Transient Magnetic Shielding of a Planar Conductive Thin Screen via Exact Image Theory

The exact image theory is here applied to study the transient shielding effectiveness of a canonical shielding configuration, i.e., a small current loop in the presence of a parallel planar thin conductive sheet. In particular, the Green function providing the electromagnetic field produced by a pulsed vertical magnetic dipole in the whole space is derived in a closed analytical form. The recently introduced definitions of transient shielding effectiveness are used to evaluate the shielding properties of the configuration. The derived results allow for calculations in a closed form for simple transient waveforms of the exciting currents.


I. INTRODUCTION
T HE study of low-frequency (time-harmonic) shielding of a loop that is axially perpendicular to a metallic planar shield of infinite extent is a canonical problem in electromagnetic shielding.In fact, two current loops separated by a parallel planar thin conductive sheet is the configuration prescribed in NSA tests [1], [2] and occurs in several EMC practical cases, such as current paths in printed circuit boards.
This canonical configuration has been extensively studied in the past.The first seminal paper by Levy appeared in the 1930 s [3], assumed thin screens, and was based on an extension of the classical eddy-current analysis of Maxwell.In the 1960 s, Moser derived exact integral expressions for the relevant shielding effectiveness (SE), which however required the numerical evaluation of Sommerfeld-type integrals [4].Quasi-near-field approximations of such expressions were then derived by Bannister for thick [5] and thin [6] screens.Moser's formulation was compared with experimental data and with Schelkunoff's transmission-line approach in [7], and was further extended to evaluate loop-to-loop coupling in [8].
Paolo Burghignoli is with the Department of Information Engineering, Electronics and Telecommunications, University of Rome "Sapienza," 00184 Rome, Italy (e-mail: paolo.burghignoli@uniroma1.it).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TEMC.2023.3326190.
Digital Object Identifier 10.1109/TEMC.2023.3326190 and in [10] for magneto-conductive thins screens.Such expressions were then generalized to screens with arbitrary thickness and constitutive parameters in [11].A unified wave impedance formula to be used in the transmission-line approach was derived in [12].However, all these investigations have considered timeharmonic sources, while only recently the problem of the evaluation of the SE for transient waveforms has started to be addressed [13], [14], [15], [16].In general, the shielding problem in the time domain (TD) can be solved passing through its frequency-domain counterpart and inverse-Fourier transforming the quantities of interest.However, such a procedure is numerically intensive and cumbersome and does not offer any physical insight into the involved transient wave phenomena.In this connection, the authors of [14], [15], and [16], as in [17], [18], and [19], adopted a modified Cagniard-de Hoop (CDH) approach to solve the electromagnetic problem directly in the TD and to obtain analytical or semi-analytical solutions.In fact, while a number of effective numerical approaches have been developed in the TD based on, e.g., the finite-difference TD method or on TD integral equations, analytical methods maintain their importance both for the derivation of benchmark solutions in canonical geometries and for the physical insight that they afford on the involved transient wave phenomena.In addition to the abovementioned classical CDH method [20], [21], [22], it is worth mentioning the double-deformation method developed by Tsang and Kong [23], [24], the Haddon leaky-wave method [25], [26], the modal asymptotic approaches of Felsen and Niu [27], [28], [29], and the TD version of the exact image theory (EIT) of Lindell and Alanen [30], [31], [32] (initially developed for the solution of the Sommerfeld problem [33], [34]) developed by Nikoskinen and Lindell [35].
In this work, we present a TD version of the EIT to analytically solve the abovementioned classical shielding problem directly in the TD (as Nikoskinen did in the 1990s for a simple half-space configuration [35]) showing that an inverse-Fourier transform of the frequency-domain quantities can be made symbolically starting from the EIT expressions.It is here shown that the TD EIT analytically solves the problem in a considerably simpler way compared with the CDH approach.In particular, easy-tocompute expressions for the transient field components of this canonical configuration are derived.In fact, the final integrals expressing such components may be computed analytically giving a closed-form solution (an example is provided considering the z-component of the magnetic field).Other configurations can be studied through the proposed formulation (such as the same shielding problem in a coplanar loop set up [36], [37]) and the TD-EIT could in principle be used to characterize other transient problems of EMC interest, such as the transient behavior of grounding systems [38].
The rest of this article is organized as follows.The proposed TD EIT formulation together with its basic assumptions is presented and described in Section II.In Section III, closedform results for the transient fields are derived.In Section IV, figures of merit for assessing the transient magnetic shielding are introduced and evaluated on the basis of the developed formulation for a specific source waveform.Numerical results are presented in Section V showing the advantages and the limits of the proposed formulations.Finally, Section VI concludes this article.

A. Problem Description and Assumptions
The configuration under analysis is sketched in Fig. 1.It consists of an infinite planar metallic plate with thickness d and conductivity σ.The plate is orthogonal to the vertical z-axis and is excited by a circular loop of radius a carrying a current i loop (t) with its axis coincident with the z-axis and placed at a distance h from it.
As is well known, when the loop radius a is much smaller than h and the free-space wavelength λ 0min at the highest frequency f max contained in the spectrum of the loop current, the transient loop source is equivalent to a transient z-directed (vertical) magnetic dipole of moment p m (t) with p m (t) = v(t) and where μ 0 is the magnetic permeability of free space.We, thus, consider a vertical magnetic dipole (VMD) of length , directed along the z-axis, located at z = h > 0 in air carrying a voltage v(t).The source is thus expressed as Moreover, if the screen is sufficiently thin, it can be modeled as a zero-thickness sheet with surface (transition) admittance G s = σd.In particular, as established in [15], the surface admittance representation is accurate provided that where f max is the high-frequency limit of the spectrum of the dipole waveform.

B. TD Potential Green's Function
As is well known, a VMD excites a purely TE z field, which in turn can be derived from a z-directed electric vector potential (in the Lorenz gauge) where ε 0 is the permittivity of free space and c 0 = 1/ √ μ 0 ε 0 .
Thanks to the rotational symmetry of the configuration around the z-axis, the potential is azimuthally symmetric, i.e., subject to appropriate boundary conditions.In the following we aim at deriving analytical expressions for f z (ρ, z, t).In general, such a potential can be expressed as a time convolution between the source waveform ε 0 v(t) and the relevant TD scalar Green function g F (r, t), i.e., the potential produced by an impulsive impressed magnetic current j m i (r, t).In particular, to derive the TD potential Green function g F (r, t) of the problem we assume an impulsive i loop (t), i.e., with
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The free-space term GF inc in (7) gives rise to the well-known incident scalar Green function for a point source in free space, i.e., where which has the following TD counterpart: The EIT is based on representing the coefficient R TE as a Laplace transform, so that also the reflected part of GF can be expressed in terms of point sources (more precisely, as an integral superposition of point sources placed at complex locations).In the present case, from (9), we may write so that we obtain and, hence, where with We, thus, have the sought EIT representation of the reflected Green function of the electric potential in terms of an exponentially converging integral.The integral is a superposition of images located at −h + jζ TE .
The space-time counterpart of GF r can be obtained directly through an inverse-Fourier transform of (15), i.e., By interchanging the integrals, we obtain where which can be evaluated in a closed form (details are reported in the Appendix) obtaining where and where J n (•) is the Bessel function of first kind and of order n and H(•) indicates the unit-step Heaviside function.
The expression in (19) can, thus, also be rewritten as where The integrals in (25) can also be evaluated in simple closed forms using the integral identity [40, 6.621.4],thus obtaining The result further simplifies for source and observation points along the same axis.In fact, for ρ = 0, we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
2) Transmission Half-Space: When z < 0, the spectral Green function for the electric potential is (29) where the TE transmission coefficient is The TD transmitted Green function for the potential could be obtained immediately from the EIT representation of the reflected potential given above.In fact, the first addend in (30) gives rise to the incident Green function g F inc (r, t), whereas the second addend gives rise to a term analogous to g F r (r, t), the only difference being that in all the expressions the term (z + h) should be replaced by (h − z).Therefore, where r − has been defined in (11) while and It can easily be shown that the result in (31) perfectly coincides with that obtained in [15].The result further simplifies for source and observation points along the same axis.In fact, for ρ = 0, we have and therefore,

C. Limiting Cases
It should be noted that in the limit G s → 0 (i.e., free space), it results whereas for G s → +∞ (i.e., PEC sheet)

III. FIELD CALCULATION
The TD magnetic potential f z can be obtained by means of a time convolution between g F (r, t) and the voltage waveform ε 0 v(t) , i.e., Taking into account that ∂/∂φ = 0, from (4), we have and In particular, it also results As an example, let us consider the z-component of the magnetic field along the z-axis (ρ = 0).First of all, from (28), it is easy to show that where Z r = z + h, T r = Z/c 0 , and having assumed v(t) = 0 for t ≤ 0.Moreover, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where and having assumed V (t) = 0 for t ≤ 0. From (43) [by using (44) and ( 45)], we obtain the z-component of the reflected magnetic field h r z as On the other hand, the g F inc term gives rise to the incident magnetic field and, in particular where Z t = | z − h| and T t = Z t /c 0 .Therefore, With the same reasoning, considering the transmitted Green function g F t , for z < 0, we obtain IV. TRANSIENT SHIELDING To assess the transient shielding properties of the screen in the considered configuration the following transient SE (peak-reduction and derivative-reduction efficiencies) are introduced [ where the magnetic fields are assumed to be calculated in (ρ, z) = (0, −h 2 ).A simple, but representative, transient current-loop behavior is that reported in Fig. 2 for which and the relevant equivalent voltage is analytically expressed as where V 0 = μ 0 πa 2 1 I 0 .With this assumption, from (50), the magnetic field is thus where h z1 (0, z, t) = 0 for t < T t , for T 2 + T t < t < T 3 + T t , and h z1 (0, z, t) = 0 for t > T 3 + T t .
Moreover, it results h z2 = 0 for t < T t , for for T 2 + T t < t < T 3 + T t , and for t > T 3 + T t .On the other hand, the incident field can be evaluated as h inc z = h z1 by simply letting Ĝs = 0 in (56)-(58).

V. NUMERICAL RESULTS
We consider two exciting circular electric current loops with a transient waveform as in Fig. 2 and with the following parameters: T 1 = 6u, T 2 = 16u, and T 3 = 20u, where u equals 10 −4 s (current #1 -slow transient) and 10 −6 s (current #2 -fast transient), respectively.The frequency spectra of the currents are reported in Fig. 3.We also consider two shields having conductivities σ = 37 MS/m (aluminum) and σ = 6.3 MS/m (stainless steel), respectively, with different thicknesses d; in both cases, the electric permittivity and magnetic permeability of the shields have been assumed to be those of vacuum, i.e.,  ε = ε 0 and μ = μ 0 .The circular current loop has a radius a = 30 cm and is placed at a distance h = 30 cm from the shields.
Figs. 4 and 5 show the normalized magnitudes of the zcomponent of the magnetic field computed at z = −30 cm as a function of time, in the case of the slow transient current loop #1 for the two considered shields, for different screen thicknesses d.The results have been computed through the proposed EIT (solid black lines) and the theory proposed by Moser [7] (dashed red lines).The results confirm the great accuracy of the proposed approach as far as the shield can be considered electrically thin according to (3).Furthermore, they show that, for observation points placed on the rotational axis of the system, representing the loop source through a magnetic dipole is a valid approximation also when the source radius a is not small with respect to the source distance from the screen h (in the shown results a = h).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  is worth noting that both the spectral integrals occurring in Moser's frequency-domain formulation and the inverse Fourier-transform integrals required to obtain the relevant TD results are defined over semi-infinite domains and are in general both oscillating and slowly converging at infinity.Their numerical evaluation is, thus, both nontrivial and computationally intensive if compared with the proposed formulation, which involves only single nonoscillating integrals over finite intervals.The latter are even amenable to closed-form evaluation for simple source waveforms, as shown in Section IV; even when this is not possible, the reduction in computational time afforded by the proposed formulation is on the order of 10 2 or more.
By increasing the screen thickness d, the distortion of the field shape increases due to the effect of the dispersion and losses that the field undergoes while passing through the shield.For clarity, the normalized waveform of the exciting current i loop (t) is also reported (dashed gray lines) to visually assess the field components' distortion.
Fig. 6 shows the normalized magnitudes of the z-magnetic field in the same previous setup for the stainless-steel shield in the case of the fast transient current loop #2.For the same shield, in Figs.7 and 8, we report the peakreduction SE SE H PR and the derivative-reduction SE SE H DR under the transient current loops #1 and #2, respectively, for different thicknesses d.Both the SE obviously increase by increasing the thickness d; it is worthy pointing out that, as stated in previous publications [14], [41], [42], [43], SE H DR is usually larger than SE H PR , and the present results confirm this trend.

VI. CONCLUSION
A TD version of the EIT has been presented to solve a canonical shielding problem directly in the TD.The problem basically consists in evaluating the TD magnetic field produced by a small circular current loop with a transient waveform in the presence of a parallel planar thin conductive sheet.It has been shown that the TD EIT analytically solves the problem in a considerably simple way and the transient field components can be easily computed through the TD Green function derived in closed form.In fact, the final integrals expressing such components may be computed analytically giving a closed-form solution (an example is provided considering the z-component of the magnetic field).We have shown that for simple transient waveforms of the exciting currents, the dominant component of the on-axis magnetic field can also be expressed in a closed form and the derived results allow for evaluating the transient SE of different configurations.Comparisons with numerical brute-force techniques based on inverse Fourier transforms confirm the validity and the efficiency of the proposed formulation.
The proposed approach can also be used to study the transient excitation of a thin conductive screen by a vertical electric dipole, which would excite a TM rather than a TE field, as well as the more complex case of a shielding configuration involving two coplanar loops orthogonal to the screen.In the latter case, the problem would not be rotationally symmetric anymore and the field would have a hybrid polarization, which requires to consider both the TE and the TM reflection and transmission coefficients of the screen.Work is in progress on the analysis of these configurations as well as on their application to the transient analysis of grounding systems.APPENDIX DERIVATION OF (21) The integral in (20) can first be rewritten as With the change of variable k 0 = k + jα + q and some algebra, the integral in (63) becomes The determination of the square root in (64) is chosen so that Im{ √ • } ≤ 0. It should be noted that the square-root function introduces branch points on the real k-axis at k bp 1,2 = ±2qρ/( Ĝs r 2 + ): the relevant branch cut can be chosen as the segment joining the two branch points and passing through the origin.
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Fig. 1 .
Fig. 1.Electromagnetic configuration considered here: a small transient electric current loop in vacuum, represented as a transient VMD, over an infinite planar screen of thickness d and conductivity σ.The adopted rectangular and cylindrical reference systems are also indicated, along with the two auxiliary distances r + and r − employed in the expressions of the transient potential and fields.

Fig. 2 .
Fig. 2. Transient behavior of the exciting current loop i loop (t).

Fig. 3 .
Fig. 3. Frequency spectrum of two exciting loop currents with slow transient waveform (black line) and fast transient waveform (red line).

Fig. 4 .
Fig. 4. Normalized magnitude of the z-component of the magnetic field produced at z = −30 cm by a circular current loop with radius a = 30 cm placed at a distance h = 30 cm from a screen with σ = 37 MS/m (aluminum), for different values of the screen thickness d: case of a slow transient loop current #1.Legend: solid black lines-EIT; red dashed lines-Moser formulation; gray dashed line-waveform of the current loop i loop (t).

Fig. 7 .
Fig. 7. Peak-reduction SE H PR and derivative-reduction SE H DR as functions of the thickness d for the stainless-steel shield in the case of the current loop #1.

Fig. 8 .
Fig. 8. Peak-reduction SE HPR and derivative-reduction SE H DR as functions of the thickness d for the stainless-steel shield in the case of the current loop #2.
real k-axis, and with the change of variable k = −js, we haveG c (q, t) = c 0 4πr + e −α + qc 0 t