A Dual Integral Equation Approach for Evaluating the Shielding of Thick Circular Disks against a Coaxial Loop

: The electromagnetic interaction between a circular disk with ﬁnite conductivity and ﬁnite thickness and a coaxial circular loop of constant current is addressed here. The ﬁnite conductivity and thickness of the material disk lead to the adoption of suitable generalized boundary conditions


Introduction
The interaction of electromagnetic fields with a circular disk (and its dual counterpart, i.e., a circular aperture in a plate) is a canonical problem in electromagnetic theory; research on this subject started in the 1950s and continues to attract the attention of the electromagnetic research community, although most studies have considered plane-wave excitation and/or perfectly-conducting (PEC) obstacles, ideally with zero thickness (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]). In connection with this, we point out that in [12][13][14] the method of analytical preconditioning together with a Galerkin spectral Method of Moments with entire domain basis functions has been used to study the scattering from an infinitesimally thin PEC disk. Such a method has been adopted in those few studies which have considered thin disks, including those with dielectric properties [15][16][17]. On the other hand, the screening of a finite source (e.g., an electric or a magnetic dipole) by means of an infinite planar conductive shield is well-documented in both the frequency [18][19][20] and time [21,22] domains. Recently, the shielding of a thin conductive disk in the presence of an ideal vertical magnetic dipole has been studied in details [23]; however, the main assumption in [23] is that the disk thickness has to be smaller than the skin depth. Such an assumption introduces an upper limit in terms of frequency, and makes the formulation essentially into a low-frequency analysis. It is worth mentioning that the dual problem (i.e., a circular aperture in a thin metallic plate) has only recently been addressed [24]. In general, for thick disks (and as such for higher frequencies) analysis requires the adoption of generalized boundary conditions (GBCs). In turn, such GBCs require the introduction of both equivalent magnetic and electric current densities along the discontinuity interface [25][26][27]. This implies that, for thick disks an integral formulation based on GBCs leads to the derivation of two sets of dual integral equations (DIEs) to be solved.
In the present work, we solve such sets of DIEs (derived in the Hankel transform domain) by expanding the spectral unknowns in two Neumann series of Bessel functions.
Moreover, for thick metallic non-magnetic disks, a new generalized boundary condition is derived which is a generalization of that used for thin metallic screens. This means that it is sufficient to introduce only an equivalent surface electric current density as unknown. In this way, only one set of DIEs needs to be solved, thereby reducing the problem in a similar way to that of thin screens (preliminary results of this work have been presented in [28]). Thus, we generalize the analysis presented in [23] considering a finite source (a coaxial loop with finite radius and constant current) and a conductive disk with finite thickness, now without any limitation in frequency and thickness.
Finally, we want to mention a possible limitation of the present study. For simplicity, the electric current in the loop is assumed to be constant. In principle, the extension of the proposed analysis to non-constant currents could be feasible; however, introducing an azimuthal dependence of the exciting current would destroy the azimuthal invariance of the problem, thereby making the incident and scattered field hybrid. We leave this analysis for a future work.
It is clear that the focus of the paper is more on modeling than on the performance of different materials. We considered copper as a typical conductive material for the purpose of presenting numerical results that illustrate the proposed method; however, the validity of the method is general and has been tested on different values of the screen parameters.
The rest of this paper is organized as follows. The electromagnetic problem is described in Section 2, while in Section 3 the adopted Mitzner GBCs are discussed and two auxiliary sub-problems are introduced, each of which leads to DIEs for which the unknown is an equivalent spectral electric or magnetic current density. The procedure for the solution of these two sets of DIEs is outlined in Section 4, where all the quantities of interest are calculated as well. In Section 5, we propose a new boundary condition for thick conductive disks with no magnetic properties. This allows for formulating the problem in terms of only one set of DIEs, with consequent savings in terms of computation time compared with the original full formulation. In addition, we propose a series expansion (which in many practical case reduces to a single term) to efficiently evaluate the involved improper integrals. It is worth pointing out that the linear algebraic system resulting from the discretization of the involved DIEs defines a Fredholm problem, and as such has guaranteed convergence. Numerical results are presented in Section 6, showing the dependence of the magnetic shielding effectiveness on different geometrical parameters; these point out possible critical situations in particular configurations. Our conclusions are drawn in Section 7.
Finally, we want to clearly point out the contribution of the paper. The present manuscript addresses thick circular plates, and its novel contribution is twofold: (i) for general magneto-conductive plates, a set of established DIEs having both electric and magnetic currents as unknowns is solved rigorously, and (ii) for nonmagnetic conductive plates, an original alternative DIE formulation having only electric currents as unknowns is introduced. This allows an original and efficient tool to be derived, allowing for the evaluation of, e.g., the shielding effectiveness of material disks against a near-field source such as a circular current loop coaxial with the disk.

Formulation of the Problem
The configuration of the electromagnetic problem is shown in Figure 1, where a circular disk of thickness d and radius a with electrical conductivity σ and relative permeability µ r is reported. The disk is placed on the plane z = 0 of a cylindrical coordinate system (ρ, φ, z) with the center at the origin. The source of the electromagnetic field is a circular current loop of radius R and constant current I 0 coaxial with the disk and placed at z = h. The time-harmonic e jωt dependence is assumed and suppressed throughout. It should be noted that thanks to the azimuthal invariance of the problem the considered source produces a TE z field (with E inc φ , H inc ρ , and H inc z ).

Pair of DIEs through Mitzner GBCs
The finite thickness of the screen makes the analysis more complex with respect to the infinitesimally-thin structure case. However, if the thickness d is sufficiently small (in particular, k 0 d 1, where k 0 is the free-space wavenumber), it is possible to shrink the thickness to zero (reducing the screen to an infinitely thin surface) and adopt the Mitzner generalized boundary conditions (GBCs) [25]. The GBCs link the tangential electric and magnetic fields at the surface interfaces as and where ζ 0 is the free-space impedance. Such GBCs have recently been used successfully in different scattering problems [16,17,27,29,30]. The coefficientsẐ S andŶ S are related to the thickness and the electromagnetic parameters of the screen, and different expressions have been proposed in the literature. In his seminal paper [25], Mitzner derived where ε cr = 1 − jσ/(ωε 0 ) indicates the relative complex permittivity. It is worth noting that the condition k 0 d 1 is not a limiting factor in all the practical configurations, where the thickness d of the screen is much smaller than the operating wavelength in air up to the microwave range.
The GBCs in (1) and (2) define a combined electrically resistive/magnetically conductive sheet with infinitesimal thickness, which supports both electric and magnetic surface currents that are related to the discontinuities across the sheet of the tangential magnetic and electric fields, respectively. Because any plane containing the vertical z-axis is an odd symmetry plane, the electric surface current density J s is azimuthally directed, whereas the magnetic current density M s is radially directed.
Following the same reasoning illustrated in [31], two sets of DIEs can be obtained in the canonical form where r = ρ/a, J 1 (·) is the Bessel function of order 1. The unknown functions are while the expressions of the functions γ i (i = 1, 2) are and The functionsJ Sφ (ν/a) andM Sρ (ν/a) appearing in (7) are the Hankel transforms of order 1 of the surface current densities J Sφ (ρ) and M Sρ (ρ), respectively, while is the z-component of the normalized spectral wavenumber [31].

Solution of the Pair of DIEs
As described in [24], the ψ i (ν) functions (which are related to either to theJ Sφ orM Sρ functions) are expanded in Neumann series of Bessel functions, i.e., thus, thanks to the Weber-Schafheitlin integral ([32], 6.574.3), (6) is spontaneously satisfied. The parameters µ i in (10) are arbitrary (provided that µ i > 0), and can be used to guarantee the convergence of the integrals arising in the analysis or to enforce a specific order of singularity of the spatial currents at the edge of the aperture [24]. In fact, all the addends in (10) are the Hankel transforms of terms proportional to where P (·,·) n (·) are Jacobi polynomials of order n [33]. By using (10), (5) becomes which is valid for 0 ≤ r < 1. By Galerkin testing (12), the dual integral equations are reduced to the matrix system where the matrix elements are provided by while it is easy to show that and From (8) and (14), it can be immediately derived that to make the integrals convergent it must result µ 1 > 1/2 and µ 2 > 1; thus, we can set, e.g., µ 1 = 1 and µ 2 = 3/2, and the coefficients in (13) may be obtained as i.e.,

A
and the known terms in (13) as By solving the algebraic system (13), the coefficients ψ (i) n are derived and used to compute the spectral currents asJ andM From the knowledge of the spectral currents, any component of the field can easily be calculated. In particular, the z-components of the scattered magnetic fields H (1)scat z and H (2)scat z beyond the disk, (i.e., for z < 0) can be calculated as and i.e., and with h (1) and such that The z-component of the incident magnetic field (i.e., that radiated by the original current loop source in free space) is instead From (28) and (29), the magnetic shielding effectiveness SE H can easily be calculated through [34]:

Single DIE for Conductive Disks
An approximate boundary condition is derived here for purely conductive disks. The Mitzner GBCs allow for exact representation of the screen of infinite extent in a spectral-domain equivalent transmission line (TL) as a T network, as shown in Figure 2 [35]. The expressions of the longitudinal impedance Z L and the transverse impedance Z T are respectively When a loop source radiates in the presence of the infinite screen, the equivalent TL of the problem in a Schelkunoff approach [34,36] is as reported in Figure 3, where Z loop is the impedance of the current loop and Z 0 is the free-space impedance η 0 . For a magneto-conductive screen, this always results in |Z L | Z 0 , and the presence of the longitudinal impedances Z L in the TL network in Figure 3 can be neglected if the result is |Z L | Z loop . The impedance of a parallel loop is a function of the distance; in particular, at the air-screen interface for a small loop, it results in [34,37] Therefore, the effects of Z L can be neglected if Thus, it should be noted that for purely conducting screens (33) is always satisfied provided that d h, which always occurs in practice. If the presence of the longitudinal impedances Z L can be neglected, then the screen is represented only by the transverse impedance Z T . This results in only one boundary condition, which is the same as in (1). Thus, we can derive a single set of DIEs which solves the problem, which consists of (5) and (6) with i = 1 andẐ T instead ofẐ S . Therefore, under assumption (33), the solution of the original problem requires the introduction of only one unknown (i.e., an equivalent surface electric current density) and the solution of only one set of DIEs; this is exactly the same formulation adopted to solve the problem under the assumption of thin screens [16,23]. In such a case, the scattered magnetic field is provided by (24), with coefficients ψ (1) n provided by the solution of (13) with i = 1. Thus, it is clear that with respect to the full formulation based on Mitzner GBCs, the savings in terms of computation time are about 75%.
Another important advantage is that for k 0 a 1 (a condition which in practice is always met in EMC) the elements of the matrix (18) have an exceptionally simple expression. In fact, using the series expression derived in [38] to express the integral term in (18), we have where Γ(·) is the Gamma function [33], p = m − n, and q = m + n. However, for k 0 a 1, only one addend is sufficient to reach convergence, such that Using the definitions of p and q and the properties of the recursive properties of the Gamma function, we finally have which makes the solution of the system dramatically faster. Finally, by following the same line of reasoning as in [24], the linear system (13) with coefficient matrix (37) can be shown to define a Fredholm operator (in particular, a second-kind Fredholm equation) in the space of square-summable sequences, to which the right-hand side of (13) belongs. Hence, the system (13) has a unique solution and guaranteed convergence, and no regularization scheme is required [39].

Numerical Results
We first consider a copper disk (conductivity σ = 5.7 × 10 7 S/m) with thickness d = 1 mm and radius a = 1.5 m in a frequency range from 100 Hz to 1 MHz. The electromagnetic source consisting of a circular current loop coaxial with the disk is assumed to be placed at height h = 30 cm (i.e., h/a = 0.2) and with a radius R = 5 cm (i.e., R a). In Figure 4, the magnetic shielding effectiveness SE H is evaluated at a symmetric point with respect to the loop position, i.e., at z = −h, and is reported as a function of the frequency.
We have used different formulations here; in particular, it can be seen that the formulation based on Mitzner GBCs (see Section 4, blue solid line) is perfectly superimposed with respect to the proposed formulation based on the new boundary condition (34) (see Section 5, cyan dashed-square line) in the entire frequency range; these results coincide with those obtained with FEKO commercial software (not reported here).
On the other hand, the formulation based on the thin-screen approximation [23] (red dashed line) is correct only up to about 5 kHz; in this low frequency range, the SE H is essentially that of an infinite metallic screen having the same thickness of the disk [18][19][20], which is reported in Figure 4 as a dot-dashed line for a fair comparison. In practice, at low frequencies and for the considered values of R and a the metallic disk is seen by the loop as if it were of infinite extent.
At high frequencies (that is, larger than 100 kHz), all the formulations (i.e., the full Mitzner BC formulation, the new proposed BC formulation, the thin-screen formulation, and FEKO) approach the PEC disk results [14], which are shown in Figure 4 as a dotted line.
However, it can clearly be seen that there exists an intermediate frequency range in which neither the thin disk formulation nor the infinite metallic disk formulation can reproduce the correct results. In such a frequency range, the finite thickness of the disk must be carefully taken into account either through the formulation based on Mitzner GBCs or the new boundary condition (34). It can be observed that in this intermediate frequency range the SE H of the infinite solid conductive thick screen and the SE H of the infinitesimally thin PEC disk are comparable; moreover, the SE H of the thick disk with finite conductivity can be much larger (almost 30 dB larger) than the SE H of the equivalent PEC disk, presenting a sharp peak.
In Figure 5, the dependence of SE H on the thickness-to-skin depth ratio d/δ is reported for the same parameters as in Figure 4. As expected, the thin disk formulation is accurate only for disk thicknesses smaller than the skin depth δ. Moreover, in this low-frequency range there is no difference with the infinite solid conductive thick screen.
The dependence of SE H on the on the source-to-screen distance h is instead shown in Figure 6. The distance h ranges from h = 20 cm (i.e., h/a = 0.13) to h = 1.5 m (i.e., h/a = 1). Thus, it can be seen that a peak of SE H may or may not be present in the intermediate frequency range, depending on h; furthermore, the asymptotic PEC limit is approached faster as h increases. Therefore, the presence of the intermediate frequency range with the presence of the SE H peak is strongly dependent on the source-to-screen distance h. It is interesting to point out that at low frequencies the SE H increases for larger values of h, while in the high-frequency range the SE H dramatically decreases with increasing h.   Finally, the dependence of SE H on the loop radius R is reported in Figure 7 for radius values between R = 5 cm and R = 50 cm. While this always results in R < a, the loop radius can be smaller or larger than the fixed source-to-screen distance h = 30 cm. In all of the cases shown here, the PEC limit is achieved at frequencies higher than about 100 KHz; however, the position and shape of the SE H peak in the intermediate range depends on R, becoming more pronounced and shifting towards high frequencies as R decreases. On the other hand, at the limit of large R the peak of SE H tends to disappear.
In summary, in the above figures the variations of SE H with frequency, thickness-toskin-depth ratio, source-to-screen distance, and loop radius are illustrated. Although these results do not span the entire parametric space, they are both relevant and representative of the typical electromagnetic response of the considered configuration. Thus, it is worthwhile to point out that although we have considered only copper as a typical conductive material, the validity of the method is general and has been tested on different values of the disk conductivity.

Conclusions
The magnetic shielding effectiveness of a finite screen consisting of a circular disk with finite conductivity and finite thickness is evaluated when a circular current loop of finite radius coaxial with the disk and with constant current is assumed as the electromagnetic field source. An original formulation is proposed to solve the problem using suitable generalized boundary conditions to take into account the finite thickness of the screen and solving the two dual integral equations derived by enforcing the boundary conditions through an expansion of the unknowns in Neumann series of Bessel functions. An alternative formulation that is valid for a purely conductive screen with no magnetic properties is proposed as well, and is computationally much faster. The magnetic shielding effectiveness is studied in different configurations while pointing out possible critical situations. Additional work is currently in progress to extend the presented analysis to non-constant exciting currents.
Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.