absolute difference of 2.35 dB in the predicted SE values. The ILCM accurately reproduces the detailed structure of the SE curves as a function of frequency and observation point. Index Terms—Circuit model, gaskets, multiple modes, shielding effectiveness (SE).
I. INTRODUCTION FREQUENTLY occurring problem in the discipline of electromagnetic compatibility (EMC) is the determination of the electromagnetic fields inside an enclosure containing apertures and/or gaskets. The problem of a rectangular aperture in a rectangular box has been approached using a number of different computational and analytic techniques. For example, in [1]–[4] the electric field integral equation (EFIE) is used to solve the problem using the method of moments (MoM). Two other common computational techniques used to solve the problem are the finite-difference time-domain (FDTD) method [5] and the transmission line matrix (TLM) method [6]. Such methods can be efficient, but are usually computationally intensive and require large amounts of computer random access memory (RAM) and hard disk space. Run times of several hours or days are not uncommon to reach a solution, despite the significant and continuing improvement that has been made in commonly
A
Manuscript received May 11, 2005; revised October 30, 2005. This work was supported by BAE Systems. The authors are with the Department of Electronics, University of York, Heslington, YO10 5DD, U.K. (e-mail:
[email protected];
[email protected]. ac.uk;
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/TEMC.2006.870703
available computing resources in recent years. (Clock speeds of 2–3 GHz are typical of computing equipment at the time of writing.) Broadly speaking, these traditional computational methods are found to agree with each other and experiment, to within a few decibels. The slowness of solution using these methods has led to a number of intermediate level circuit models (ILCMs) being devised to describe the interior excitation of a box with an aperture, irradiated by a plane wave [7]–[11]. The ILCM technique considerably simplifies the original excitation problem by providing a relatively simple equivalent circuit representation of the box and aperture. The equivalent circuit is then usually solvable in a few lines of computer code, using only minimal RAM and hard disk space. Although such ILCMs are often limited in their generality (for example, only TE10 mode excitation is considered in [7]–[9]), they provide accurate results in fractions of a second rather than several hours. The ILCM technique has also been applied successfully to provide rapid solutions to the coupling between elements such as monopoles, dipoles, loops, and transmission lines inside rectangular or cylindrical boxes [12]–[16]. In [12], [13], and [16], a multimode treatment is presented that extends the validity of the ILCM to cope with high frequencies and large boxes, where many higher order modes propagate inside the box in addition to the TE10 mode. The multimode treatment is extended to cope with an aperture in a box in [17]. Since an aperture in even a perfectly electrically conducting (PEC) box allows electromagnetic energy to enter the box, with possible resulting EMC problems, efforts will often be made to close apertures with a shielding conductive material. Normally this takes the form of a gasket. The performance of a gasket is a function of the PEC box geometry in addition to the constitutive parameters and geometry of the gasket itself. Despite these complications, one of the first useful quantitative investigations into the use of a sheet material as an electromagnetic (EM) shield was made by Schelkunoff [18]. Section IV describes the basic result of this theory, which is often used as a reference point for more sophisticated models. The theory assumes that the shield/gasket is a resistive sheet of infinite extent that is irradiated by a plane wave. The plane wave assumption is a limitation if we consider a PEC box containing the shield to be in the near field of an interfering electric or magnetic source. The latter cases are dealt with in [19], with the treatment being extended to cope with arbitrary distances from the shield in [20]. The treatment considers the near field of an electric or magnetic dipole in terms of an expansion of plane waves
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KONEFAL et al.: FAST CIRCUIT MODEL DESCRIPTION OF SE OF BOX WITH IMPERFECT GASKETS OR APERTURES
travelling obliquely to the shield/resistive sheet; oblique incidence of a plane wave on the sheet is dealt with in [21]. A very useful extension of the Schelkunoff result, applicable to general quadric surfaces (plates, spheres, ellipsoids, circular and elliptic cylinders) constructed from imperfectly shielding material shells, was developed by Kaden and is described in [22]. Some basic results of this theory (originally published in German) are summarized in [23]. More accurate and general computational techniques for analyzing gaskets/thin resistive sheets in PEC boxes have been developed. For example, a general MoM treatment of penetrable resistive, dielectric, or ferrite materials, including embedded or cladded materials, is given in [24]. Finite-element (FE) and FDTD implementations of the shielding effect of a thin resistive sheet are described in [25] and [26]. EM diffusion through thin resistive sheets and composite material shells is modeled in [27], [28] using the TLM method. In view of the lengthy computer solution times involved in such traditional numerical methods, attempts have been made to improve computational efficiency in the treatment of materials with such “general” constitutive parameters (i.e., non-PECs). For example, the fast adaptive integral method (AIM) is used in [29] to analyze scattering from metallic and more general resistive/dielectric surfaces by adapting the EFIE in a suitable manner. Using a combination of field and circuit theory, the shielding effectiveness (SE) of a completely closed enclosure constructed from an imperfect conductor is efficiently modeled in [30] using a “boundary impedance matrix” concept. Despite the success of researchers’ efforts in modeling materials with such general constitutive parameters, be they composite shells or individual gaskets, such full wave numerical solvers are fundamentally computationally intensive, and therefore nearly always take a long time to reach solution. We note also a distinct lack of experimentally measured data in the literature with which to compare such models, presumably because of the difficulty in characterizing a practical gasket (e.g., its transfer impedance or constitutive parameters as a function of frequency). In this paper, we combine the Schelkunoff theory of [18] with the technique of ILCM described in [7]–[17] and [31]. The result is a very fast ILCM which will predict the SE of a rectangular box possessing panels with joints sealed by one or more gaskets of known geometry and constitutive parameters. The ILCM can readily cope with rectangular gaskets or thin resistive sheets positioned anywhere in the irradiated front face of the box, and will give the SE at any arbitrary position within the box. Although we consider here only one polarization and direction of travel for the incident plane wave, extension of the model to cope with arbitrary polarization and incidence angle is feasible. The model can readily be used at high frequencies (we have set an upper limit of 3 GHz for convenience, where the box length corresponds to three free-space wavelengths) without deterioration in accuracy, provided an appropriate number of modes are taken into account. The model has been developed in such a way that existing ILCM techniques for modeling the presence of elements such as dipoles, monopoles, or loops inside the box can easily be incorporated into the circuit, though for simplicity we present only the results for an empty box here.
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Fig. 1. (a) Realistic implementation of gasket in TLM. (b) Corresponding approximation used in ILCM and TLM.
The ILCM is valid if the gaskets are “good” in the sense that an incoming wave is attenuated reasonably well (e.g., by a factor of 10 or more), but the model will not be valid at low frequencies or if the conductivity of the gasket material is so low that the gasket resembles an aperture. Some further guidelines on model validity are given in Section VI. The circuit model is rapid, taking typically 0.65 s to process 750 frequency data points when coded in C++. This contrasts with a typical run time of four and a half hours for the numerical technique of TLM modeling. Despite its simplicity, the ILCM is remarkably accurate, showing an overall rms difference in SE values of 3.90 dB compared with TLM and a mean absolute difference of 2.35 dB. Visual examination of the curves in Section VI supports the validity of the circuit model, with the vast majority of the many features in the TLM simulations reproduced by the circuit theory. Sections II and III describe the basic shielding problems at hand, and illustrate the equivalent ILCM circuit that represents and simplifies the problems. Section IV summarizes the basic shielding result of Schelkunoff [18], while Section V describes the various reference numerical TLM simulations carried out. The TLM results are compared with the results of the ILCM in Section VI, with some conclusions being drawn in Section VII. II. SOME BASIC GASKET PROBLEMS Fig. 1(a) illustrates a realistic physical layout, which is readily modeled in TLM, for the use of a gasket. Unfortunately, it is not possible to implement this arrangement directly in the present ILCM. Instead, the ILCM uses an approximation to this arrangement, as illustrated in Fig. 1(b). Fig. 2 illustrates the use of a single EMC gasket in one edge of the lid of a rectangular box of length d. Figs. 3 and 4 illustrate the use of a gasket in more realistic situations, where there are effectively four long rectangular strips of gasket material holding a metal panel in place. In Fig. 4 the entire front panel is held in place by gasket material, this being a typical experimental scenario. Fig. 5 shows a scenario whereby a thin resistive sheet is used to
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Fig. 5.
Fig. 2.
Rectangular resistive sheet in center of front panel.
Illustration of model for a single gasket in rectangular box.
Fig. 6. Rectangular panel in top right hand corner of front face, surrounded by gasket.
Fig. 3.
Rectangular panel in center of front face, surrounded by gasket.
Fig. 4.
Entire metallic panel in front face surrounded by gasket.
cover an aperture in a box, which, for example, could represent a conductive coating over a display. Finally, Fig. 6 illustrates the use of a rectangular panel placed asymmetrically in the front face of the box, and held in place by four strips of gasket. In each of Figs. 2–6, the box is of size a(x) × b(y) × d(z) m3 , and is considered to be constructed from a perfect electrical conductor (PEC). The gasket material has thickness dg , a finite conductivity σg , permittivity εg , and permeability µg . The box ˆ polarized plane wave of peak is irradiated from the left by a y ˆ direction. magnitude Eyinc , travelling in the z Both of the arrangements in Fig. 1(a) and (b) are implemented in TLM in the case of Fig. 3 for comparison with the ILCM in Section VI. The conversion from Fig. 1(a) to (b) is made in both the width and height of the front panel in Fig. 3. As we shall see, this subtle difference in implementation of the gasket [Fig. 1(a) and (b)] is found to have only a minor effect on the SE of the box
as predicted by the numerical technique of TLM (less than 3 dB rms difference). The gaskets in the remaining Figs. 2, 4–6 are therefore modeled as in Fig. 1(b) in TLM, to give a direct comparison with the ILCM. In fact, in TLM simulations a perfectly electrically conducting (PEC) wall is idealized to have infinitesimal thickness. The thickness dg of the box wall in Figs. 1 and 2 is thus for illustrative purposes only (though of course dg has a literal meaning as far as the gasket material itself is concerned). Also in TLM, the transmission and reflection properties of a gasket of small but finite thickness dg are implemented within an idealized wall of infinitesimal thickness. The box dimensions a, b, and d can thus be considered as either internal or external dimensions. However, for any practical experiment (with finite thickness box walls) carried out to verify the TLM results, the dimensions a, b, and d must clearly be the internal dimensions. III. EQUIVALENT CIRCUIT FOR BOX WITH GASKET A. Field Expansion in Box It is readily shown [32] that the electric fields inside a rectangular box can be expanded in terms of its permitted forward and reverse travelling TE and TM basis modes. For example, for an infinitely long waveguide in the half space z > 0, or alternatively for a waveguide of finite length d (as in Figs. 2–6) with a perfectly absorbing back wall in the plane z = d, the fields Ey and Ex inside the box can be expanded in terms of forward traveling waves as Ey (r) =
∞ ∞ m =1 n =0
×
TE Cfmn exp(−γmn z)
mπ a
mπx nπy sin cos a b
KONEFAL et al.: FAST CIRCUIT MODEL DESCRIPTION OF SE OF BOX WITH IMPERFECT GASKETS OR APERTURES
+
∞ ∞
137
TM Cfmn exp(−γmn z)
m =1 n =1
nπ
mπx nπy sin cos b a b ∞ ∞ TE Cfmn exp(−γmn z) Ex (r) = − ×
m =0 n =1
× +
nπ b ∞
cos
∞
mπx
mπ a
nπy sin b
TM Cfmn exp(−γmn z)
m =1 n =1
×
a
(1)
cos
mπx a
nπy sin . b
(2)
The propagation constant γmn of the TEmn /TMmn modes in the above expansions is given by γmn = + µ0 ε0 (ωc2 − ω 2 ) (3) mπ 2 nπ 2 ωc2 µ0 ε0 = + . (4) a b If we have a rectangular aperture extending from x = xl to x = xh and y = yl to y = yh in the front face of the box (the plane z = 0), we can ultimately express the coefficients C in (1) and (2) in terms of the field in the aperture Eyaperture under steady state conditions. If we assume (a) the usual boundary condition of zero tangential electric field on the inner PEC surface of the front plate with rectangular slot; (b) that Ey in the aperture is in fact independent of y, i.e., Eyaperture = Eyaperture (x); and (c) that the field Ex in the aperture is approximately zero, we can show [17] that the coefficients C in (1) and (2) are given by −1 m π TE TE m π nπ n Cfmn 2N Cfm0 a b N0 a = (5) TM − nbπ maπ 0 Cfmn
Fig. 7. Equivalent circuit model for box with gasket (only two modes shown).
B. Circuit Representation Using Analogous Transmission Line Theory Using the concept of an analogous transmission line to represent each waveguide mode inside the boxes of Figs. 2–6 [12], [31], the equivalent circuit representing the modal excitation of a typical box with a gasket in its front face is given in Fig. 7. Only two modes are shown in Fig. 7 for simplicity, though in practice it is necessary to include as many propagating modes as are present below the highest frequency of interest. In Fig. 7 the main body and conducting back wall of the (n ) (n ) box are represented by an impedance ZT = Zc tanh(γ (n ) d) (n ) for each mode (n), where Zc and γ (n ) are the characteristic impedance and propagation constant of the mode/analogous transmission line, respectively. γ (n ) (≡ γmn ) is given by (3) (n ) while Zc is given by TE Zmn = jγω(nµ)0 TE modes (n ) Zc = (8) (n ) TM Zmn = jγω ε 0 TM modes (n )
where
xh
= xl
and
Nn =
b TE mπ a C N0 fm0 a 2 mπx dx Eyaperture (x) sin a
yh −
yl , n πny h= 0
b − sin n πby l , n π sin b
(6)
n ≥ 1.
(7)
Assumptions (b) and (c) are good if the aperture is “slot like” with (yh − yl ) (xh − xl ) and if (yh − yl ) is much shorter than a fraction of a wavelength of radiation at the frequency considered. Under these circumstances, the slot can be approximated as a transmission line supporting TEM waves (hence, Ex ≈ 0 in the slot). Alternatively, if the aperture is filled with a “good” gasket material (i.e., the gasket attenuates the electric field significantly as it traverses the gasket), assumptions (b) and (c) will be good irrespective of the slot size and shape, if the ˆ polarized as in Figs. 2–6. Assumption (a) is incident field is y of course always good.
ZT is simply the impedance looking into a short-circuited section of transmission line of length d, characteristic impedance (n ) Zc , and propagation constant γ (n ) . The PEC wall at the back of the box, which induces reverse travelling modes inside the box, provides the short circuit. The resulting multiple reflections between the planes z = 0 and z = d for each mode are dealt with naturally by the circuit model of Fig. 7, and the fields for an infinitely long box given by (1) and (2) are altered to include the reverse travelling modes. Each mode (n) is excited (n ) by a voltage source Vwg that depends on the field Ein present on the inner surface of the gasket. Ein results from penetration of the external field through the gasket, and is nonzero for an imperfect gasket. Ein of course depends linearly on the external field Eyinc incident upon the box, and is the corresponding field which should be used to replace Eyaperture (x) appearing in (6) for a gasket free aperture. In the latter case, intermode coupling will take place in the aperture, which complicates the problem somewhat. This problem is dealt with in [17]. However, for a reasonably good gasket we will normally have |Ein | |Eyinc |, and the gasket viewed from the inside of the box will appear as an approximate short circuit. Under these circumstances we can
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ignore intermode coupling at the gasket and treat the voltage (n ) source Vwg as a short circuit as far as reverse travelling modes incident upon the front panel from within the box are concerned. This simplifies the problem considerably. (n ) The relationship between the voltage source Vwg and its corresponding modal amplitude coefficient C in (1) and (2) is established by insisting that the power flow down the waveguide due to mode (n) should equal the power flow down the analogous transmission line for mode (n). For an infinitely long waveguide we find
n π 2 TE TE mπ 2 − Z mn + Cfmn TE modes j ω µ0s b (n ) a
= Vwg .
2 2 mπ TM − 1 Cfmn + nbπ TM modes u γ mn a (9) where [12] s=
TE Zmn 2αmn
12 (10)
ab −jωµ0 γmn αmn =
2
2 2 8 mπ + nbπ a mπ 2 (1 − δm 0 )(1 + δn 0 ) × a nπ 2 + (1 + δm 0 )(1 − δn 0 ) b u=
1
(11) (12)
1
Z TM ) 2 (2αmn mn
ab −ωε0 jγmn = 2 2 . αmn mπ 8 + nbπ a
(13)
Here δk l is the Kronecker-delta symbol, equal to zero for TE k = l and unity for k = l. The characteristic impedances Zmn TM and Zmn of the analogous transmission lines for TE and TM modes, respectively, are given by (8). Once again, the circuit of Fig. 7 will deal naturally with the multiple reflections that occur in the box once the conducting wall in the plane z = d is introduced. In this case, it is a simple matter to express the forward (Vf0 ) and reverse (Vr0 ) traveling voltage waves for the (n ) analogous transmission line in terms of Vwg , i.e., (n )
Vf0 =
Vwg
1 − exp −2γ (n ) d
Vr0 =
Vwg
. 1 − exp +2γ (n ) d
(14)
(n )
(15)
For TE modes, the forward and reverse voltage waves are related to the forward and reverse traveling electric fields Eyfor and Eyrev for the waveguide mode by Eyfor (r) = −
Vf0 jωµ0 sfx exp(−γmn z) TE γ Zmn mn
(16)
Eyrev (r) = −
Vr0 jωµ0 sfx exp(+γmn z) TE γ Zmn mn
(17)
where mπ mπx nπy γmn fx = 2 2 sin cos . mπ nπ a a b + a b (18) For TM modes, the corresponding relationships are Eyfor (r) = −Vf0 ugy exp(−γmn z)
(19)
(20) Eyrev (r) = −Vr0 ugy exp(+γmn z) nπ γmn gy = 2 2 mπ nπ b + b a nπy mπx cos . (21) × sin a b By summing the fields due to forward and reverse waves for all the TE and TM modes considered, the overall total field EyTot (r) at any point inside the box may be calculated. The SE at that point is then given by EyTot (r) (22) SE(r) = −20 log10 Eyinc (Other definitions for SE may also be used, but since we are completely ignoring Ex and Ez this definition will suffice for comparison with numerical simulation.) While the theory of this section deals with a single rectangular gasket extending from x = xl to x = xh and y = yl to y = yh , the principle of superposition allows us to cater for multiple rectangular gaskets. Thus, in each of Figs. 3, 4, and 6, we can account for all four gaskets by working out the equivalent modal voltage (n ) source Vwg(k ) for each individual gasket (k = 1, . . . , 4) from (9). The equivalent circuit for an experimental configuration with four gaskets is then precisely the same as in Fig. 7, where 4 (n ) (n ) = Vwg(k ) . (23) Vwg k =1
The circuit model of Fig. 7 is readily extended to model the presence of dipoles, monopoles, or loops inside the box, using existing ILCM techniques [12]–[14], though for simplicity we treat only an empty cavity here. It is clear from (3)–(22) that the SE at any point within the box illustrated in Fig. 2 (with a single long, thin rectangular gasket) can be expressed in terms of the field Eyaperture (x) of (6), which becomes the field appearing on the inner surface of the gasket that fills the aperture. In the case of a “good” gasket material filling the aperture, the field Eyaperture (x) is approximately equal to Ein considered earlier in this section, a constant value of field (i.e., independent of x and y) that appears over the internal surface of the gasket as the external field Eyinc penetrates the gasket. To a good approximation, the value of Ein is unaltered by the reverse travelling modes impinging on the gasket from within the box; such modes see an approximate short circuit. It remains to calculate the field Ein on the inner surface of the gasket when an external incident field Eyinc impinges on its exterior surface. This is done in Section IV.
KONEFAL et al.: FAST CIRCUIT MODEL DESCRIPTION OF SE OF BOX WITH IMPERFECT GASKETS OR APERTURES
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mation to the aperture field Eyaperture (x) on the inner surface of the gasket in Fig. 2 and in (6). This approximation is justified provided the metal box surrounding the gasket resembles an ˆ “infinite sheet,” allowing uninhibited current flow in the y ˆ polarized incident field), as in the original direction (for a y theory of [18]. Note that at low frequencies the ILCM will therefore not be valid, since we cannot set up a constant current on a galvanically isolated box in the presence of a static electric field. This point is discussed further in Section VI, where we attempt to provide some guidelines on the low frequency limit for validity of the ILCM. V. SIMULATIONS PERFORMED USING TLM
Fig. 8.
(27)
In order to test the circuit model in Section III, some reference numerical simulations were carried out using the TLM method. Figs. 2–6 illustrate the different simulations that were carried out. In each of the simulations a mesh size of 5 mm was used, with the box dimensions being a = 0.3 m, b = 0.12 m, d = 0.3 m. From the point of view of resonant behavior of the box, these must be considered as internal dimensions, despite the fact that in TLM the PEC and gasket boundaries effectively have infinitesimal thickness. Table I gives further information on the nature of the gaskets used in Figs. 2–6, and the points of observation inside the box. With the exception of Fig. 5, the width wg of the rectangular gaskets is set at 5 mm. In Figs. 3 and 5, the “hole” in the front face of the metallic box is a centrally placed 5 cm square. In Fig. 5 this hole is filled with a resistive sheet as in Table I, while in Fig. 3 the hole is filled with a square metal panel of side 4 cm, and surrounded by gaskets of width wg = 5 mm. As mentioned in Section II, two different implementations of the gasket in Fig. 3 were simulated in TLM, according to the physical arrangements illustrated in Fig. 1(a) and (b). In practice, the approximate arrangement in Fig. 1(b) is found to produce only minor degradation in the accuracy of the results. In Fig. 6, the square hole, gaskets, and metal panel of Fig. 3 have been physically relocated to an asymmetric position in the top right hand corner of the front face of the box. Some adjustments to the thickness dg of the gasket and its conductivity σg have also been made. Indeed, apart from the dimensions of the box, the TLM simulations carried out in Table I exhibit a number of different values for dg and σg . Some limitations on the latter two quantities are discussed in Section VI. For simplicity, the gasket is assumed to be nonmagnetic (µg = µ0 ) and to have a permittivity equal to that of free space (εg = ε0 ). The TLM simulations use the thin conducting boundary model of [28], which accurately represents the penetration of an external field through the gasket. In performing the reference TLM simulations, the aim was to consider a variety of gasket types, positions, and field observation points without producing an excessive amount of data. In all cases the incident field consists of ˆ polarized wave travelling in the z ˆ direction. ay
(28)
VI. COMPARISON WITH CIRCUIT MODEL
Field penetration of plane wave through a thin conducting wall.
IV. SOLUTION FOR Ein For a gasket with known (measured) frequency-dependent transfer impedance ZTnsf (ω) (in units of Ωm), the resulting transmitted field Ein through the gasket due to an incident field Eyinc (see Fig. 8) is given approximately by 2ZTnsf (ω) inc Ey (24) ZFS wg where wg is the width of the gasket as illustrated in Fig. 1 and ZFS = 377 Ω is the impedance of free space. Equation (24) is a good approximation provided the gasket is a “good” one, i.e., it reflects most of the incident energy upon the gasket (i.e., the electric field reflection coefficient ρE ≈ −1). However, note that for reasons discussed below and in Section VI, (24) cannot be used in the low frequency limit, where the ILCM will not be valid. In the case of a homogeneous gasket of known constitutive parameters we can apply the theory of Schelkunoff [18]. The basic problem is illustrated in Fig. 8, where a field Eyinc is normally incident on an infinite conducting medium of thickness dg and impedance Zg given by jωµg Zg = . (25) jωεg + σg Ein =
Here εg , µg , and σg are the permittivity, permeability and conductivity of the gasket material, respectively. The resulting transmitted field Ein through the gasket is given by
1 − ρ221 exp(−γg dg ) inc E Ein = (26) 1 − ρ221 exp(−2γg dg ) y where Zg − ZFS Zg + ZFS γg = jωµg (jωεg + σg ).
ρ21 =
The value of Ein given by (26) [or (24) for an experimentally determined value of ZTnsf (ω)] is used as a reasonable approxi-
In this section, the results of the ILCM presented in Section III are compared with results of numerical modeling using the
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TABLE I SUMMARY OF NUMERICAL SIMULATIONS CARRIED OUT USING TLM, FOR COMPARISON WITH CIRCUIT MODEL
Fig. 9. Comparison of the shielding effectiveness predicted by the ILCM with the numerical prediction of TLM for the box, gasket, and position indicated in Case 1(b) of Table I and Fig. 2. (σ g = 100 Sm−1 , dg = 1.5 mm, robs = (22.25, 5.75, 22.25) cm.)
Fig. 10. Comparison of the shielding effectiveness predicted by the ILCM with the numerical prediction of TLM for the box, gasket and position indicated in Case 2(a) of Table I and Fig. 3. (σ g = 300 Sm−1 , dg = 1.0 mm, robs = (14.75, 11.75, 15.25) cm.)
TLM method. The TLM simulations performed are summarized in Figs. 2–6 and Table I. The circuit model and TLM values for SE as defined in (22) are compared in Figs. 9–13 for the five different cases, where it can be seen that agreement between the ILCM and numerical simulation is generally very good, both qualitatively and quantitatively. The upper frequency range considered here is 3 GHz, where many higher order modes are present. In fact, the circuit model of Section III includes all modes up to m = 5, n = 5, a total of 55 modes (ignoring TE0n modes which are not excited). This choice ensures that all propagating modes below 3 GHz are included in the ILCM of Fig. 7. Many of the higher order modes will be evanescent for much of the frequency range below 3 GHz, and this is reflected
by the fact that γmn is real. This does not however cause any difficulties with the theory, which can readily accommodate evanescent modes. For larger boxes and/or higher frequencies it is necessary to increase the limits on m and n, and this is easily implemented in the ILCM presented here. As a general rule, it is best to at least include all propagating modes up to the highest frequency of interest (in our case 3 GHz), though the inclusion of some higher order evanescent modes can improve the results further (see for example [12]). Table II summarizes the statistics for the agreement between the circuit model and TLM in Figs. 9–13 and the remaining cases in Table I; the remaining cases have not been plotted here for the sake of brevity. The normalized cross correlation coefficient
KONEFAL et al.: FAST CIRCUIT MODEL DESCRIPTION OF SE OF BOX WITH IMPERFECT GASKETS OR APERTURES
ρdB (0) is defined here as +∞ ρdB (ω0 ) = −∞ +∞ −∞
Fig. 11. Comparison of the shielding effectiveness predicted by the ILCM with the numerical prediction of TLM for the box, gasket, and position indicated in Case 3(b) of Table I and Fig. 4. (σ g = 500 Sm−1 , dg = 0.6 mm, robs = (22.25, 5.75, 22.25) cm.)
Fig. 12. Comparison of the shielding effectiveness predicted by the ILCM with the numerical prediction of TLM for the box, gasket, and position indicated in Case 4(a) of Table I and Fig. 5. (σ g = 100 Sm−1 , dg = 1.0 mm, robs = (14.75, 11.75, 15.25) cm.)
Fig. 13. Comparison of the shielding effectiveness predicted by the ILCM with the numerical prediction of TLM for the box, gasket, and position indicated in Case 5(a) of Table I and Fig. 6. (σ g = 100 Sm−1 , dg = 0.8 mm, robs = (14.75, 11.75, 15.25) cm.)
141
S1 (ω)S2 (ω − ω0 )dω +∞ S12 (ω)dω −∞ S22 (ω)dω
(29)
and is always less than or equal to unity. S1 and S2 are the SE responses in decibels. The overall rms difference between the curves in Figs. 9–13 and the remaining cases in Table I is 3.90 dB, with a mean absolute difference of 2.35 dB and a correlation coefficient ρdB (0) = 0.9981. On visual examination of the curves, it is clear that the agreement is very good. The rms difference emphasizes the larger differences between the curves S1 (ω) and S2 (ω), while the mean absolute difference weights the differences equally. If S1 (ω) and S2 (ω) were identical then we would have found ρdB (0) = 1. Indeed, we would find ρdB (0) = 1 even if S1 (ω) = C0 S2 (ω) for some arbitrary constant C0 , so that ρdB (0) gives some measure of the similarity of the shapes of the responses S1 (ω) and S2 (ω) in decibels. The values obtained here for the three chosen figures of merit in Table II are remarkable considering the wide dynamic range (∼100 dB) exhibited in the SE of the box over the frequency range 4 MHz–3 GHz. We note from Section II that by implementing the gasket in Fig. 3 [Case 2(a) in Table I] as in Fig. 1(a) in TLM, instead of the approximate manner illustrated in Fig. 1(b), the agreement with the ILCM [which itself uses the approximation of Fig. 1(b)] is only slightly degraded. With the TLM configuration of Fig. 1(a), the rms difference between TLM and the ILCM for Case 2(a) in Table I is 5.95 dB, with a mean absolute error of 4.63 dB and correlation coefficient ρdB (0) = 0.9972. This compares with the improved figures of 3.46 dB (rms difference), 1.77 dB (mean absolute difference), and ρdB (0) = 0.9986 in Table II, obtained when both TLM and the ILCM implement the gasket of Fig. 3 in the approximate manner shown in Fig. 1(b). The time taken for the circuit model to reach a solution is much smaller than that taken by the numerical method of TLM. The data in each of Figs. 9–13 consist of 750 frequency points separated by approximately 4 MHz, with TLM taking over 4 h to reach solution (Pentium III, 750 MHz). In contrast, on the same computer the circuit model running in MATLAB (a relatively slow interpreted language) took just 14 s to reach solution, faster by over three orders of magnitude. Indeed, when compiled in C++ the circuit model provides the same data in approximately 0.65 s. This represents a speed of solution over 20 000 times faster than TLM, with no significant deterioration in accuracy. At the same time, the circuit model requires only minimal RAM (∼812 kB) and hard disk space (∼17 kB). This contrasts with 20 MB of RAM and 9 MB of hard disk space required by TLM. In many of the results presented in Figs. 9–13, there is a tendency for the TLM and circuit model results to diverge at the low frequency end, say below 200 MHz. The reasons for this are twofold. Firstly, it should be recognized that a gasket with any nonzero conductivity σ will ultimately provide infinite SE at DC (0 Hz). Charge will flow in the gasket and PEC box in such a way
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TABLE II SUMMARY OF AGREEMENT OF CIRCUIT MODEL WITH TLM
as to internally cancel any external electric field Eyinc applied to the box. In some of the TLM results (e.g., Fig. 12) we see a tendency for the curve to approach large SE values for low frequencies. It must be borne in mind however that a practical TLM simulation must be truncated in the time domain at some suitable point. Usually this is chosen when the Gaussian pulse response at a point has decayed by about 40 dB of its highest value. In fact, the TLM simulations performed in Figs. 9–13 (30 000 time steps) resulted in about 35 dB decay of the Gaussian pulse, which is sufficient for most of the higher frequencies. However, the effect of prematurely curtailing the time domain response has a detrimental effect on the Fourier transformed frequency-domain response, particularly at the low frequency end. This can be true even when the time domain response has decayed by more than 40 dB. We therefore should not place too much faith in the low-frequency TLM results of Figs. 9–13, which are likely to be compromised by the finite simulated time-domain response. Indeed, by slightly varying the precise moment of termination of the Gaussian pulse response in the TLM simulations described here, significant variations (∼6 dB) in the Fourier transformed frequency response are possible below a frequency of about 200 MHz, with even greater variations at lower frequencies. The response at higher frequencies is not altered significantly by varying the termination point. The frequency of 200 MHz should therefore be considered a lower limit of validity for the TLM simulations. Secondly, we note that the ILCM itself will not be valid at low frequencies. For example, the field Ein on the inner surface of the gasket in (26) and Fig. 8 approaches the value Ein = {2/(2 + σg dg ZFS )}Eyinc as ω → 0. However, this will only be possible at DC if a DC current is allowed to flow in the gasket material. Whilst the theory of [18] allows this to happen for an infinitely large gasket sheet, it clearly cannot happen for a gasket positioned in a box that is galvanically isolated in the presence of a static DC field. We should therefore also be sceptical about the low-frequency results of the ILCM, and
try to define a lower limit on the frequency for validity of the model. Free circulation of current through the gasket will be possible if the metal box surrounding the gasket resembles an “infinite sheet,” as mentioned in Section IV. For example, if the box length d in Fig. 2 corresponds to a half wavelength of free ˆ polarized plane wave space radiation, it is feasible that for a y incident on the front face (z = 0), current can flow freely around the perimeter of the box, with current nodes at (y = b, z = d/2), (y = 0, z = d/2) and current maxima on the front and back faces of the box (z = 0, z = d). On the other hand, if the perimeter length 2(b + d) is much less than a wavelength (e.g., 2(b + d) ≤ λ/10), the circulation of current will be severely restricted and the ILCM will not be valid. In practice, the assessment as to whether current flow is restricted or not depends on the precise geometry of the box and the position of the gasket itself. For the box dimensions and gasket positions considered here, we would expect the ILCM to be valid at frequencies where the perimeter 2(b + d) of the box is greater than a free-space wavelength of radiation so that natural current nodes and maxima can be established on the box. The current nodes are unlikely to occur on the front face of the box containing the gasket, since currents on this face are directly excited by the incident radiation. The low-frequency limit for validity of the ILCM is therefore found from the condition 2(b + d) ≥ λ, or f ≥ c/{2(b + d)} = 357 MHz. It can be seen from Figs. 9–13 that agreement of the ILCM with TLM above this frequency is generally very good. If the conductivity σg of the gasket is so low that the gasket resembles an aperture, mode coupling in the aperture must be taken into account and the ILCM will not be valid. Similarly, if the gasket thickness dg is so small that Ein ∼ Eyinc in (26), the gasket will again resemble an aperture, and the model will not be valid. Another condition for validity of the ILCM therefore, which encompasses both of the restrictions on σg and dg mentioned here, is that in (26) we must have |Ein /Eyinc | 1. This condition is fulfilled in all the cases considered in Table I.
KONEFAL et al.: FAST CIRCUIT MODEL DESCRIPTION OF SE OF BOX WITH IMPERFECT GASKETS OR APERTURES
Typical indicative values for the lower limits on σg and dg can be found by setting |Ein /Eyinc | ≈ 1/10 in (26). We note that the lower limits on σg and dg are not independent (high σg permits lower dg and vice versa) and that their values are also frequency dependent. However, in the worst case low-frequency limit ω → 0 we obtain (Ein /Eyinc ) = 2/(2 + σg dg ZFS ) from (26). Setting this ratio to be less than or equal to 0.1 we obtain the following criterion for validity of the ILCM dg σ g ≥
18 ≈ 0.05Ω−1 . ZFS
(30)
This sets a lower limit on the value of σg for a given dg and vice versa. For example, if dg = 1 mm then we must have σg ≥ 50 Sm−1 . If dg = 0.1 mm then we must have σg ≥ 500 Sm−1 , etc. The worst case low frequency criterion embodied in (30) is also valid for all other frequencies, where the gasket material itself becomes a better attenuator. Note that for overall validity of the ILCM, (30) must be satisfied in addition to the frequency constraints mentioned earlier in this section. VII. CONCLUSION An ILCM has been developed to model the plane wave excitation of a rectangular metallic box containing a single gasket or multiple rectangular gaskets in its front face. The ILCM can incorporate as many higher order modes as are necessary to adequately describe the box excitation at the highest frequency of interest, and is therefore suitable for use at high frequencies and with large boxes. Both propagating and evanescent modes are permitted. The gaskets may be positioned anywhere in the front face of the box, and indeed may even form thin resistive sheets over a relatively large rectangular aperture in the front face. The gaskets must be “good” in the sense that they attenuate incoming electric fields reasonably well (i.e., by a factor of 10 or more). This condition is fulfilled if the gasket thickness dg and conductivity σg are not too small, in which case the ratio |Ein /Eyinc | in (26) is much less than unity. If the condition |Ein /Eyinc | 1 is not satisfied, the gasket resembles an aperture where mode coupling can take place, and the ILCM will not be valid. The lower limits on dg and σg can be defined by the necessary condition for validity dg σg ≥ 0.05 S. Irrespective of the choice of σg and dg , the ILCM is not valid in the low frequency limit. For “normal” shaped boxes with d > b in Figs. 2–6, the ILCM is valid for frequencies f ≥ c/{2(b + d)}, where c is the speed of light in free space. Solution times for the ILCM are significantly less than those exhibited by traditional numerical techniques, with significantly less computer resources being required. In our simulations, covering a variety of gasket sizes and positions, the ILCM coded in MATLAB was over 1000 times faster than TLM, whilst exhibiting an rms difference of 3.90 dB, mean absolute difference of 2.35 dB and correlation coefficient of 0.9981 over eight data sets. When coded in C++ the ILCM was found to run over 20 000 times faster than TLM. Visual examination of the curves in Section VI shows that the circuit model successfully predicts the vast majority of the features in the TLM simulations of SE. Indeed, it is known that the latter features can be highly sensitive
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to observation position and therefore to the spatial resolution used in the TLM simulations. An rms difference of less than 4 dB between the ILCM and TLM is therefore quite remarkable. Indeed, this is better than the level of agreement that can usually be expected between a numerical TLM simulation and an experimental measurement. The main problem lies in the fact that two resonant peaks that are slightly displaced in frequency can lead to a large rms difference, when in reality there is good qualitative and quantitative agreement between the two curves, which is evident from visual inspection. ACKNOWLEDGMENT The authors wish to thank I. MacDiarmid of BAE Systems. REFERENCES [1] R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 6, pp. 500– 505, Jun. 1985. [2] Y. Xingchao, R. F. Harrington, and J. R. Mautz, “The pseudo-image method for computing the electromagnetic field that penetrates into a cavity,” Archiv fur Elektronik und Uebertragungstechnik, vol. 41, no. 5, pp. 307–317, Sep.–Oct. 1987. [3] T. Wang, R. F. Harrington, and J. R. Mautz, “Electromagnetic scattering from and transmission through arbitrary apertures in conducting bodies,” IEEE Trans. Antennas Propag., vol. 38, no. 11, pp. 1805–1814, Nov. 1990. [4] R. F. Harrington and J. R. Mautz, “Electromagnetic coupling through apertures by the generalized admittance approach,” Comput. Phys. Commun., vol. 68, pp. 19–42, Nov. 1991. [5] S. V. Georgakopoulos, C. R. Birtcher, and C. A. Balanis, “HIRF penetration through apertures: FDTD versus measurements,” IEEE Trans. Electromagn. Compat., vol. 43, no. 3, pp. 282–294, Aug. 2001. [6] P. Sewell, J. D. Turner, M. P. Robinson, D. W. P. Thomas, T. M. Benson, C. Christopoulos, J. F. Dawson, M. D. Ganley, A. C. Marvin, and S. J. Porter, “Comparison of analytic, numerical and approximate models for shielding effectiveness with measurement,” Proc. Inst. Elect. Eng., vol. 145, no. 2, pp. 61–66, Mar. 1998. [7] M. P. Robinson, T. M. Benson, C. Christopoulos, J. F. Dawson, M. D. Ganley, A. C. Marvin, S. J. Porter, and D. W. P. Thomas, “Analytical formulation for the shielding effectiveness of enclosures with apertures,” IEEE Trans. Electromagn. Compat., vol. 40, no. 3, pp. 240–248, Aug. 1998. [8] R. Azaro, S. Caorsi, M. Donelli, and G. L. Gragnani, “Evaluation of the effects of an external incident electromagnetic wave on metallic enclosures with rectangular apertures,” Microw. Opt. Technol. Lett., vol. 28, no. 5, pp. 289–293, Mar. 2001. [9] T. Konefal, J. F. Dawson, and A. C. Marvin, “Improved aperture model for shielding prediction,” in Proc. IEEE Int. Symp. EMC, vol. 1, Boston, MA, Aug. 2003, pp. 187–192. [10] R. Azaro, S. Caorsi, M. Donelli, and G. L. Gragnani, “A circuital approach to evaluating the electromagnetic field on rectangular apertures backed by rectangular cavities,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2259–2266, Oct. 2002. [11] I. Belokour and J. LoVetri, “A 2D-transmission line model for the EM field estimation inside enclosures with apertures,” in Proc. IEEE Int. Symp. Electromagnetic Compatibility, Minneapolis, MN, Aug. 2002, pp. 424– 429. [12] T. Konefal, J. F. Dawson, A. C. Denton, T. M. Benson, C. Christopoulos, A. C. Marvin, S. J. Porter, and D. W. P. Thomas, “Electromagnetic coupling between wires inside a rectangular cavity using multiple mode analogous transmission line circuit theory,” IEEE Trans. Electromagn. Compat., vol. 43, no. 3, pp. 273–281, Aug. 2001. [13] , “Electromagnetic field predictions inside screened enclosures containing radiators,” in Proc. IEE Conf. Electromagnetic Compatibility, York, U.K., Jul. 1999, vol. 464, pp. 95–100. [14] A. Nanni, D. W. P. Thomas, C. Christopoulos, T. Konefal, J. Paul, L. Sandrolini, U. Reggiani, and A. Massarini, “Electromagnetic coupling between wires and loops inside a rectangular cavity using multiple mode transmission line theory,” in EMC Europe 2004, Int. Symp.
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Electromagnetic Compatibility, Eindhoven, The Netherlands, Sep. 2004, vol. 2, pp. 609–614. T. Konefal, J. F. Dawson, A. C. Denton, T. M. Benson, C. Christopoulos, A. C. Marvin, S. J. Porter, and D. W. P. Thomas, “Electromagnetic fields produced by PCB stripline and microstrip inside a screened rectangular enclosure: A circuit approach,” Proc. 4th European Symp. Electromagnetic Compatibility, Brugge, Belgium, Sep. 2000, vol. 1, pp. 587–592. I. D. Flintoft, N. L. Whyman, J. F. Dawson, and T. Konefal, “Fast and accurate intermediate level modelling approach for EMC analysis of enclosures,” IEE Proc. Sci. Meas. Technol., vol. 149, no. 5, pp. 281–285, Sep. 2002. T. Konefal, J. F. Dawson, A. C. Marvin, M. P. Robinson, and S. J. Porter, “A fast multiple mode intermediate level circuit model for the prediction of shielding effectiveness of a rectangular box containing a rectangular aperture,” IEEE Trans. Electromagn. Compat., vol. 47, no. 4, pp. 678–691, Nov. 2005. S. A. Schelkunoff, “The impedance concept and its application to problems of reflection, refraction, shielding and power absorption,” Bell Syst. Tech. J., vol. 17, pp. 17–48, Jan. 1938. H. Nagao, A. Nishikata, and Y. Shimuzu, “Errors and correction factors for Schelkunoff’s shielding effectiveness formula,” Electron. Commun. Jpn, Part 1: Commun., vol. 78, no. 10, pp. 38–50, Oct. 1995. A. Nishikata and A. Sugiura, “Analysis for electromagnetic leakage through a plane shield with an arbitrarily oriented dipole source,” IEEE Trans. Electromagn. Compat., vol. 34, no. 3, pp. 284–291, Aug. 1992. T. B. A. Senior and J. L. Volakis, “Sheet simulation of a thin dielectric layer,” Radio Sci., vol. 22, no. 7, pp. 1261–1272, Dec. 1987. H. Kaden, Wirbelstrome und Schirmung in der Nachrichtentecknik. Berlin, Germany Gumam: Springer-Verlag, 1959. J. C. G. Field, “An introduction to electromagnetic screening theory,” in IEE Colloquium Screening and Shielding, London, U.K., Nov. 1983, pp. 1/1–1/15. S. W. Gilmore and A. K. Dominek, “Electromagnetic scattering by an inhomogeneous penetrable material with an embedded resistive sheet,” Ann. Telecommun., vol. 50, no. 5–6, pp. 573–581, May–Jun. 1995. J. M. Jin, J. L. Volakis, C. L. Yu, and A. C. Woo, “Modelling of resistive sheets in finite element solutions (EM scattering),” IEEE Trans. Antennas Propag., vol. 40, no. 6, pp. 727–731, Jun. 1992. L. K. Wu and L. T. Han, “Implementation and application of resistive sheet boundary condition in the finite difference time domain method (EM scattering),” IEEE Trans. Antennas Propag., vol. 40, no. 6, pp. 628–633, Jun. 1992. A. Mallik and C. P. Loller, “The modelling of EM leakage into advanced composite enclosures using the TLM technique,” Int. J. Numer. Model.: Electron. Networks, Devices Fields, vol. 2, no. 4, pp. 241–248, Dec. 1989. J. A. Cole, J. F. Dawson, and S. J. Porter, “Efficient modelling of thin conducting sheets within the TLM method,” in IEE 3rd Int. Conf. Computation in Electromagnetics, Bath, U.K., Apr. 1996, pp. 45–50. E. Topsakal, M. Carr, J. Volakis, and M. Bleszynski, “Galerkin operators in adaptive integral method implementations,” IEE Proc. Microw., Antennas Propag., vol. 148, no. 2, pp. 79–84, Apr. 2001. A. Nishikata and S. Kiener, “Shielding effectiveness of non-perfect conducting enclosures: Characterisation of walls and numerical applications,” in Proc. 10th Int. Zurich Symp. EMC, Zurich, Switzerland, Mar. 1993, pp. 617–622. D. F. Williams, L. A. Hayden, and R. B. Marks, “A complete multimode equivalent circuit theory for electrical design,” J. Res. Nat. Inst. Stand. Technol., vol. 102, no. 4, pp. 405–423, Aug. 1997. R. E. Collin, Field Theory of Guided Waves, 2nd ed. Oxford, U.K.: Oxford Univ. Press, 1991, ch. 5, pp. 329–410.
Tadeusz Konefal was born in Nottingham, U.K., in 1962. He received the B.S. degree in physics from the University of Sheffield, U.K., in 1987, and the D.Phil. degree in nondestructive testing from Cranfield Institute of Technology, U.K., in 1993. Since 1991, he has worked in the Electronics Department, University of York, York, U.K., in both the Applied Electromagnetics and Communications Groups. Currently, he is a Research Fellow in the Applied Electromagnetics Group, working on intermediate level modeling design tools for EMC. Dr. Konefal was awarded the prize for the best oral paper at the 4th European Symposium on EMC in Brugge, Belgium, in September 2000.
John F. Dawson (M’90) received the B.S. and D.Phil. degrees from the University of York, York, U.K., in 1982 and 1989, respectively, both in electronics. He is a Senior Lecturer and member of the Applied Electromagnetics and Electron Optics Research Group, University of York. His research interests include numerical electromagnetic modeling, electromagnetic compatibility prediction for circuits and systems, electromagnetic compatibility test environments, and optimization techniques for EMC design.
Andrew C. Marvin (M’85) received the B.Eng., M.Eng., and Ph.D. degrees from the University of Sheffield, U.K., in 1972, 1974, and 1978, respectively, all in electrical and electronic engineering. He is a Professor of Applied Electromagnetics and Leader of the Physical Layer Research Group, University of York, York, U.K. and Technical Director of York EMC Services Ltd, U.K. His main research interest is in EMC measurement techniques. Dr. Marvin is currently Chairman of COST Action 286 (EMC in Diffused Communications Systems). He is a member of the IEE, where he is in the Executive Committee of the IEE Professional Network for EMC and represents the U.K. on URSI Commission A (Electromagnetic Metrology). He contributes to the joint CISPR/IEC task force on the use of TEM cells for EMC measurements and is an Associate Editor of the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY.
Martin P. Robinson received the B.A. and M.A. degrees from the University of Cambridge, U.K., in 1986 and 1990, respectively, both in natural sciences, the M.S. degree in medical physics from the University of Aberdeen, U.K., in 1990, and the Ph.D. degree in dielectric imaging from the University of Bristol, U.K., in 1994. He worked for two years at the National Physical Laboratory, U.K. and for three years at Bristol Oncology Centre, U.K. He joined the University of York, York, U.K., in 1993, where he is currently a Lecturer in electronics. His research interests include design for EMC, dielectric measurements, and the interaction of electromagnetic radiation with biological tissues.
Stuart J. Porter (M’93) received the B.S. and D.Phil. degrees in physics from the University of York, York, U.K., in 1985 and 1991, respectively. He is a Lecturer and Member of the Applied Electromagnetics Research Group, Department of Electronics, University of York. His research interests include computational electromagnetics, particularly as applied to radio frequency and microwave problems, computational and computer-aided tools for EMC design, antenna design, application of evolutionary computation optimization methods to antenna design and EMC, and computational acoustics.
