The Effect of Apertures Position on Shielding Effectiveness of Metallic Enclosures based on Modal Method of Moments

In this study, an effective numerical method suitable for determination of electric field Shielding Effectiveness (SE) of rectangular enclosure with multiple rectangular apertures is presented. Assuming appropriate electric field distribution on the aperture, Electromagnetic fields inside the enclosure are determined using rectangular cavity Green’s function. Electromagnetic fields outside the enclosure 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. The integral equation is solved for unknown aperture fields using the Method of Moments. From the aperture fields the electromagnetic SE of the rectangular enclosure is determined. The numerical results of the proposed technique are in very good agreement with data available in the literature and experimental results. It is shown that apertures’ position and shape, aperture’ number, polarization have noticeable effect on the electric field SE.


INTRODUCTION
Most aircraft used a series of cables, chains, cranks and mechanical mechanisms to operate the systems which gave the aircraft its ability to fly in the past.Nowadays many mechanical devices have been replaced or augmented with avionics.Electronic devices have increasingly been designed and used for flight critical aircraft control systems, because of their ability to accurately control complex functions and increase reliability.Electronic circuits, however, not only respond to their internal electrical signal flow, but may respond to any input which can couple into the wire bundles, wires, IC leads and electrical junctions.The Electromagnetic Environment (EME) is one of these inputs that has access to all these electronic circuits and may result in disabling effects called Electromagnetic Interference (EMI).The aircraft skin and structure have also evolved.The classic aircraft is made of aluminum and titanium structure with an aluminum skin.Modern technology and the desire to develop more efficient aircraft have driven the introduction of carbon-epoxy structure, carbon-epoxy skins and aramid fiber-epoxy skins in civil aircraft.Aluminum may be a good EM shield against High Intensity Radiated Field (HIRF) and hence electronic circuits are provided inherent protection.However, some composites are poor EM shields, causing HIRF to irradiate the electronic systems on such aircraft with relatively little attenuation.So electromagnetic shielding is an important technique in Electromagnetic Compatibility (EMC).It can restrain electromagnetic energy radiation and prevent the electromagnetic interference effectively.
Nowadays metallic shielding enclosures are frequently employed to suppress its directive radiation effects.Unfortunately, in practical applications, the integrity of these metallic enclosures is often compromised by all kinds of apertures that are used to accommodate visibility, ventilation or access to interior components, such as input and output connections, heat dispersion panels, control panels, visual-access windows, etc.What is more, those openings allow exterior electromagnetic energy to penetrate to the inside space, where they may couple onto Printed Circuit Boards (PCBs), then cause the inner field resonance and the shielding performance being degraded.So it is very important to know the EM Shielding Effectiveness (SE) of shielding enclosures in the presence of these apertures.The EM SE study may also help in locating these apertures at proper places to reduce the EM emission or improving the immunity of electronic components present inside the metallic enclosure.
Several analytical and numerical techniques to estimate SE 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 formulations even though provide a much faster means of calculating SE are based on various simplifying assumptions whose validity may be questionable at high frequencies.Robinson et al. (1996Robinson 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;Li et al., 2000), Finite Element Method (FEM) (Benhassine at 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 utilized 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 et al., 2010) and for finite wall thickness (Dehkhoda et al., 2009).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).
In the majority of the aforementioned methods, there are few studies on the influence of apertures' position on shielding effectiveness of metallic enclosures with apertures.In this study the modal MoM solution is formulated, the apertures are replaced by equivalent magnetic current sources, employing the surface equivalence principle and boundary conditions at each end of the aperture, matching the tangential electromagnetic fields across the apertures, coupled integral differential equations with the magnetic currents at the apertures as unknown variables are obtained.The coupled integral differential equation in conjunction with the method of moment is then solved for unknown magnetic current amplitudes.The electric field SE has been calculated at many points inside the enclosure with multiple rectangular apertures sat different places.The electric field SE due to horizontal as well as vertical polarizied incident fields is also studied.A very good agreement among the results of the proposed technique, results available in the literature and experimental results is observed.It is shown that aperture position has noticeable effect to the electric field SE, especial for single aperture case.

ELECTROMAGNETIC PROBLEM AND THE FORMULATION OF MODAL MOM
The shielding effectiveness of an enclosure is defined as: where, ‫ܧ‬ ୧୬୲ : The electric field at a given point inside the enclosure ‫ܧ‬ ୣ୶୲ : The electric field at 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 ‫ܮ‬ × ܹ .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 where, ‫ܧ‬ ୟ୮୲ is the tangential electric field induced on the apertures and n is the aperture normal vector: where, ܷ and ܸ are the unknown amplitudes of the pq th mode of magnetic current on the outer of the r th aperture, ܷ ≠ 0 and ܸ ≠ 0 and ܷ = ܸ = 0 otherwise.‫ܮ‬ ܽ݊݀ ܹ are the length and width of r th aperture, ‫ݔ‬ and ‫ݕ‬ are center coordinates of the r th aperture.‫ݔ‬ ො, ‫ݕ‬ ෝ are the unit vectors in x, y directions.
The unknown amplitudes ܷ , ܸ are determined by setting up coupled integral equations.
Using the equivalence principle, the equivalent magnetic currents are: Electromagnetic field due to incident wave: Figure 2 is the definition of incidence and polarization angles.
The incident time harmonic plane wave illuminating the rectangular apertures on the cavity can be written as: where, ݇ • ‫ݎ‬ = ݇ sin ߠ ‫∅ݏܿݔ‪ሺ‬‬ + ‫∅݊݅ݏݕ‬ ሻ + ‫݇ݖ‬ ‫ߠݏܿ‬ ݇ = Free space wave number ߠ ,∅ = Angles of incident plane wave ߙ = Polarization of the incident plane wave From Eq. ( 7), the x-, y-and z-components of the incident magnetic field may be written, respectively as: For normal incidence, with ߙ = 0, ߠ = 0 and ∅ = 0, the incident field in the z = 0 is given by ‫ܪ‬ 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: where the electric vector potential ‫ܨ‬ 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 ( 14) -( 16) ϕ ௬ is the Fourier transform of Φ ௬ and ψ ௬ is the Fourier transform of Ψ ௫ .
Electromagnetic 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 ( 17)-( 19): 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 ( 20)-( 22): 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.

Derivation of integral equation:
The total tangential fields inside the cavity from apertures are written as: Applying the continuity of tangential magnetic field on the z = 0 plane yields: Now selecting Ψ ′ ′ ′ ௫ as a testing function and use of Galerkin's method reduces the (25) to: where,

‫ݕ݀ݔ݀‬
Similarly, selecting −Φ ′ ′ ′ ௬ as a testing function and use of Galerkin's method reduces the (26) to: where, Sci. Eng. Technol., 6(17): 3125-3136, 2013 3129 (26) as a testing function and use of Galerkin's method reduces the (25) to: as a testing function and use of Galerkin's method reduces the (26) to: Equation ( 27) and ( 31) can be written in a matrix form as: The matrix Eq. ( 35) can be numerically solved for the unknown amplitudes of equivalent magnetic currents induced on the apertures due to given incident field.From the knowledge of these amplitudes electromagnetic field inside as well as outside the enclosure can be obtained.

VALIDATION OF THE PRESENT
In this section, for the validation of the presented method, we consider a rectangular enclosure of size (30 × 12 × 30 ܿ݉) with a rectangular (10×0.5 cm) located at the center of the front wall (15 cm, 6 cm, 0), as illustrated in

RESULTS AND DISCUSSION
Single aperture case: In next section, we discuss electric field SE results calculated at three different points(x = 15 cm, y = 6 cm, z = 25 cm, 15 and 5 cm respectively) inside enclosure versus Fig. 3: Geometry of 30× 12 × 30 ܿ݉ enclosure with a single aperture at (15 cm, 6 cm, 0) Equation ( 27) and ( 31) can be written in a matrix The matrix Eq. ( 35) can be numerically solved for the unknown amplitudes of equivalent magnetic currents induced on the apertures due to given incident field.From the knowledge of these amplitudes romagnetic field inside as well as outside the

PRESENT TECHNIQUE
, for the validation of the presented we consider a rectangular enclosure of size rectangular aperture of size 0.5 cm) located at the center of the front wall ( 15. 3. The enclosure is illuminated by a normal incident plane wave at 0 Assuming only expansion mode on the aperture and considering only dominant mode inside the cavity.The shielding effectiveness is calculated at the center of eld shielding obtained using ) is plotted in Fig. 4  App.Sci. Eng. Technol., 6(17): 3125-3136, 2013 3130 eld SE calculated at the center of 30×12×30 cm enclosure with a 10×0.5 cm aperture located z = 0 plane illuminated by vertical polarized plane wave Enclosure illuminated by vertical polarized plane wave Enclosure illuminated by horizontal polarized plane wave eld SE versus frequency at three different points (x = 15 cm, y = 6 cm, z = 25 cm, 15 and 5 cm inside 30×12×30 cm enclosure with one 10×0.5 cm aperture located at three different places (y = 6 cm, x = 24 cm, 20 and located at 15×6 cm in eld SE versus frequency at three different points (x = 15 cm, y = 6 cm, z = 25 cm, 15 and 5 cm, respectively) inside 30×12×30 cm enclosure with one 10×0.5 cm aperture located at three different places (y = 6 cm, x = 24 cm, 20 and apertures having same area situated at six different places and illuminated by vertical and horizontal polarized plane wave respectively.We also consider (30×12×30) cm³ enclosure with two types of square aperture of size (10×0.5 cm, 2.23×2.23 cm).
Figure 5 and 6 present the plots of electric field SE versus frequency at three different points(x = 15 cm, y = 6 cm, z = 25, 15 and 5 cm, respectively) inside 30×12×30 cm enclosure with one 10×0.5 cm aperture.In Fig. 5, the aperture located at three different places (y = 6 cm, x = 24 cm, 20 and 15 cm, respectively) in z = 0 plane.In Fig. 6, the aperture located at those places (x = 15 cm, y = 10 cm, 8 and 6 cm, respectively) in z = 0 plane.(a) Enclosure illuminated by vertical polarized plane wave, (b) enclosure illuminated by horizontal polarized plane wave.
In Fig. 5a, we observe a similar behavior for three different places below the first resonance frequency of the cavity, but there are two rapid increase for the calculation point of z = 50 mm because of modal structure of the fields, one up to about 60 dB appear at about 0.82 GHz for the aperture located at x = 24 cm, the other up to about 40 dB appear at about 0.74 GHz for the aperture located at x = 20 cm.We also note that there are little difference for electric field SE of the apertures located at x = 20 cm and x = 15 cm respectively, but for the aperture located at x = 24 cm, the electric field SE are larger than others, especially for the calculation point of z = 5 cm, there are about 10 dB difference.
In Fig. 5b, we note that there are similar trend for apertures located at x = 20 cm and x = 24 cm respectively and the first resonance frequency is also about 0.7 GHz, but for the aperture located at x = 15 cm, it is absolutely different.inside 30×12×30 cm enclosure with one 10×0.5 cm aperture located at three different places (x = 15 cm, y = 10 cm, 8 and 6 cm, respectively) in z = 0 plane larger than that of vertical below 0.3 GHz, between 0.4 to 0.6 GHz and 0.75 to 1 GHz, the difference has up to about 35 dB.For the electric field SE of the aperture located at x = 15 cm, it has no resonance below 1 GHz and almost all electric field SE are larger than 60 dB, for the aperture located at x = 15 cm, its SE has even up to about 78 dB at 0.5 GHz.From discussed above, we can safely draw the conclusion that the enclosure with 10×0.5cmaperture has better electric field SE facing the horizontal polarization plane wave, especially for the aperture located at x = 15 cm.In Fig. 6a, we note that three SE curves have little difference below 1 GHz, which indicates that different y values have little effect on the electric field SE of the enclosure with one 10×0.5 cm aperture illuminated by vertical polarized plane wave.In Fig. 6b, we note that there are similar trend for apertures located at y = 8 cm and y = 10 cm respectively and the first resonance frequency is also about 0.7 GHz.But for the aperture located at y = 6 cm, it is absolutely different, it has no resonance below 1 GHz and almost all electric field SE are larger than 60 dB, the maximum of SE has even up to about 78 dB at 0.5 GHz, which indicate that the place of apertures except for the center of the front wall have minimal effect on the electric field SE.
Figure 7 and 8 present the plots of electric field SE versus frequency at three different points(x = 15 cm, y = 6 cm, z = 25 cm, 15 and 5 cm respectively) inside 30×12×30 cm enclosure with one 2.23×2.23 cm aperture.In Fig. 7, the aperture located at three different places (y = 6 cm, x = 24 cm, 20 and 15 cm respectively) in z = 0 plane.In Fig. 8 Compares Fig. 7 and 8, we find that the enclosure has better electric field SE illuminated by horizontal polarized plane wave and the electric field SE calculated at z = 5 cm are affected more than other points by the places of apertures.Especial for the SE of enclosure with aperture located at x = 15 cm and calculation point at z = 15 cm and z = 25 cm in Fig. 7b, we note that almost all the SE are more than 60 dB, at 0.4 to 0.85 GHz, the SE has up to about 70 dB.Compares Fig. 7a and 8a with Fig. 5a and 6a, we observe that there is another rapid increase even up to about 80dB at about 0.6 GHz for the electric field SE calculated at z = 25 cm because of boundary effect of the wall.The other obvious difference is that there are better electric field SE for the calculation points of z = 15 cm below 0.5 GHz in Fig. 7a and 8a, which indicates that the shape of aperture has some effect on the electric field SE at some calculation points for different polarization wave.In Fig. 8, there is fewer difference for the electric field SE of apertures located at y = 8 cm and y = 6 cm due to the aperture's different places, but for the SE of apertures located at y = 10 cm, it is absolutely different; the maximum of the difference is about 10 dB.For single aperture case, we can conclude that the electric field SE of enclosure is larger when the apertures are far from the center of the wall.

CONCLUSION
In this study, an efficient evaluation approach based on modal MoM technique is presented to evaluate electric field SE of enclosures with multiple apertures at various places.Expressing electromagnetic fields in terms of cavity Green's function inside the enclosure and the free space Green's function outside the enclosure, integral equations with aperture tangential electric fields as unknown variables are obtained by enforcing the continuity of tangential electric and magnetic fields across the apertures.Using the Method of Moments, the integral equations are solved for unknown aperture fields.From these aperture fields, the electromagnetic fields inside a rectangular enclosure due to external electromagnetic sources are determined.
Numerical results on electric field SE of a rectangular enclosure with apertures are validated with data available in the literature and measurement.It is also shown that effect factors of electric field SE include calculation points, aperture' place, number and shape, polarization.Compared with single slit case, electric field SE of multiple apertures are less effected by apertures' position.The apertures' place have different effects on the electric field SE of different calculation point and polarization.When the enclosure with apertures is illuminated by horizontal polarization plane wave, it has better electric field SE, especial for the aperture located at the center of wall.The electric field SE of enclosure is larger when the apertures are far from the center of the wall.And electric field SE near the aperture is lower than that at location inside the enclosure far away from the aperture below the resonance frequency.Therefore, single aperture should be located far from the center of the wall and low frequency sensitive apparatus inside the enclosure should be placed at the points far away from the apertures, which can improve the ability of electromagnetic compatibility.These useful results gained in this study have the important practical significance to improving the electric field SE of shielding cavity.

Fig. 1 :
Fig. 1: Geometry of rectangular enclosure with rectangular apertures exposed to a normal incident plane wav Fig. illuminated by a normal incident plane wave at 0 polarization.Assuming only expansion mode on the aperture and considering only dominant mode inside the cavity.The shielding effectiveness is calculated at the center of the cavity.Electric field shielding obtained using expression (35) is plotted in Fig results from Robinson et al. (1998) the numerical data obtained using the agrees well with the earlier published results.Experimental data from Robinson reproduced in Fig. 4.