Analytical Shielding Metrics-Based Shielding Configuration Guideline for ELF Magnetic Field Mitigation

This paper presents a simple guideline for configuration of the shielding materials that mitigates the extremely low-frequency (ELF) magnetic fields generated by power facilities located close to our daily life activities. Generally, materials with high permeability and conductivity are used to mitigate the magnetic field; however, in the current source region, before passing through the shielding material, the magnetic field may be increased by the configuration of the shielding material. To assess the effect of the shielding configuration in the current source and shielding regions, metrics are newly introduced, which were obtained based on the analytical solution for infinite width shields. In addition, the analytical solution of the shielding pipe wrapping a current source was deduced by solving the cylindrical Helmholtz equation. The shielding pipe is an important factor that can bring about changes in the metrics introduced in this study. The simple shielding guidelines suggested from these analyses help determine strategies for designing shields that can mitigate the magnetic field in both the current source and shielding regions.

In recent decades, many studies have been conducted on 30 the technical possibility and effects of magnetic field sup-31 pression countermeasures applied near transmission or dis-32 tribution lines [11]. These can be broadly classified into two 33 categories: 1) creation of cancelling magnetic fields, such as 34 those resulting from cable installation geometry [12], [13], 35 [14] and passive shielding loops [15], [16]; 2) use of shielding 36 materials, such as those with high permeability and conduc-37 tivity [17], [18], [19], [20], and a suitable shielding material 38 installation geometry [21], [22], [23]. 39 The primary emphasis of methods using shielding mate-40 rials focuses only on the mitigation of the magnetic field 41 in the shielding region without considering the magnetic 42 field in the current source region, which could affect 43 the cable management operators or monitoring system. 44 Shield effectiveness was introduced to evaluate the magnetic 45 field shield, which is suitable for describing the degree of 46 shielding from changes in the electrical properties of the 47 shielding material, such as permeability and conductivity. 48 time-harmonic magnetic field. A is defined as the magnetic 97 field B = ∇ × A and the electric field E = −∇φ e − ∂A/∂t, 98 where the scalar function, φ e , denotes an arbitrary electric 99 scalar potential which is a function of position, and t is a 100 time [24]. The inhomogeneous Helmholtz equation for A, 101 which is a govern equation of the mathematical model, can 102 FIGURE 1. Basic shielding configuration. A horizontal current source is located under shielding materials with infinite width (y s < 0). be described as where ω is the angular frequency. The curl of A was defined 106 as B, while the divergence of A which is independent of 107 its curl has a liberty [25]. To get the greatest mathematical 108 convenience for (1), let Substituting (2) into (1), (1) for A in each region n can be 111 simplified as follows: 112 ∇ 2 A n + p 2 n A n = 0 (3) 113 A constant p n = ω √ µ n n √ 1 − jσ n /(ω n ) is called com-114 plex propagation of the medium. A n has only a z-component 115 because the current source does not vary with z. Thus, A is 116 expressed as a function of x and y and not of z. 117 ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + p 2 n A n (x, y) = 0 (4) 118 The uniqueness theorem for time-harmonic electromagnetic 119 waves states that the solution satisfies Maxwell's equations 120 and that its boundary conditions are unique. This means 121 that all approaches to Maxwell's equations express that the 122 Helmholtz equation has the same and unique solution [26]. 123 The method of separating variables (also known as the Fourier 124 method) can be applied to the partial differential equation 125 of (4). By letting A n (x, y) = A x,n (x) · A y,n (y), we substitute 126 into (4) to obtain Each of the terms with the second derivation in (5) must 129 be equal to a constant because they must be independent of 130 each other's denominator variables (x and y) and similarly 131 for the third term. When defining that the first and second 132 terms are separation constants, −k 2 and γ 2 n , respectively, (5) 133 is separated as follows: where k means the wave number along the x-axis and k ∈ 137 (0, ∞) and the wave number along the y-axis γ n = k 2 − p 2 n .

138
The general solution to (6a) and (6b) is as follows: where G s , U s , C n and D n are unknown coefficients.

142
A(x − x s , y) = A(−x + x s , y) because of its symmetric struc-143 ture, as shown in Fig. 1 The magnetic field for the A n with only z-component is  C n−1 e −γ n−1 T n + D n−1 e γ n−1 T n 176 = C n e −γ n T n + D n e γ n T n (11b) 177 The coefficients can be rewritten in matrix form as where W n = µ n−1 γ n /(µ n γ n−1 ). To avoid the confusion due 185 to the index of interfaces and regions, Table 1 lists the index 186 examples for the parameters used in this paper.

187
In the multilayer interface, the coefficient matrix [M n ], 188 which is a 2-by-2 matrix on the right-hand side of (12), 189 combines the system's incident and outcome waves. 220 Although the shielding effectiveness is related to the magnetic The sign of the real part for M indicates whether interfer-262 ence is constructive or destructive. If ''+'' is obtained, the 263 magnetic field intensity in the current source region increases 264 along the vertical line of Interface 1 passing through x = x s 265 and y = 0.

III. PARAMETRIC ANALYSIS FOR SHIELDING MATERIALS 267
A parametric analysis was performed to study the influence 268 of the geometrical and electrical parameters of the shielding 269 material. Table 2 lists the electrical properties of the materials 270 used for the parametric analysis. Here, the relative permeabil-271 ity and permittivity are expressed as µ r and r , respectively, 272 and the magnetic material, air-gap, and conductive material 273 are expressed in abbreviated form as MM., ari., and CM., 274 respectively. The analysis conditions for the electrical prop-275 erty combinations of these materials are listed in Table 3  In Cases 1 and 2, the contours of the dB-scale (20 log 10|H |) 283 for the maximum magnetic field intensity at an arbitrary 284 location are plotted in Fig. 2. When the positions of MM.1 285 and CM.1 are exchanged, the distribution of the magnetic 286 field intensity in the shielding region hardly changes, whereas 287 that in the current source region changes. Fig. 3 shows a 288 plot of magnetic field intensity along the vertical line passing 289 through x = 0. It is clear from Fig. 3 that a magnetic mate-290 rial maintains a low magnetic field intensity over the entire 291 region occupied by this material, whereas a conductive mate-292 rial causes the magnetic field intensity to drop drastically. 293  abutting Interface 1 determines the sign of M ,re . Although 310 the magnetic field region covered in this study is assumed to 311 be near-field due to the 50 or 60 [Hz] operating frequency, 312 the characteristic impedance and reflection coefficient for 313 far-field can be qualitatively applied even for the near-field. 314 The characteristic impedance increases as the permeability 315 increases, whereas it decreases as the conductivity increases. 316 The characteristic impedance of Region 1 with high per-317 meability is bigger than that of the current source region. 318 Thus, the reflection coefficient has the ''+'' sign. Because 319 the reflection coefficient for the far-field is defined based on 320 the electric field, it has the opposite sign when defined based 321 on magnetic field intensity, as in M of this study. Changes 322 in M ,re and SE M for relative permeability and conductivity 323 will be discussed in more detail in subsections B and C of 324 this section. The parametric analysis is performed with only 325 one parameter modified at a time and the other dimensions 326 maintained at the previously defined reference values.

328
As shown in Fig. 5, the magnetic field intensity in the shield-329 ing region decreased as the relative permeability decreased, 330 regardless of the arrangement of the shielding material. 331 The high relative permeability of the shielding material 332 concentrates more magnetic fields per unit cross-sectional 333 area than materials with relatively low relative permeabil-334 ity. The magnetic field intensity in the shielding region can 335 decrease because of this impact. A comparison of the result of 336 Figs. 5a and 5c shows that the magnetic field intensity in 337 the current source region is more affected by the change in 338 the relative permeability of the immediately adjacent region. 339 As shown in Fig. 5a, for the MM.2-air.-CM.1 case, the 340 magnetic field intensity is reduced because the relative per-341 meability of MM.2 is 10 times higher than that of MM.1. 342  However, as shown in Fig. 5c, even if the relative permeabil- The M ,re and SE G,dB values with a change in the thickness 387 of each shielding material are shown in Fig. 9, where the 388 y-location, y q , to obtain SE G,dB is 0.031 [m]. As the thickness 389 increases, more magnetic field energy is stored in the shield 390 or the conduction loss increases more. The impact reduces the 391 magnetic field intensity in the shielding region regardless of 392 the arrangement of the shielding material, as shown in Fig. 9. 393 As mentioned above, M ,re is the parameter that affects the 394 magnetic field intensity in the current source region, and 395 M ,re changes according to the thickness of the material adja-396 cent in this region. Fig. 9a shows that the sign of M ,re varies 397 based on MM.1 thickness at 1.65 × 10 −3 [m], and this means 398 that the magnetic field intensity in the current source region 399 can reduce more beyond a certain thickness. As shown in 400 Fig. 9c, the increasing thickness of CM.1 adjacent in the cur-401 rent source region rather increases the magnetic field intensity 402 in this region. In addition, unlike MM.1, M ,re of CM.1 has 403 the ''+'' sign on all thickness range. Fig. 10 shows the impact 404 for air-gap thickness. M ,re has a constant value regardless of 405 the air-gap thickness. SE G,dB changes with increasing air-gap 406 thickness, but the impact of this thickness is insignificant as 407 the difference between the maximum and minimum of SE G,dB 408 in Fig. 10b is merely 0.09 [dB].

410
Prior to using shielding materials to reduce the mag-411 netic field intensity in the shielding region, the current 412 source was wrapped with a shielding pipe [27], [28]. 413 Under this condition, the incident magnetic field intensity at 414 Interface 1 changes. Fig. 11 shows the basic configuration of 415 the current source and its shielding pipe, which mitigate the 416 magnetic field intensity incident on Interface 1. Owing to the 417 uniform and infinitely long structure in the z-direction, 418 the magnetic vector potential of the cylindrical coordinate, 419 A n,wp has only the z-component, and is expressed only as 420 a function of the radial distance, r, from the center of the 421 VOLUME 10, 2022       A magnetic alloy with conductivity of 7.25 × 10 5 [S/m], 469 is employed in Region 1 to reduce the magnetic field intensity 470 in the current source region. Its nonlinear characteristics are 471 shown in Fig. 12, including its relative permeability and the 472 incident magnetic field intensity at which magnetic satura-473 tion begins. The maximum relative permeability is 36,233 at 474 |H | = 5.66 [A/m], indicating maximum shielding efficiency. 475 Region 2 is occupied by aluminum whose the electrical 476 property is the same as that of CM.1 in Table 2. Here, the 477 air-gap is ignored because its thickness has little effect on the 478 change in the magnetic field intensity as shown in Fig. 10. 479 The thicknesses of Regions 1 and 2 are equal to 0.005 [m]. 480 The shielding pipe discussed in this section is made of copper 481 (σ = 3.8 × 10 7 [S/m]), and its geometrical parameters are 482 r 1 = 0.05 [m] and r 2 = 0.001 [m]. The magnetic field inten-483 sity generated by the cable decreases drastically as it passes 484 through the shield pipe, and then it is incident on Interface 1. 485 When I s = 750 [A], the incident magnetic field intensity 486 calculated in accordance with distance r from the cable center 487 using (21) is shown in Fig. 13. Here, r = 0.91 [m] corre-488 sponding to |H | = 5.66 [A/m]. To simplify calculations, the 489 current source that generates this magnetic field is assumed to 490 be an equivalent point source. In this shielding configuration, 491 M ,re and SE M ,dB obtained by (15) and (17)