Study of Shielding Properties of a Rectangular Enclosure with Apertures Having Different Shapes but Same Area Using Modal Method of Moments

In this study, electric field Shielding Effectiveness (SE) of rectangular enclosure with apertures illuminated by vertical polarization plane wave has been studied by using modal method of moment technique. Electric field SE of enclosure with different shape apertures but same area has been c alculated at three different points inside enclosure. To achieve this, assuming appropriate electric field distribution on the aperture, fields inside the cavity are determined using rectangular cavity Green’s function. Electromagnetic fields outside the cavity and scattered due to the aperture are obtained using the free space Green’s function. Matching the tangential magnetic field across the apertures, the integral equation with aperture fields as unknown variables is obtained. A very good agreement among the results of the proposed technique, results available in the literature and experimental results is observed. The simulation results show that the electric field SE is seriously affected by calculation points, aperture shape and the number of aperture. It has been shown that usual assumption made in EMC literature that lower electric field SE near the aperture than at location inside the enclosure farther away from the aperture is not always true to square aperture at some frequency and square aperture has higher electric field SE than rectangular aperture even though they have same area.


INTRODUCTION
Shielding enclosures are utilized to hinder electromagneticleakage from electronic equipment and as well to protectsensitive instruments against external interference and hence meet the Electromagnetic Compatibility (EMC) standard.The ability of a shielding enclosure to do thisis characterized by its shielding effectiveness which is the ratio of the field strength in thepresence and absence of the enclosure for both electric andmagnetic fields.SE is primarily affected by the wavepenetration through apertures and slots used toaccommodate visibility, ventilation or access to interiorcomponents, such as input and output connections,heat dispersion panels, control panels, visual-access windows, etc. Whatis more, those openings allow exterior electromagnetic energy to penetrate tothe inside space, where they may couple onto Printed Circuit Boards (PCBs), then resulting inthe inner field resonance and the shielding performance being degraded.
Several analytical and numerical methods to calculate shielding effectiveness of metallic enclosures with apertures have been suggested in past years.However, each technique has advantages and disadvantages with respect to other techniques.Pure analytical methods are accurate but can be applied only to various simplifying geometries with some approximations whose validity may be questionable at high frequencies.
For electrically small apertures, (Bethe, 1944) derived a small hole theory, which states that the electromagnetic fields at both sides of a zero-thickness wall with an electrically small aperture represented by an electric and a magnetic dipole illuminated by a plane wave.The relationship between the dipole moments of these dipoles to the original fields is described by polarizability that are functions of the size and shape of the small apertures.Bethe's solution is applicable only to simple geometries (McDonald, 1985) and for complex shapes, numerical techniques are needed (Arvas and Harrington, 1983).Mendez generalized Bethe's theory, for EM radiation from rectangular enclosures with an aperture excited by a center-fed thin dipole and a square loop (Mendez, 1974;Mendez, 1978).However, his formulation is limited to frequencies below the lowest enclosure resonance.Small hole approximation and its generalization application are the primary studies on the subject.Robinson et al. (1996) and Robinson et al. (1998) introduced a very simple analytical method based on transmission line model.However, this approach is limited by the assumption of thin apertures, simple geometries, negligible mutual coupling between apertures and fields can be calculated only at points in front of the aperture.The transmission line model was later extended to include higher order cavity modes (Belokour et al., 2001) and the effects of loading due to electrical circuits within the enclosure (Thomas et al., 1999).
In addition, some efficient and reliable numerical techniques for the electromagnetic analysis and many numerical tools have been applied to the analysis of shielding effectiveness, such as finite Difference Time Domain (FDTD) (Jiao et al., 2006;Nuebel et al., 2000), finite Element Method (FEM) (Benhassine et al., 2002;Carpes et al., 2002), Method of Moments (MoM) (Wallyn et al., 2002;Wu et al., 2011), Transmission Line Matrix (TLM) (Podlozny et al., 2002;Attari and Barkeshli, 2002) and hybrid method (Wu et al., 2010) are utilised with good accuracy over a broad frequency band at the cost of large amount of computer memory and CPU time.
Deshpande introduced a moment method technique (modal MoM) using entire domain basis functions to represent apertures fields and therefore the magnetic currents on the apertures, which can evaluates the SE of a zero thickness enclosure exposed to a normally incident plane wave accurately at the center inside enclosure (Deshpande, 2000).This method has been further modified for obliquely polarimetric incident plane wave (ALI et al., 2005;Jayasree, 2010) and for finite wall thickness (Dehkhoda et al., 2009).As modal MoM is very fast and is an analytical solution in series form.It can yield arbitrary accuracy for increasing number of cavity modes (expanded in cavity Green's function) and increasing number of aperture modes in the rectangular aperture.These modes areexpanded in bases that form a complete set for this infinite Qstructure.It is useful for statistical investigation.Nowadays the EM shielding effectiveness has becoming a hot research point in the electromagnetic compatibility area (Bahadorzadeh and Moghaddasi, 2008;Robertson et al., 2008;Faghihi and Heydari, 2009;Koledintseva et al., 2009;Wang and Koh, 2004;Hussein, 2007;Morari et al., 2011;Kim et al., 2008;Bahadorzadeh and Moghaddasi, 2008;Fang et al., 2008).
In most of the aforementioned methods, the shielding effectiveness is calculated at the center of the enclosure.There are few studies that calculatedSE at other points inside enclosure and few SE studies in view

Electromagnetic problem and the formulation of modal MOM:
The shielding effectiveness of an enclosure is defined as: where, is the electric fieldat a given point inside the enclosure, is the electric fieldat the same point in the absence of the enclosure.Therefore, the problem of estimation of shielding effectiveness is essentially the problem of calculating the cavity fields excited by a plane wave incident from free space upon the shielding enclosure.
Figure 1 shows a rectangular enclosure with rectangular apertures exposed to a normal incident plane wave.The dimensions of the cavity are .There are r number of apertures and the dimensions of the rth aperture are .The orientation of the reference axes is also shown with the origin at the lower right corner of the front wall.

Apertures fields and equivalent magnetic currents:
In the Modal MoM formulation, we assume that the apertures are relatively small compared to the walls in which they are located and are placed far enough away from the edges of the enclosure.In addition, the edge diffracted fields are neglected.These assumptions enable us to use image theory and equivalence principles, using the surface equivalence principle, the apertures both internal (Region I) and external (Region II) to the enclosure can be replaced by equivalent magnetic currents of: where, is the tangential electric field induced on the apertures and n is the aperture normal vector.Using the equivalence principle, the equivalent magnetic currents are: where, Electromagnetic field outside enclosure: Consider the aperture on the z=0 plane, the scattered EM field outside due to the rth aperture can be determined by solving electric vector potential: (7) Where the electric vector potential is given by: Superposition of the scattered electromagnetic field due to all apertures on the z = 0 plane gives the total scattered field as (Deshpande, 2000): In expressions ( 10)-( 12) ф epgy ourier transform of Φ and ψ is the Fourier transform of Ψ .
Eletromagnetic field inside enclosure The equivalent magnetic currents, present on the apertures of the enclosure, radiate electromagnetic fields inside the enclosure.The total electromagnetic field at any point inside enclosure is obtained by a superposition of fields due to each equivalent magnetic current source.
Considering the x-component of the magnetic current and using dyadic Green's function, The total magnetic field inside the enclosure is then obtained from (Deshpande, 2000) as: In ( 13)-( 15),

Ψ , sin cos
Likewise, considering the y-component of the magnetic current and using the proper boundary conditions, The total magnetic field inside the enclosure is then obtained from (Deshpande, 2000) as: In ( 16)-( 18),

Φ , co s sin
For a unique solution the electromagnetic fields in various regions satisfy continuity conditions over their common surfaces.The tangential electric fields over the apertures are continuous.The tangential magnetic over the apertures must also be continuous, thus yielding coupled integral equations with the magnetic currents as known variables.The coupled integral equation in conjunction with the method of moments can be solved for the amplitudes of magnetic currents.the center of the front wall (15, 6cm, 0), as illustrated in Fig. 2. The enclosureis illuminated by a normal incident plane wave at 0 polarization.Assuming only expansion mode on the aperture and considering only inside the cavity.the shielding effectiveness is calculated at the center of the cavity.electric field shielding obtained using expression (31) is plotted in Fig. 3 along with the results from (Robinson et al., 1998).It is observed that the numerical data obtained using the present method agrees well with the earlier published results.Experimental data from (Robinson et al., 1998) is also reproduced in Fig. 3.

RESULTS AND DISCUSSION
In this section, in order to study the electric field shielding effectiveness versus different calculation points in the enclosure, a 30 12 30 cm enclosurehaving two types of apertures with the same area is selected.The enclosure is illuminated by a normal incident plane wave at 0 polarization.
Figure 4 presents the plot of electric field shielding effectiveness versus three different points inside the enclosure with two types of apertures.We note that increasing the frequency decreases the shielding effectiveness and the resonance frequency of the cavity is about 0.7 GHz.Also note that the maximum and minimum of electric field SE below the resonance frequency occur at z = 250 mm and z = 50 mm respectively in Fig. 4a, the maximum difference of electric field SE is almost 20dB, which is in accordance with the usual assumption made in EMC literature that lower electric field SE near the aperture than at location inside the enclosure farther away from the aperture, but when the frequency exceed the resonance frequency, this result is wrong.
The most interesting observation that can be made for electric field SE in Fig. 5b is that the electric field SE under 0.46 GHz at z = 50 mm is higher than the others, which is not similar with the usual assumption.We also note that electric field SE increase rapidly to more than 40dB at 0.75GHz on the z = 50cm point because of modal structure of the fields.A particular case is that the electric field SE of 2.23×2.23 cm aperture at z = 250mm has also a rapid increase at about 0.59GHz.
In this section, we discuss electric field SE results calculated at three different points versus different shapes apertures.We also consider a 30cm 12 30 cm enclosure with rectangular aperture of size (10 0.5 cm, 5.0 1.0cm and 2.23 2.23 cm) located at the center ofthe front wall (15 cm, 6 cm, 0) respectively.The enclosureis illuminated by a normal incident plane wave at 0 polarization.Fig. 5a, b and c presents three plots of electric field SE versus three different shapes aperture with same area respectively.We observe a similar changed trend in Fig. 5a, b and c that all the resonance frequency is about 0.7 GHz, electric field SE increase in turn from 10.0cm×0.5cmaperture to 2.23×2.23cmaperture and electric field SE have a rapid increase up to about 50dB at 0.75GHz at z = 50 mm because of modal structure of the fields.Figure 5 confirms the electric field SE dependence upon distance from the apertures.We also note that the apertures with different shape but same area have different effect on electric field SE.For one aperture case, from discussed above, we can conclude that square aperture have higher electric field SE than rectangular aperture when their area is same.
Figure 6 presents six plots of electric field SE versus two types of apertures with the same area at different calculation points in the cavity.We observe that the electric field SE have a similar changed trend in two types of apertures at all frequency and the electric field SE of two apertures is higher than that of one aperture while they have the same area.The results indicate that two apertures is a better select to improve the shielding ability of an enclosure than one aperture while they have the same area.

CONCLUSION
In this study, an investigation on the effects of different calculation point inside the enclosure and different shape apertures on the electric field shielding effectiveness of metallic rectangular enclosures illuminated by vertical polarized plane waves has been carried out.For this purpose, the modal MoM solution is formulated and by employing the surface equivalence principle and boundary conditions at each end of the aperture, the problem is solved.Numerical results on electric field shielding effectiveness of a rectangular enclosure are good agree with data available in the literature.The results of the present analysis show that electric field SE of 10.0 0.5 cm aperture is in accordance with the usual assumption made in EMC literature that lower electric field SE near the aperture than at location inside the enclosure farther away from the aperture.But for the electric field SE of 2.2 2.23 cm aperture, the assumption is not true.Additionally, square apertures has higher SE than other shape apertures, even though they have the same area.For the same area of aperture, the electric field SE of two apertures is higher than that of one aperture.These useful results gained in this study have the important practical significance to improving the electric field shielding effectiveness of shielding cavity.

ACKNOWLEDGMENT
This study was supported by National Science funds of Civil Aviation Flight University of China (serial number: J2007-23) and Open Funds of CAAC academy of flight technology and safety.

Fig
Fig. 1: Geometry of rectangular enclosure with rectangular apertures exposed to a normal incident plane wave of apertures shape.In this paper, the modal MoM solution is formulated and by employing the surface equivalence principle and boundary conditions at each end of the aperture.Electric field SE is calculated at three different points inside enclosure with different shapes apertures, such as different rectangular apertures, square apertures.A very good agreement among the results of the proposed technique, results available in the literature and experimental results is observed.It is also shown that different calculation points severely affects the electric field SE, lower electric field SE near the aperture than at location inside the enclosure farther away from the aperture below the resonance frequency.Additionally, square apertures has higher SE than other shape apertures, even though they have the same area.
amplitudes of the pqth mode of magnetic current on the outer of the rth aperture, are the length and width of rth aperture, and are center coordinates of the rth aperture., are the unit vectors in x, y directions.The unknown amplitudes , are determined by setting up coupled integral equations.

Fig. 4 :
Fig. 4: Electric field shielding calculated at different points inside 30×12×30 cm enclosure with two types of apertures located at 15×6 cm in z = 0 plane illuminated by vertical polarized plane wave a: 10.0 0.5 cm aperture; b: 2.23 2.23 cm aperture