Loop-to-Loop Magnetic Coupling Through a Circular Aperture in a Planar PEC Screen of Finite Thickness

An analytical formulation for magnetic field penetration through a circular aperture in a thin perfect electric conductor (PEC) screen with finite-thickness is developed. Especially, we propose an analytical formula for the magnetic coupling between two circular loops separated by the screen and placed coaxially with the aperture. The formula is verified by finite element simulations and experimental data. The thickness effect is explained by the cut off attenuation of TE01 mode in the aperture. The influences of loop-screen-loop distance and loop radius on the coupling intensity are also investigated.


I. INTRODUCTION
Shielding of low frequency magnetic field using metal screens (enclosure, sheets, wire meshes) has been important for many decades and is now even more indispensable with the burgeoning number of RF transmitters and sensitive devices of all types [1], [2], [3], [4], [5], [6]. Frequently, a screen has to contain apertures or slots for ventilation or cabling purposes. These clearly may reduce the shielding performance. Previous researches show that apertures are the primary path for magnetic field leakage at higher frequencies, and by contrast magnetic diffusion through metal sheet is dominant at lower frequencies [7], [8], [9], [10]. In term of analytical formulation of magnetic field leakage through an aperture in a planar sheet, Bethe's small aperture coupling theory is widely adopted [7], [11], [12]. Where, the aperture is modelled as a magnetic dipole with its moment proportional to the tangential component of external applied magnetic field. However, this treatment requires the premise that the applied field distributes uniformly across the The associate editor coordinating the review of this manuscript and approving it for publication was Luyu Zhao . aperture surface. For example, when the emitting loop and the receiving loop are coplanar and perpendicular to the infinite slotted conducting plate [13], the tangential component of the magnetic field in the slit is considered to be uniform and Bethe's small aperture coupling theory can be applied.
For the case where the loops are coaxial with each other and parallel to the solid shielding planar plate, Ref. 14 studies the effects of loop radius and loop-to-loop distance for the infinite screen; and Ref. 15 studies the loop-to-loop coupling in the case of a shielding plate with limited radius. In a recent paper [16], a circular aperture on the screen is investigated where the applied field is produced by a circular current loop placed coaxially with a circular aperture. Then the tangential magnetic field on the aperture surface will be nonuniform and omnidirectional. Especially, the analytical expression for the longitudinal component of the shielded magnetic field is obtained if only the emitting loop radius or the loopto-aperture distance is much larger than the radius of the aperture. However, this research has two limitations. The first one is the lack of analytical formula for the loop-toloop coupling. This requires an integration of magnetic flux density through the receiving loop. However, a check of the This letter aims to relax the two limitations. First, noting that the field distribution far from the aperture (Eq. 28, Ref. 16) is similar to that produced by a quadrupole, an analytical formula of the magnetic vector potential is derived. Then, the formula for the integration can be obtained easily. Second, the thickness is considered with the attenuation effect of waveguide below cut off, which results in an overall exponential decay of the shielded field. Further, the effects of changing the receiving loop radius and moving the screen longitudinally are analyzed. Such analysis is of practical interest. In addition, we show that the tangential component of magnetic field on the illuminated side of the aperture can be approximated as the unshielded field, which can be applied to evidently reduce the computing time and memory in numerical simulation.

II. MODEL AND THEORY
As shown in the Fig. 1, a circular aperture of radius a is drilled in a planar PEC screen of infinite extension and with thickness t. The emitting loop, placed coaxially with the aperture, has a radius r 1 and carries a time harmonic current I with frequency f . A receiving loop with the radius r 2 is coaxial with the aperture, too. The distance from the upper surface of the screen to the emitting loop is z 1 , and to the receiving loop is z 2 . The coordinate origin is at the center of the upper surface of the aperture and the z-axis is coincident with the axis of the aperture. It should be noted that the analytical model should meet quasi-static conditions: the sizes of the two loops and the loop-to-loop distance are electrically small relative to the free space wavelength λ (size < λ/2π). In practice, the sizes are usually less than 1 m, and hence up to 30 MHz (λ = 10 m) the configuration is still electrically small. Then, the above magnetic shielding problem can be treated well using quasistatic formulation. Especially, when the thickness t tends to zero and the aperture is small (a < r 1 or a < z 1 ), an analytical formula for magnetic field on the z-axis (z > 0) is proposed in [16], Further, if z > 2a, Eq. 1 is approximated as Here, equation (2) is explained quantitively as the results of a magnetic quadrupole placed in the aperture center (the coordinate origin), as shown in the Fig. 2. Wherein, the magnetic quadrupole consists of two close magnetic dipoles with the same moment m but opposite directions (+z and −z). The spacing h of the two dipoles tends to zero, while the product of m and h keeps unchanged. It is easy to prove, if we let then the field generated by the quadrupole will be identical to that expressed by Eq. 2. In other words, the leakage field is equivalent to the free space field of the quadrupole. Then, the corresponding magnetic field intensity and vector potential can be written in cylindrical coordinator system (ρ, φ, z) as For a receiving loop of radius r 2 and with a distance z 2 from the screen, the magnetic flux through this loop is Attention now turns to the screen thickness t. In the aperture, the field will experience exponential attenuation like e −αt resulted from the effect of waveguide below cutoff. Considering the configuration is axisymmetric and transverseelectric, the attenuation coefficient is related to the TE0n modes (n = 1, 2, 3, . . .). In order to obtain a simple approximate analytical formula for calculating shielding effectiveness (SE), we overestimated the leakage field and select the TE01 mode with the weakest attenuation among them to describe the shielded field [17], [18], [19]. Then, the attenuation coefficient α is expressed as where, c is the speed of light in free space.
With this attenuation considered, both the magnetic field (4) and magnetic flux (6) should be modified as The SE of the screen is usually defined as the ratio of the induced voltage in the receiving loop with the screen absent to that with the screen in presence. This voltage based on definition can be replaced equivalently with mutual inductance between the two loops like wherein, M s refers to the mutual inductance with the screen present and is calculated as s /I . M 0 denotes the mutual inductance with the screen absent, and its expression is well known [20].
In practice, a screen is usually made of metal material like aluminum, copper and steel. These screens can behavior well like a PEC screen when magnetic diffusion effect is weaker than aperture leakage effect, which typically happens when frequency is above tens of or hundreds of kHz, depending on the concrete geometry and material parameter [16].

III. VALIDATION
First, the field point on the z-axis is considered. Here, the loop-to-point SE can be calculated by the ratio of the magnetic field intensity observed at a given position with the enclosure removed to that observed at the same point with the enclosure loaded. The SE-z/a curves are plotted in the Fig. 3 for different r 1 /a, z 1 /a, and t/a. The calculated results are obtained using the magnetic field in Eq. 8. Meanwhile, the simulation results with f = 0 Hz from a 2D axis-symmetrical finite-element method (FEM) model built in the COMSOL are also provided for comparison. It should be noted that under the PEC boundary condition (the normal magnetic field component on the surface of the PEC screen is zero), the results are frequency-independent within the quasi-static range [16]. The results show that the approximate analytical formula (8) is in good agreement with the FEM results when  r 1 /a > 2, z 1 /a > 2, and z/a > 2. However, beyond the range of this condition, it cannot be said that the FEM results under all parameters are inconsistent with the analytical results. For example, when r 1 /a =1, z 1 /a =1, and z/a > 2, the calculation results also agree well with the FEM. Therefore, we set the conditions of ''r 1 /a > 2, z 1 /a > 2, and z/a > 2'' as a conservative description of the scope of application for the analytical formula.
Second, the loop-to-loop SE is considered. Combined with the above analysis, Fig. 4 shows the calculated (Eq. 9) and FEM-simulated (PEC boundary condition and f = 0 Hz) SE-r 2 /a curves with r 1 /a = 2 and z 1 /a = 2. It can be seen that the calculation results of the loop-to-loop SE formula are in good agreement with the FEM simulation for different t/a in the case of z 2 /a > 2.

IV. APPLICATIONS A. EFFECT OF THE LOOP-TO-SCREEN DISTANCE
For an infinite solid planar conducting screen, the emittingloop-to-screen distance z 1 has no effect on the SE under the premise that the loop-to-loop distance b (b = z 1 + z 2 ) is fixed [1]. However, for the perforated planar PEC screen, z 1 has an impact.
When the observation point is on the z-axis, the shielded field (2) can be expressed as Take the derivative of (10) and we get the conclusion that if 0 < b ≤ 3.6r 1 , H sz increases with z 1 ; if b > 3.6r 1 , with the increase of z 1 , the trend of H sz is to increase first, then decrease, and finally increase. In general, the magnetic field increases as the screen moves closer to the receiving side.The magnetic field curves changing with z 1 under various b are shown in the Fig. 5.
The calculated H sz from (10) is in good agreement with the FEM results in the range of z 1 > 2a and z 2 > 2a. It can VOLUME 10, 2022  be seen that the trend of H sz is consistent with the above conclusions. For larger loop-to-loop distance, the range of distances in which ''cal'' and ''FEM'' are consistent also expands. Practically, the distance between rings is often large, so the analysis results are consistent with FEM in a wide range.
For the loop-to-loop situation, we calculate the magnetic flux s (Eq. 6) through the receiving loop with different loop-to-loop distance in the Fig. 6, and the data has been normalized for a more intuitive display. It shows that when r 2 /r 1 = 1/4 and 1/2, s generally increases with the screen moving close to the receiving loop. In the case of r 1 = r 2 as (d) shows, the curves of s are symmetrical. When the loop-to-loop distance is relatively large such as b = 5r 1 , the screen in the middle position can produce the highest SE; and when b < 3r 1 , the screen in the middle produces the lowest SE.
Some experiments were carried out to investigate the effect of the emitting-loop-to-screen distance on SE when the loopto-loop distance is fixed. The SE of a perforated finite conductivity conductor (FCC) screen (1 m × 1 m, 3.8 × 10 7 S/m, t = 1 mm, and a = 2 cm) is measured by the handmade loops shown in Fig. 7(a) with f = 300 kHz. The emitting loop (10 turns) is 2.5 cm in loop radius, 2 mm in wire radius and the receiving loop (15 turns) is 0.5 cm in loop radius, 1 mm in wire radius. Here, we define z 1 and z 2 as the distance from the upper surface of the screen to the middle position of the emitting and receiving loops respectively as shown in Fig. 7(b). Further, some measurement results are displayed for z 1 + z 2 = 7 cm (Fig. 8(a)) and 10 cm (Fig. 8(b)). We also calculated the SE of the loop-to-loop model (using Eq. 9) and the loop-to-point model (using Eq. 8 with ρ = 0).
When the radius of the receiving loop is relatively small, the loop-to-point SE and loop-to-loop SE agree well. The measured and FEM simulated SEs are in good agreement with each other and generally decrease with increasing z 1 . The calculated results are no longer accurate when z 2 < 2a. For example, as z 2 approaches 0 and the shielded field in Eqs. 2 and 6 approaches infinity, which leads to wrong SE results (close to 0 or even negative).
It should be noted that the loops in Fig. 7 are different from that used in the Ref. 16. Where the emitting loop has a metal frame, which makes it difficult to determine the equivalent radius of the practical loop when it is treated as an equivalent ideal loop. In contrast, the handmade loops have relatively small wire radius and there is no attached frame structure, which can reduce the uncertainty in the parameters (r 1 , r 2 , z 1 , and z 2 ). Due to this improvement, the present experimental configuration has better agreement with our theoretical model.
It is also worth noting that the screen can be regard as PEC with f = 300 kHz. The reason is that the same FCC screen has experimentally behaved like PEC even for f = 100 kHz (see Fig. 6(b) in Ref. 16).

B. THE MAXIMUM COUPLING RADIUS
Here, we investigate the effect of the receiving loop radius r 2 on the coupling (mutual inductance M s ) of the two loops. Keeping other parameter unchanged, the maximum coupling occurs when r 2 = 2 3z 2 ≈ 0.82z 2 (11) Besides, (11) can also be obtained by solving the r 2 when the direction of H sz changes, that is, to find the zero point of H sz in (4). Equation (11) was experimentally verified using the receiving loops with different radius in the Fig. 9. Considering that the mutual inductance is proportional to the induced voltage, the coupling is studied by testing the induced voltage in the experiment with r 1 = 6 cm, z 1 = 5 cm, a = 2 cm, t = 1 mm, and f = 300 kHz.   The normalized induced voltage curves are shown in the Fig. 10. The experimental maximum coupling radius r 2−m and the theoretical result r 2 (Eq. 11) under different z 2 are listed in the Tab. 1. We can see that r 2−m is basically consistent with r 2 .

C. BOUNDARY CONDITIONS FOR THE APERTURE SURFACE
When the screen thickness is equal to zero, the tangential component of the magnetic field on the aperture surface is equal to that produced by the current loop alone without the screen [16]. For the PEC screen with a thickness t (t = 0), we would like to investigate whether the tangential magnetic field H aρ of the aperture surface on the illuminated side is still equal to the unshielded. Therefore, the illuminated magnetic field of a screen with different thickness and the unshielded field in free space are calculated using FEM simulation. The results are shown in the Fig. 11 and the ratio of the unshielded to the shielded field is within the range 0.85-0.99 (−1.41 to −0.09 dB). This shows that the tangential component on the illuminated side of the aperture can be approximated as the unshielded one.
Using the unshielded field as a boundary condition of the aperture surface on the illuminated side, the field H sz and H sρ on the non-illuminated side is calculate by the FEM simulations. The FEM results H sz and H sρ are from the original FIGURE 11. FEM results of the tangential magnetic field H aρ (a = 1 cm, I = 1 A). Where, ''shielded'' and ''unshielded'' represent the shielded field on the illuminated aperture surface side and the unshielded field in free space.

FIGURE 12.
Curves of FEM simulation magnetic field on the non-illuminated side of a perforated PEC screen (a = 1 cm, I = 1 A, r 1 / a = 3, z 1 /a = 3, and t/a = 0.5).
model of perforated PEC screen in the Fig. 1. Fig. 12 shows that using the boundary condition that the tangential magnetic field on the illuminated aperture surface is equal to the field without shielding can produce the same field with the original model, through which the range of field domain can be cut down.

D. APPLICATION TO THE SCREEN WITH A FINITE CONDUCTIVITY
Here, the dependence of SE on frequency is investigated by measurements and FEM simulations. Meanwhile, the calculated SE for perforated FCC screen is obtained by the superposition principle: the field corresponding to the perforated FCC screen is the sum of the field for the solid FCC screen and the perforated PEC screen [7], [16]. The SE of a solid FCC screen is calculated using the ratio of magnetic flux in [21], and that of the perforated PEC screen is calculated by Eq. 9. SE of the perforated FCC screens with different thicknesses (t = 2 and 5 mm) is also measured by the SE test-bench setup [16] and simulated using FEM software.
For the perforated FCC screen, the SE versus frequency exhibits two-stage behavior: it increases when f < f 0 (diffusion effect) but is nearly unchanged when f > f 0 (aperture effect). Where, f 0 is the intersection frequency of the solid FCC screen curve and the perforated PEC screen curve. From Fig. 13, f 0 is about 15 kHz and 10 kHz for t = 2 and 5 mm, respectively. It can be seen that f 0 is smaller with larger thickness.

V. CONCLUSION
The leakage magnetic field from a circular aperture in a PEC screen, when excited by a circular current loop placed coaxially with the aperture, is expressed analytically using equivalent magnetic quadrupole. An analytical formula is proposed to predict the magnetic coupling between two circular loops separated by the PEC screen and placed coaxially with the aperture. Comparisons with FEM simulations show the field expression is applicable for observation points beyond two times of aperture radius away from the aperture. Effect of the screen thickness is explained by the exponential attenuation of TE01 mode below cut off in the aperture. With fixed loopto-loop distance, the SE reduces with the screen moving close to the receiving loop for smaller receiving loop. By contrast, for larger receiving loop and loop-to-loop distance, the SE is highest when the screen is in the middle of the two loops. Keeping other parameter unchanged, the maximum coupling occurs when the receiving loop radius equals 0.82 times of the receiving-loop-to-screen distance. The tangential component of magnetic field on the illuminated side of the aperture can be approximated as the unshielded field, and hence in numerical simulation the range of field domain can be cut down.