Near-Field Magnetic Shielding in the Frequency Range from 20 Hz to 100 MHz

Abstract

This work is part of our larger research activity that has as its ultimate goal the experimental validation of the theoretical predictions regarding the electromagnetic shielding of conductive materials, in the frequency range 20 Hz–18 GHz. The present article deals with electromagnetic shielding in the range of 20 Hz to 80 MHz, i.e., in a near, magnetically generated field. For this frequency range, there is currently no normative document or in-depth research regarding the experimental validation of theoretical predictions, so the works published to date do not have an adequate degree of confidence. The paper presents the theoretical basis, i.e., the equations that describe the phenomenology of electromagnetic shielding, equations that are unanimously accepted in the scientific community in the field. The employed methodology and laboratory equipment that allowed obtaining the experimental results are also presented. The main component is an adapted zero gauss chamber with three concentric layers (two of Mu-metal and one of steel). The experimental results are compared with the theoretical ones for analytically calculable materials: copper, aluminum, Monel, and graphite. The agreement of these data validates the experimental method, which can then be used also for materials that are not analytically calculable, like composite materials, multilayer materials, or textiles.

1. Introduction

Electromagnetic shielding is a method of protection against unwanted electromagnetic radiation. Applied correctly, this method ensures the electromagnetic compatibility (EMC) of electric and electronic devices and equipment as well as the protection of living systems against electromagnetic (EM) fields. Electromagnetic interference (EMI) shielding refers to shielding the EM radiation in the radio frequency (RF) spectrum, while magnetic shielding refers to attenuating the magnetic field, i.e., the magnetic field in the near-field region of a magnetic field source.

EM shielding consists of separating the EM field source or the device to protect by using a barrier made of a suitable material. Depending on the required level of attenuation and the EMI frequency range, materials with certain macroscopic parameters (ε, µ, σ) will be used. Materials used for EM shielding include metals, carbons, ceramics, concrete, polymers, textiles, and composites [1].

Metals are the most commonly used materials for EM shielding. Copper and aluminum have high electrical conductivity and are very effective in the RF range, while magnetic metals like steel, nickel, and nickel alloys can be used for low-frequency magnetic shielding. Plain metals can be used in the form of sheets, thin layers, or meshes [2,3,4].

Carbon materials like graphite, graphene, carbon fibers, carbon nanofibers, and carbon nanotubes are also very suitable candidates for EM shielding applications due to their very good electrical conductivity, their ability to absorb electromagnetic radiation in a wide frequency range, and other properties (ease of processing and fabrication, low density, corrosion resistance, thermal stability, high-temperature resistance, etc.) [1,5,6,7,8,9,10,11,12].

Ceramic materials have lower conductivity which makes them less popular than carbons or metals for EM shielding applications [1]. However, ceramics show good magnetic and dielectric properties as well as high operation temperatures and good mechanical properties, which makes them excellent choices for applications in harsh environments where other materials may lose their properties [13]. Examples of ceramic materials for EM shielding include ferrites, MXenes, metal–organic frameworks (MOFs), perovskite ceramics, sulfide ceramics, and associated composites [13]. Due to their high magnetic permeability, ferrites and ferrite-based composites are used as EM wave absorbers having a wide frequency bandwidth up to GHz frequencies [13,14,15,16,17,18,19,20,21,22,23]. In fact, MXenes are not ceramics but a class of two-dimensional (2D) transition metal carbides, nitrides, and carbonitrides [13]. Due to their high electrical conductivity, good mechanical stiffness, and high-temperature stability, as well as the tunability of their properties, MXenes are often used to create ceramic composites with enhanced properties for EMI shielding applications in harsh environments [13,24,25,26,27,28,29]. By incorporating magnetic ions or nanoparticles, MXenes can gain magnetic properties, which could make them suitable for shielding applications at lower frequencies [30,31,32]. MOFs are nanoporous materials made from metal ions and organic ligands. Their structural versatility, including ultra-high specific surface areas, tunable porosity, and customizable compositions, makes them suitable for creating or enhancing ceramic and other materials with absorption properties in the microwave frequency range [13,33,34,35,36]. Perovskite ceramics are a class of ceramic compounds whose unique structure, thermal stability, and capability to customize electromagnetic properties make them promising materials for EMI shielding [37,38,39,40,41]. Depending on the elements in their composition, sulfide ceramics can have excellent electrical properties. Copper sulfide is a good example of such material, which, besides its EMI shielding capabilities, also shows flexibility and transparency [42]. Also, by combining sulfide ceramics with materials with better conductivity, one can obtain hybrid or composite materials with excellent results in the microwave range [43,44].

Regular concrete is dielectric, but in combination with conductive (steel fibers, graphite) or magnetic materials (ferrites), it can achieve a high shielding effectiveness in wide frequency ranges [45,46,47,48,49].

Polymers for electromagnetic shielding include intrinsically conductive polymers (e.g., polypyrrole) and combinations of intrinsically conductive polymers or standard polymers with conductive and/or magnetic fillers. These materials are lightweight, resistant to corrosion, can be processed in various forms, and, by adjusting the type and concentration of fillers, their electromagnetic, mechanical, and thermal properties can be customized for each application [50,51,52,53].

Textile materials for EM shielding are mainly used in the microwave range, as they are made by endowing regular fabrics with conductive properties by introducing electrically conductive yarns and/or by coating fabrics with conductive materials [54,55,56].

By combining two or more materials, new, composite materials are obtained with better characteristics than those of the individual materials alone [57]. In addition to shielding capabilities, corrosion resistance, and mechanical or thermal properties can also be enhanced by combining different materials. Composites for electromagnetic shielding are made by embedding conductive and/or magnetic fillers in a polymer, ceramic, or metal matrix [57,58,59,60]. Different forms (particles, fibers, flakes, sheets, platelets, or tubes) of metals, carbons, MXenes, and certain oxides are extensively used as filler materials [58].

There are two main types of methods for testing the shielding capabilities of materials: methods that use waveguides and methods using antennas or electric/magnetic field probes. Waveguides like coaxial TEM cells and hollow waveguides are used as sample holders and simulate the far-field conditions [61,62,63,64,65,66]. Coaxial TEM cells are used for frequencies up to a few GHz, while the hollow waveguides can work up to hundreds of GHz. The methods based on antennas or electric/magnetic field probes are used to test the shielding capabilities of materials in both near-field and far-field conditions (up to 18 GHz). These methods include free-space methods [3,66,67,68,69,70,71] and methods based on the IEEE-STD-299 standard for determining the shielding effectiveness of enclosures [66,72,73,74].

Generally, the most widely used frequency range for testing materials is 20 Hz–18 GHz. For this field, there are currently no standards covering the entire range and no experimentally validated and confirmed method. The IEEE 299 standard is dedicated to testing shielded enclosures made of various materials and has certain inconsistencies and difficulties regarding materials testing, so it is currently less used. The ASTM ES7 and ASTM D-4935 standards apply only in the plane wave range up to a few GHz and only for electrically thin samples [62,63,75].

This work is part of our larger research activity that has as its ultimate goal the experimental validation of the theoretical predictions regarding the electromagnetic shielding of conductive materials, with or without magnetic properties, in the frequency range 20 Hz–18 GHz. The present article deals with electromagnetic shielding in the range of 20 Hz to 100 MHz, i.e., in a near, magnetically generated field.

As stated above, there are two important classes of methods for determining SE as a function of frequency: free-space methods and methods based on TEM cells and shielded enclosures. For the plane wave area, closed system methods (TEM cells or shielded enclosures) are used almost exclusively, while for the magnetic near-field zone, only free-space systems have been used so far. From an in-depth bibliographic study, it was found that our work is the first time that these SE = F(f) measurements in the magnetic near-field zone are made in a “closed system,” using a Zero-Gauss-Chamber-type enclosure with three layers (2 layers of mu-metal and one of soft steel). Free-space methods have the disadvantage of using large material surfaces to avoid the occurrence of magnetic field diffraction at the boundaries, in addition, the higher the frequency, the stronger the diffraction phenomenon. As proof, in [76,77], when the maximum frequency was 20 MHz, the area of the shield to be tested was 0.04 m², compared to 0.027 m² in the case of the present article, while when measurements were made up to 120 MHz, the area was about 1 m². It is obvious that from the shielding materials manufacturers’ point of view, it is desirable that the areas of the test samples be as small as possible. A sample size as small as possible was pursued, based on the above, but on the other hand, the minimum size is conditioned by the size of the transmitting/receiving antennas (100–120 mm). Another advantage of the Zero-Gauss Chamber is that it allows us to define a dynamic range for the entire measurement system. Until now, the dynamic range calculus for the measurement systems employed for determining the electromagnetic field attenuation of materials in this frequency range has not been addressed in specialized articles, although it is extremely important. This is because in free space a dynamic range cannot be defined for the measurement system (because of the diffraction phenomenon).

The paper aims to experimentally validate the unanimously accepted theory for the electromagnetic shielding of conductive materials, with or without magnetic properties, in the above-mentioned frequency range, through a method that can be subsequently used for standardized measurements.

2. Materials and Methods

The materials used in this work for shielding effectiveness tests, in the 20 Hz–80 MHz frequency range, were chosen so that they are defined in terms of macroscopic parameters (σ, µ_r ≈ 1), that is, to be analytically calculable in accordance with the existing theory, and, at the same time, to represent the materials most used for electromagnetic shielding. The exception is electrical steel, which is not precisely defined in terms of electrical conductivity (σ) and even less so for magnetic permeability (µ_r). Even if µr is defined for certain frequencies (mainly 50 Hz), its frequency dependence in the specified range is not known. These materials are listed in

Table 1. The list of materials used for shielding effectiveness tests.

Material	Manufacturer	Thickness [µm]	Electrical Conductivity [S/m]	Relative Magnetic Permeability (µr)
Copper 1	Unknown	35	3.92 × 107	≈1
Copper 2	Unknown	200	3.46 × 107	≈1
Aluminum	Unknown	10	2.51 × 107	≈1
Monel alloy 400	Goodfelow, Huntingdon, UK	25	1.07 × 106	≈1
Toray carbon paper	Toray Industries, Neu-Isenburg, Germany	500	9.92 × 103	≈1
Graphite foil 1	ProGraphite GmbH, Untergriesbach, Germany	200	4.25 × 104	≈1
Graphite foil 2		500	6.49 × 104	≈1
Graphite foil 3		1000	8.81 × 104	≈1
Electrical steel	Cogent Power Ltd., Newport, UK	330	≈2 × 106 [78,79]	unknown

.

The electromagnetic shielding effectiveness of a material, sometimes called insertion attenuation and denoted SE (shielding effectiveness), or SE_dB if given in decibels, can be written as the following functional dependence:

SE = F(f, r, t, σ, µ, ε),

(1)

where f is the frequency of the electromagnetic radiation, r is the distance between the source of electromagnetic radiation (the transmitting antenna) and the shield, and t is the thickness of the material sample, while σ, µ, and ε are the macroscopic material parameters (electrical conductivity, magnetic permeability, and dielectric permittivity, respectively).

By definition, the experimentally determined value of SE_dB is

$${SE}_{dB}=20\log {\displaystyle \frac{E_0}{E_1}}=20\log {\displaystyle \frac{H_0}{H_1}},$$

(2)

where E₀ and H₀ represent the field values (electric/magnetic) measured by the receiving antenna without the material sample, while E₁ and H₁ represent the field values (electric/magnetic) measured by the receiving antenna with the material sample placed between the antennas, keeping all other parameters identical.

The mathematical model used for the analytical calculation of the electromagnetic shielding of a material was initiated by Schelkunoff [80,81], and it is based on the isomorphism of the second-order equation of electromagnetic waves with the propagation equations in a transmission line. Based on this analogy, which includes the phenomena of reflection, multiple reflections, and absorption, Equation (3) was developed. This algorithm is currently unanimously accepted in the scientific community, and it is based on the following arguments:

-

the equation has been experimentally verified on certain frequency domains;

-

it is verified within the phenomenology of the Fabry–Perot interferometer in the framework of electromagnetic optics.

$$SE=e^{\gamma t}·{\displaystyle \frac{\left|Z_0+{\underset{\_}{Z}}_m\right|·\left|Z_0+{\underset{\_}{Z}}_m\right|}{4Z_0\left|{\underset{\_}{Z}}_m\right|}}·\left|1-{\left({\displaystyle \frac{{\underset{\_}{Z}}_m-Z_0}{{\underset{\_}{Z}}_m+Z_0}}\right)}^2{e}^{-2\gamma t}\right|.$$

(3)

Equation (3) may also be written as [82]

$$SE=e^{\gamma t}·{\displaystyle \frac{\left|1+K^2\right|}{4\left|K\right|}}·\left|1-{\left({\displaystyle \frac{K-1}{K+1}}\right)}^2{e}^{-2\gamma t}\right|.$$

(4)

Equations (3) and (4) describe the SE_dB for the entire frequency range, including both the near-field region (magnetic- or electric-generated field), also called the Fresnel zone, and the far-field region, called the Fraunhofer zone.

In Equations (3) and (4) t is the material thickness, Z₀ is the impedance of free space, Z_m is the material impedance, γ is the propagation constant, and K expresses the impedance mismatch at the interfaces of the material shield and is given by the expression [82]:

$$K={\displaystyle \frac{Z_0}{Z_m}}={\displaystyle \frac{k\sqrt{{\mu }_0/{\epsilon }_0}}{\left(1+j\right)\sqrt{\pi f\mu /\sigma }}},$$

(5)

where k, for low impedance magnetic fields (the case of this article), is [82]

$$k={\displaystyle \frac{2\pi r}{\lambda }}=2\pi rf\sqrt{{\mu }_0{\epsilon }_0},$$

(6)

with λ being the radiation wavelength and r the distance from the magnetic field source to the material shield.

Thus, for low impedance magnetic fields, Equation (5) is [82]

$$K_H=r\sqrt{2\pi f\sigma {\mu }_0/{\mu }_r}.$$

(7)

The shielding effectiveness (SE) is usually given in decibels by using the relation

SE_dB = 20 log SE = A_dB + R_dB + B_dB

(8)

Thus, the three terms in Equation (8) are:

• Absorption loss term:

$$A_{dB}=20{\mathit{log}}_{10}{e}^{\gamma d},$$

(9)

with $\gamma =\alpha +j\beta =\sqrt{j\omega \mu \left(\sigma +j\omega \epsilon \right)}$.

Because for metals and other high conductivity materials, $\sigma \gg \sigma \epsilon$, it follows that $\gamma =\left(1+j\right)\sqrt{\pi f\mu \sigma }\mathrm{o}\mathrm{r}\alpha =\beta =\sqrt{\pi f\mu \sigma }$, and

$$A_{dB}=8.686\alpha d=8.686d\sqrt{\pi f\mu \sigma }.$$

(10)

• Reflection loss term (for magnetic field):

$$R_{H,dB}={\displaystyle \frac{{\left(1+K_H\right)}^2}{4K_H}},$$

(11)

• Multiple reflections term (for magnetic field):

$$B_{H,dB}=\sqrt{1-2{\left({\displaystyle \frac{K_H-1}{K_H+1}}\right)}^2\cos 2\beta d{e}^{-2\alpha d}+{\left({\displaystyle \frac{K_H-1}{K_H+1}}\right)}^4{e}^{-4\alpha d}}.$$

(12)

Figure 1 shows Equation (4) plotted as a function of frequency and the material thickness for copper at a distance of 10 cm between the transmitting antenna and shield. The parametric curves in Figure 1 are given in the 10 Hz–30 GHz frequency range, while this article deals with the low-frequency near-field magnetic shielding in the 20 Hz–80 MHz frequency range. In Figure 1, the near-field region (magnetic- or electric-generated field) is highlighted.

The first step in establishing the methodology is to confirm the identity between the curves presented above, curves unanimously accepted in the scientific community in the field, in relation to those resulting from analytical calculations for different distances r. The analytical calculations were performed with the Matlab software version R2014b program using Equations (8) and (10)–(12).

The second step consists of establishing the general conception/structure for the employed measurement setup (Figure 2).

Taking into consideration that the reference articles for this work [76,77] use measurement systems that involve materials/shields with relatively large surfaces, which, from the materials makers/providers’ point of view, is an impediment—a shielded measurement enclosure, specifically an adapted Zero-Gauss Chamber, was employed in this work. This conception allows the characterization of conductive materials, with or without magnetic properties, having a disk shape with a diameter of 185 mm. On the other hand, taking into account the extended measurement range (20 Hz–80/100 MHz), two measurement systems were used in this work: the low-frequency measurement system (Figure 3a) and the high-frequency system (Figure 3b). This entails using two generators and two sets of antennas (transmitting and receiving antennas). In order to ensure the identity of the transmitted field in both situations, with and without material between the transmitting and receiving antennas (H₀ and H₁, respectively), the current in the transmitting antenna was monitored with the help of a current probe and an oscilloscope.

The methodology for determining the shielding effectiveness of materials consists of measuring the magnetic near-field without a shield (H₀) and with the shield placed between the transmitting and receiving antennas (H₁) and calculating SE_dB according to Equation (2).

The shielded enclosure (Figure 4) employed for determining SE_dB of materials is cylindrical and has an outer diameter of 300 mm and a length of 500 mm. It is made of three concentric, spaced apart, shields, two made of mu-metal with a thickness of 2 mm and one of soft steel with a thickness of 15 mm. The front closure of the shielded enclosure is made with mu-metal circular rings and a circular ring with a Morse-taper fitting made of soft steel. The front side of the shielded enclosure is provided with a holder that allows fitting the disk-shaped material samples with a diameter of 185 mm and a maximum thickness of 5 mm. On the back side there is a waveguide-type element for passing the cable from the receiving antenna.

The equipment used for measurements in this work is given in

Table 2. Equipment used for measurements in this work.

Instrument	Model	Specifications	Manufacturer
Oscilloscope	MDO 3102	1 GHz/dual channel	Tektonix, Beaverton, OR, USA
Current probe	TCP0030A	5/30A, 120 MHz	Tektonix, Beaverton, OR, USA
Current probe	TCP0150	25/150A, 20 MHz	Tektonix, Beaverton, OR, USA
Low frequency signal generator	E 0501	1 Hz–1 MHz	IEMI, Bucharest, Romania
High speed bipolar amplifier	HAS 4014	DC—1 MHz/200 VA	NF Corporation, Yokohama, Japan
Analog signal generator	E 8257D	250 kHz–40 GHz	Agilent Technologies, Santa Clara, CA, USA
Power amplifier	SMX 50	9 kHz–1 GHz/50 W	Instruments for Industry, Ronkonkoma, NY, USA
Spectrum analyzer	FSP	9 kHz–13.6 GHz	Rohde&Schwarz, Munich, Germany
Laptop	Lenovo	Intel CORE I7	Lenovo
Precision impedance analyzer	4294A	40 Hz–110 MHz	Agilent Technologies, Santa Clara, CA, USA

.

The magnetic transmitting and receiving antennas utilized in measurements are presented in Figure 5.

Except T1, the utilized antennas are perfectly defined by the specifications declared by the manufacturers. The T1 antenna is a coil with an outer diameter of 80 mm and a length of 30 mm, consisting of 13 turns of enameled copper wire with a diameter of 1.9 mm. The low frequency magnetic field emitted by this coil is calculated with the relation [83]:

$$B={\displaystyle \frac{{\mu }_{0}IN}{2L}}\left[{\displaystyle \frac{x_2}{\sqrt{x_2^2+R^2}}}-{\displaystyle \frac{x_1}{\sqrt{x_1^2+R^2}}}\right].$$

(13)

The magnetic field generated by T1 at 10 cm is approx. 33 µT at a current of 4.5 A.

In order to determine the frequency range where it can be used as a transmitting antenna, T1 was tested with the help of an impedance analyzer at frequencies up to 110 MHz (Figure 6).

The T1 antenna has an inductivity of approximately 16.2 µH, and it shows a resonance at 17 MHz, but in this work it was used at frequencies well below its resonant frequency (maximum 300 kHz).

There are 2 important calibrations for the measurement system: determining the dynamic range of the measurement system so that the maximum attenuation of the material under test is below this limit, and the alignment/coaxiality of the antennas (transmission and reception). The latter is simply solved by successive, iterative alignment on the 2 axes, aiming to obtain a maximum of the received field. It is important to mention that perfect alignment of the antennas is not indispensable provided that their relative position is maintained during the measurement of H₀ and H₁.

The main error sources are:

-

antenna misalignment, including their plan-parallelism, which is checked every time according to the above-mentioned procedure;

-

ambient magnetic noise (only 50 Hz and its harmonics are detectable), which is eliminated by avoiding 50 Hz, 100 Hz, 150 Hz, 200 Hz, and 300 Hz frequencies, as can be seen in the article;

-

current fluctuations in the transmitting antenna, which unfortunately can only be reduced by using high-performance amplifiers or complex electronic systems.

3. Results

3.1. The Dynamic Range of the Shielded Enclosure

First, the dynamic range of the whole measurement system (signal generator, transmitting antenna, shielded enclosure, receiving antenna, and the appropriate measurement instruments) was determined in terms of shielding effectiveness (SE_dB). Since the dynamic range of the generators and measurement instruments can be calculated from their datasheets, it follows that the dynamic range of the entire system is determined by the shielding capabilities of the enclosure, which are determined experimentally. This extent was also determined by using Equation (2), where H₀ is the magnetic field received inside the shielded enclosure with the front side opened and H₁ is the magnetic field received inside the closed shielded enclosure. The shielded enclosure’s front side was closed by using caps made of the same materials that the shielded enclosure is made of (mu-metal and soft steel). The results for two consecutive measurements are given in

Table 3. Measurement data obtained on testing the dynamic range of the shielded enclosure.

Frequency	Signal level						SE
	Enclosure Open			Enclosure Closed			Meas. 1	Meas. 2	Avg.
	Meas. 1	Meas. 2	Avg. 1	Meas. 1	Meas. 2	Avg.
	[µT]	[µT]	[µT]	[µT]	[µT]	[µT]	[dB]	[dB]	[dB]
20 Hz	10.85	10.8	10.82	0.0018	0.0017	0.00175	75.60	76.06	75.83
40 Hz	11.02	11.05	11.03	0.0007	0.0006	0.00065	83.94	85.30	84.60
70 Hz	10.32	10.37	10.34	0.0005	0.0006	0.00055	86.29	84.75	85.49
120 Hz	10.56	10.62	10.59	0.0005	>0.0005	>0.0005	86.49	86.54	86.52
180 Hz	10.25	10.15	10.20	0.0005	>0.0005	>0.0005	86.24	86.15	86.19
220 Hz	10.53	10.48	10.50	0.0005	>0.0005	>0.0005	86.47	86.43	86.45
280 Hz	10.27	10.27	10.27	0.0005	>0.0005	>0.0005	86.25	86.25	86.25
500 Hz	10.15	10.34	10.24	0.0005	>0.0005	>0.0005	86.15	86.31	86.23
1 kHz	9.43	9.58	9.50	0.0005	>0.0005	>0.0005	85.51	85.65	85.58
2 kHz	8.20	8.36	8.28	0.0005	>0.0005	>0.0005	84.30	84.46	84.38
5 kHz	6.44	6.67	6.55	0.0005	>0.0005	>0.0005	82.20	82.50	82.35
10 kHz	5.58	5.64	5.61	0.0005	>0.0005	>0.0005	80.95	81.05	81.00
20 kHz	4.89	4.95	4.92	0.0005	>0.0005	>0.0005	79.81	79.91	79.86
50 kHz	4.62	4.73	4.67	0.0005	>0.0005	>0.0005	79.31	79.52	79.42
	[dBm]	[dBm]	[dBm]	[dBm]	[dBm]	[dBm]	[dB]	[dB]	[dB]
100 kHz	−39.90	−39.80	−39.85	−118.80	−120.00	−119.40	78.90	80.20	79.55
200 kHz	−34.50	−34.60	−34.55	−116.50	−121.00	−118.75	82.00	86.40	84.20
500 kHz	−27.60	−27.60	−27.60	−115.40	−118.70	−117.05	87.80	91.10	89.45
1 MHz	−22.50	−22.60	−22.55	−111.30	−115.30	−113.30	88.80	92.70	90.75
2 MHz	−16.00	−15.80	−15.90	−113.60	−114.00	−113.80	97.60	98.20	97.90
5 MHz	−12.70	−12.60	−12.65	−116.50	−118.00	−117.25	103.80	105.40	104.60
8 MHz	−18.00	−17.80	−17.90	−122.30	−125.00	−123.65	104.30	107.20	105.75
10 MHz	−15.60	−15.40	−15.50	−122.90	−127.00	−124.95	107.30	111.60	109.45
20 MHz	−20.10	−20.20	−20.15	−129.90	−134.80	−132.35	109.80	114.60	112.20
30 MHz	−23.20	−22.90	−23.05	−133.70	−138.20	−135.95	110.50	115.30	112.90
50 MHz	−29.70	−28.20	−28.95	−130.00	−130.50	−130.25	100.30	102.30	101.30
60 MHz	−28.50	−28.90	−28.70	−129.50	−127.00	−128.25	101.00	98.10	99.55
70 MHz	−29.90	−30.00	−29.95	−129.40	−131.50	−130.45	99.50	101.50	100.50
80 MHz	−30.70	−30.70	−30.70	−131.20	−136.00	−133.60	100.50	105.30	102.90

¹ Avg. = The average of the two measurements (Meas. 1 and Meas. 2).

, and the dynamic range diagram is given in Figure 7.

This dynamic range shows the maximum SE_dB values that can be determined for the materials tested with the measurement setup shown in Figure 2.

Taking into consideration that the conductive materials with or without magnetic properties used in near-field magnetic shielding applications, with technically accepted thicknesses, have SE_dB values starting from a maximum of 60 dB in low frequency, reaching up to 100 dB at high frequency (80 MHz), it is considered that the obtained dynamic range diagram is satisfactory for the purpose of this work.

3.2. Material Testing in the 1 kHz–100 MHz Frequency Range

3.2.1. Copper

In order to validate the method proposed in this article, magnetic shielding effectiveness (SE_dB) was determined as a function of frequency for analytically calculable materials (namely isotropic, homogeneous materials with µ_r ≈ 1 and known conductivity): copper, aluminum, Monel, Toray carbon paper, and graphite.

Table 4. Magnetic shielding effectiveness values experimentally obtained for copper 35 µm.

Frequency	1 kHz	2 kHz	5 kHz	10 kHz	20 kHz	50 kHz	100 kHz	200 kHz	500 kHz	1 MHz
SE_H [dB] (measured)	1.79	2.98	5.02	8.43	12.84	20.30	28.10	30.90	38.70	44.40
SE_H [dB] (calculated)	1.32	2.56	5.69	9.42	14.13	21.26	27.00	32.88	40.75	46.75
Relative Error [%]	26.46	14.23	−13.34	−11.80	−10.08	−4.73	3.92	−6.40	−5.31	−5.30
Frequency	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz
SE_H [dB] (measured)	50.30	58.30	62.70	65.00	70.70	79.10	84.60	88.30	92.6	95.7
SE_H [dB] (calculated)	52.78	60.87	65.21	67.37	74.96	80.39	88.62	91.97	94.99	97.76
Error [%]	−4.93	−4.41	−4.00	−3.64	−6.02	−1.63	−4.75	−4.16	−2.58	−2.15

shows the experimentally obtained SE_dB values for 35 µm thick copper in the frequency range from 1 kHz to 80 MHz, and Figure 8 the corresponding diagram.

Except for the measurements at 20 MHz and 50 MHz, the absolute error was less than 2.5 dB. The relative error values can be seen in the bottom graph in Figure 8; between 50 kHz and 80 MHz the relative error is less than 7%, the average relative error over the same interval being 4.26%. Therefore, it can be stated that the near-field magnetic shielding theory has been experimentally validated for 35 µm thick copper with an error less than 5%.

3.2.2. Conductivity Dependence of SE_dB in the Range 100 kHz–80 MHz

To analyze the conductivity dependence of SE_dB from an experimental point of view, comparative measurements were made for copper 35 µm (presented above), aluminum 10 µm, Monel 25 µm, and Toray carbon paper 500 µm. The obtained experimental results are presented in

Table 5. SE_dB values experimentally obtained for aluminum 10 µm.

Frequency	100 kHz	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz
SE_H [dB] (measured)	13.53	19.80	29.10	32.00	37.20	44.00	47.70	49.40	54.50	56.80	64.10	63.50	66.20	67.10
SE_H [dB] (calculated)	13.49	18.75	26.25	32.12	38.06	45.97	50.05	51.98	58.00	61.53	65.99	67.60	68.96	70.15
Error [%]	0.32	5.28	9.79	−0.37	−2.31	−4.49	−4.92	−5.22	−6.42	−8.32	−2.95	−6.45	−4.17	−4.54

,

Table 6. SE_dB values experimentally obtained for Monel 25 µm.

Frequency	100 kHz	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz
SE_H [dB] (measured)	1.66	3.60	8.80	14.80	19.60	27.28	31.33	33.17	37.26	40.17	46.60	46.80	49.70	50.50
SE_H [dB] (calculated)	2.50	4.65	9.28	13.97	19.28	26.81	30.78	32.68	38.63	42.13	46.55	48.13	49.47	50.63
Error [%]	−50.69	−29.11	−5.50	5.63	1.63	1.73	1.75	1.47	−3.68	−4.88	0.11	−2.84	0.47	−0.25

and

Table 7. SE_dB values experimentally obtained for Toray carbon paper 500 µm.

Frequency	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz
SE_H [dB] (measured)	0.67	1.75	3.90	6.47	13.08	16.72	18.65	23.00	26.07	32.80	33.00	36.02	37.23
SE_H [dB] (calculated)	0.99	2.36	4.39	7.58	13.44	16.96	18.71	24.35	27.75	32.11	33.69	35.03	36.21
Error [%]	−48.32	−34.95	−12.50	−17.11	−2.79	−1.44	−0.30	−5.85	−6.43	2.10	−2.09	2.74	2.75

and the corresponding diagrams in Figure 9.

The average relative errors were 4.68% for aluminum, very close to that of copper 35 µm (4.26%), 7.84% for Monel, and 10.7% for Toray carbon paper. The increased error values at low frequencies for Monel (below 500 kHz) and Toray carbon paper (below 2 MHz) are due to the fact that materials with low conductivity have a much lower shielding effectiveness, so that the absolute measurement errors of the system for Monel and Toray carbon paper (less than 2 dB) are reflected as high values of relative error at lower frequencies.

3.2.3. The Dependence of SE_dB on the Shield Thickness in the Range 100 kHz–100 MHz

In order to assess the effect of the shield thickness on the shielding effectiveness, high purity graphite foil samples were chosen with thicknesses of 200, 500, and 1000 µm. The experimental data are presented in

Table 8. SE_dB values experimentally obtained for 200 µm thick graphite foil.

Frequency	100 kHz	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz	100 MHz
SE_H [dB] (measured)	0.65	1.35	3.20	6.25	10.99	18.16	22.35	23.93	28.48	31.10	37.55	37.50	40.45	41.34	42.44
SE_H [dB] (calculated)	0.84	1.63	3.82	6.74	10.81	17.47	21.21	23.03	28.83	32.28	36.67	38.25	39.59	40.75	42.71
Error [%]	−28.54	−21.07	−19.30	−7.81	1.62	3.83	5.10	3.74	−1.23	−3.80	2.34	−2.00	2.13	1.42	−0.64

,

Table 9. SE_dB values experimentally obtained for 500 µm thick graphite foil.

Frequency	100 kHz	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz	100 MHz
SE_H [dB] (measured)	2.15	4.75	10.00	15.30	20.84	28.61	32.84	34.78	39.52	43.80	51.77	52.63	56.76	58.65	62.10
SE_H [dB] (calculated)	3.02	5.49	10.54	15.43	20.87	28.49	32.51	34.45	40.64	44.51	49.94	52.11	54.08	55.88	59.13
Error [%]	−40.66	−15.47	−5.37	−0.86	−0.14	0.42	1.00	0.95	−2.84	−1.61	3.54	0.99	4.73	4.72	4.78

and

Table 10. SE_dB values experimentally obtained for 1000 µm thick graphite foil.

Frequency	100 kHz	200 kHz	500 kHz	1 MHz	2 MHz	5 MHz	8 MHz	10 MHz	20 MHz	30 MHz	50 MHz	60 MHz	70 MHz	80 MHz	100 MHz
SE_H [dB] (measured)	4.80	9.23	16.34	22.04	28.32	36.67	41.99	44.66	53.88	60.67	74.45	77.70	83.80	87.60	94.10
SE_H [dB] (calculated)	6.99	11.13	17.83	23.44	29.31	37.61	42.42	44.96	54.44	61.34	71.72	75.96	79.81	83.34	89.71
Error [%]	−45.72	−20.56	−9.15	−6.35	−3.51	−2.56	−1.02	−0.67	−1.04	−1.11	3.67	2.24	4.77	4.86	4.66

and Figure 10.

The average relative errors were 6.97% for graphite foil 200 µm, 5.87% for graphite foil 500 µm, and 7.65% for graphite foil 1000 µm.

3.3. Measurements in the Range 20 Hz–300 kHz

In the first stage, the purpose was to make a comparison between the shielding effectiveness values of copper 35 µm (presented above) and copper 200 µm (

Table 11. SE_dB values experimentally obtained for 200 µm thick copper.

Frequency	280 Hz	500 Hz	1 kHz	2 kHz	5 kHz	10 kHz	20 kHz	50 kHz	100 kHz	200 kHz	300 kHz
SE_H [dB] (measured)	1.50	2.70	4.95	8.81	15.26	21.47	26.84	34.35	40.42	46.80	50.10
SE_H [dB] (calculated)	1.85	3.18	5.74	9.49	15.88	21.35	27.09	34.89	40.9	47.06	50.84
Error [%]	−23.39	−17.64	−16	−7.74	−4.05	0.58	−0.93	−1.58	−1.19	−0.56	−1.48

) and the corresponding diagrams shown in Figure 11. The average relative error for copper 200 µm over the 280 Hz–300 kHz range was 6.83%.

Considering the results obtained for analytically calculable materials (with well-determined σ and µ_r ≈ 1), it can be considered that the experimental method presented here validates the theory in the 20 Hz–100 MHz frequency range. Consequently, this experimental method can be used with a sufficiently high degree of confidence for determining SE_dB as a function of frequency for analytically calculable materials.

When it comes to materials that cannot be accurately analytically calculated (like multilayers, textiles, and composites), certain variables occur (multiple interfaces, frequency-dependent magnetic permeability, etc.) that are not accounted for in the theoretical model. In this respect, electrical steel with a thickness of 330 µm and σ ≈ 2 · 10⁶ S/m was chosen as an example. The experimental data for electrical steel are given in

Table 12. SE_dB values experimentally obtained for 330 µm thick electrical steel.

Frequency	20 Hz	40 Hz	70 Hz	120 Hz	180 Hz	220 Hz	280 Hz	500 Hz	1 kHz
SE_H [dB] (measured)	20.45	20.92	20.98	21.1	21.17	21.34	21.5	21.71	22.79
Frequency	2 kHz	5 kHz	10 kHz	20 kHz	50 kHz	100 kHz	200 kHz	300 kHz
SE_H [dB] (measured)	24.35	28.28	33.93	39.68	47.99	52.47	54.79	54.94

, and the corresponding diagram is shown in Figure 11 by comparison with the thin (35 µm) and thick copper (200 µm) samples presented above.

It is observed in Figure 11 that, at very low frequencies, electrical steel shows a higher SE_dB than thick copper due to the influence of magnetic permeability, as shown by Equations (1) and (10)–(12).

Next, the effect of overlapping two shielding material samples—electrical steel and thick copper—was experimented on, and the results are presented in Figure 12.

This experiment demonstrates that the method described here can also be used to determine the magnetic shielding effectiveness of multilayer structures (e.g., combinations between ferromagnetic materials and materials with high conductivity).

4. Discussion

Analyzing the data and graphs presented in this article, it can be stated that the described experimental setup has succeeded in validating the accepted theory of electromagnetic shielding for conductive materials, with or without magnetic properties, in the frequency range from 20 Hz to 100 MHz. The average relative errors fall mainly within a range of 4.5–7.5%, which, for magnetic field measurements, constitutes a performance in relation to the studied literature. The large errors in the low-frequency area for conductive materials are simply explained by the fact that, when the measured field has small amplitudes, the absolute error is reflected with large values in the relative error.

The frequency dependence of SE_dB as well as the dependence of this quantity on the shielding material’s conductivity and thickness have been validated in accordance with the theory shown above. The constructed cell has a good dynamic range, which in the specified frequency range starts from values of approx. 70 dB and exceeds 100 dB at high frequencies.

The fact that analytically incalculable materials could also be tested represents an accomplishment and enables these materials to be characterized with a high degree of confidence from now on.

The main improvements that can be made to the presented measurement system include increasing the power of the high-frequency generator from the current 50 W value to 100 W or even 200 W, which will implicitly lead to an increase in the measurement dynamic range for SE_dB and a better accuracy of the readings.

Also, diminishing errors will be possible through a more efficient system for positioning the transmitting antenna relative to the receiving one. This is because the measured amplitude of the magnetic field depends significantly on the parallelism, the coaxiality of the two antennas, and the distance from the transmitting antenna to the shield surface.

In order to demonstrate the applicability of the presented method in the case of analytically incalculable materials, an electrical steel sample was chosen as an example (Figure 12). Once we have demonstrated the agreement between the theoretical predictions and experimental results for conductive materials, it follows that the method can be applied to any type of shield because Equation (2) experimentally defines the shielding effectiveness of any type of material. Thus, each laboratory will be able to perform hundreds of tests on all kinds of materials. However, this research is beyond the scope and purpose of this article.

So, in Figure 12, the SE = F(f) diagrams for Cu, electrical steel, and the two overlapped are presented comparatively. It was found that the placement order of the overlapped shields is not important. This experiment was chosen, only as an example, for the following reasons:

-

Electrical steel, like any material with magnetic properties, cannot be analytically calculated because the dependence of magnetic permeability on frequency is not known;

-

These materials are of great importance in shielding low-frequency magnetic fields, unlike conductive ones with no magnetic properties, as can be seen in the figure (approx. 20 dB);

-

The combination between Cu/Al and magnetic materials is often used in shielding practice, this structure showing superior characteristics compared to the components taken separately, as can be seen from the presented diagrams.

The presented methodology allows comparative studies for different combinations of conductive materials and magnetic materials with the certainty that these tests are valid relative to theory and not as hitherto.

5. Conclusions

This article reports the results obtained during one stage of our ample research activity concerning the experimental validation of the accepted theory regarding the electromagnetic shielding of conductive materials, with or without magnetic properties, in the frequency range 20 Hz–18 GHz.

Until now, this verification has been carried out on narrow frequency ranges and with limitations in terms of measurement dynamic range and material thickness. So far, for electrically thick shields (t >> δ, where δ is the skin depth), the theory has not been confirmed at all. Achieving this objective would have particularly important practical and theoretical implications. As an example, the presented theory (Equations (3) and (4)) shows that a perfectly closed shielded enclosure made of a conductive material, e.g., thin copper (t ≈ 100 µm), shows a radiation attenuation of hundreds of dB at very high frequencies, of the order of tens of GHz, i.e., the electromagnetic field practically does not penetrate the enclosure. This has never been demonstrated experimentally.

This article deals with the 20 Hz–100 MHz frequency range. The most relevant articles in the field directly related to this article are [76,77]. In [76] the authors follow the same algorithm of comparing experimental values with those resulting from applying the theory. Experiments are performed on 770 µm thick copper foils and 20 µm thick aluminum foils with relatively large surfaces (0.04 m²) in the 20 kHz–20 MHz frequency range. The diagrams obtained by them show good agreement with the theoretical predictions, but no analysis of the errors is given. In [77], authors work in an extended frequency range (10 Hz–120 MHz), divided into two subranges (10 Hz–150 kHz and 100 kHz–120 MHz), for materials with measurable conductivity and with large surfaces (approx. 1 m²). No analysis of the errors is given, but large errors are observed in the diagrams at frequencies higher than 80 MHz. Furthermore, the measurements in the two subranges do not match. Moreover, in the second frequency subrange, the measurement dynamic range for the values consistent with the theory is less than 30 dB. For higher values of SE_dB, above 30 dB, the results cannot be taken into account.

Hence, the progress made compared to previous research can be summarized as follows:

• the use of much smaller material sizes for measurements than those used until now (~0.027 m²);
• expanded measurement range (between 20 Hz and 100 MHz);
• using a shielded enclosure with a declared dynamic range of 70–105 dB in the specified measurement range.

However, the most important achievement is the fact that by managing to demonstrate the quasi-identity between the values predicted by the theory and those determined experimentally, as a function of both the frequency and the macroscopic parameter σ, as well as a function of the shield thickness, the presented method can be considered validated. Consequently, this method can also be used to determine the electromagnetic shielding effectiveness of materials that are not analytically calculable (composite materials, multilayer materials, or textiles), which are characterized by undefined macroscopic parameters.

From this point of view, this work paves the way for research and experiments that, even if not yet subject to a standard, benefit from an experimentally validated method, which has not happened until now.