Evaluation of Absorbing Material Shielding Effectiveness Using a Baum–Liu–Tesche Based Method

With the rapid development of electronic technology, the degree of integration and miniaturization in electronic devices are constantly improving. As a result, the problem of the space electromagnetic environment is getting more attention [1]-[2]. For this issue, the shielding cavity is often used to carry out electromagnetic shielding. Typically, different shaped holes are used for heat dissipation to reduce electromagnetic interference within the shielding cavity. However, the existence of holes creates a coupling channel that disturbs electromagnetic wave propagation, resulting in a sharp weakening of the shielding ability of the cavity [3].

At present, some analytical methods, such as the Bethe small-hole coupling [4], the Robinson equivalent circuit [5], and the Baum–Liu–Tesche (BLT) equation [6], offer the advantage of high calculation efficiency with fewer resources demanded in the shielding effectiveness (SE) calculation of the coated porous cavity [7]. However, the field distribution in the cavity must be calculated from the electromagnetic field, making the computational process more complicated. Based on the transmission line theory, an equivalent circuit model was established to determine electromagnetic SE of metal cavities covered with medium conductivity materials [8]. The position and resonance effect on SE were calculated and analyzed in this work. Alternatively, an analytical model for SE evaluation in irregular cavities under plane wave irradiation was developed using coupled electromagnetic topology theory and the BLT equation [9]. The factors affecting the SE of cavities were explored. Nevertheless, the study on the analysis of inner coating materials in the shielding cavity is still limited in this field.

The mainstream absorbing materials coated on the inner wall of the shielding cavity mainly include graphene, silicon carbide (SiC), polytetrafluoroethylene (PTFE), and Ni/rGO composites. Graphene offers advantages such as low density, corrosion resistance, good thermal stability, high conductivity, and a simple process, making it a new type of effective electromagnetic shielding or microwave absorption material [10]. SiC has strong radiation resistance, high working intensity, high thermal conductivity, good oxidation resistance, and good chemical corrosion resistance [11]. PTFE is a high-performance coating with good corrosion resistance, temperature resistance, wear resistance, and aging resistance [12]. On the other hand, Ni/rGO composite is an electromagnetic wave absorption material that combines Ni nanoparticles with graphene [13]. As mentioned above, graphene, SiC, PTFE, and Ni/rGO are considered potential shielding materials to be studied in this research.

Due to the impedance mismatch between the absorbing material and the surrounding free space, the coupling of the electromagnetic energy can be absorbed and dissipated through conversion of heat into power when the electromagnetic wave enters the absorbing material [13]. To achieve effective absorption, two conditions must be satisfied:

• i.

  When an incident electromagnetic wave enters the absorbing material from a free space, most of the wave must enter the interior of the material rather than reflect back.

• ii.

  Electromagnetic waves entering the material must be effectively converted into thermal energy for power dissipation.

By employing conductive or magnetic materials to construct a shielding enclosure, electromagnetic isolation can be achieved in a specific area. It means that the electromagnetic fields inside the area can be contained without leakage, while external radiating electromagnetic fields are blocked from penetration. Maxwell's equations provide the fundamental theoretical basis for addressing electromagnetic compatibility issues, formulating the intrinsic relationships between electric fields, magnetic fields, current density, and charge density. Indeed, electromagnetic waves are dynamic manifestations of electromagnetic fields, representing periodically varying electromagnetic fields propagating through space. The oscillation directions of the electric and magnetic fields are mutually perpendicular and orthogonal to the direction of wave propagation. The expressions for electric field strength E_x and magnetic field strength H_y are expressed as follows [14]: (1) Ex(t,z)=Exmcos (ωt−kz)ex {{\boldsymbol{E}}_x}(t,z) = {E_{xm}}\cos \;(\omega t - kz){{\boldsymbol{e}}_x} (2) Hy(t,z)=Hymcos (ωt−kz)ey=kExmμω(ωt−kz)ey {{\boldsymbol{H}}_y}(t,z) = {H_{ym}}\cos \;(\omega t - kz){{\boldsymbol{e}}_y} = {{k{E_{xm}}} \over {\mu \omega}}(\omega t - kz){{\boldsymbol{e}}_y} where the electromagnetic wave propagates along the negative z-axis direction. E_xm is the amplitude of the electric field in the x-axis direction, H_ym is the amplitude of the magnetic field in the y-axis direction, t represents time, ω is the angular frequency of the electromagnetic wave, and k=ωμε k = \omega \sqrt {\mu \varepsilon} is the wavenumber, which denotes the phase change per unit distance of electromagnetic waves. In free space, the wavenumber k is given by k1=ωμ0ε0 {k_1} = \omega \sqrt {{\mu _0}{\varepsilon _0}} , μ₀ is the relative permeability in free space, ɛ₀ is the relative permittivity of free space, and k₁ is a real number. When electromagnetic waves propagate into the cavity and are absorbed by the absorbing material, the wavenumber k is given by kr=ωμrεr {k_r} = \omega \sqrt {{\mu _r}{\varepsilon _r}} , where ɛ_r is the relative permittivity of the absorbing material, and μ_r is the relative permeability of the absorbing material [15].

Combined with the law of conservation of electromagnetic field energy [16], it is deduced as (3) we=wm, D=εE, B=μHwe=12E⋅D=12ε|E|2wm=12H⋅B=12μ|H|2 \left\{{\matrix{{{w_e} = {w_m},\,\,\,{\boldsymbol{D}} = \varepsilon {\boldsymbol{E}},\,\,\,{\boldsymbol{B}} = \mu {\boldsymbol{H}}} \hfill \cr {{w_e} = {1 \over 2}{\boldsymbol{E}} \cdot {\boldsymbol{D}} = {1 \over 2}\varepsilon |{\boldsymbol{E}}{|^2}} \hfill \cr {{w_m} = {1 \over 2}{\boldsymbol{H}} \cdot {\boldsymbol{B}} = {1 \over 2}\mu |{\boldsymbol{H}}{|^2}} \hfill \cr}} \right. where w_e and w_m represent the electric field energy density and magnetic field energy density, respectively. E denotes the electric field intensity, H denotes the magnetic field intensity, D is the electric displacement vector, B is the magnetic flux density, ɛ is the permittivity, and μ is the permeability.

When an electromagnetic wave propagates through free space and impinges on the interface of another medium, the free space impedance Z₀ is obtained from (3), as shown (4). The surface impedance Z_S of the wave-absorbing material is given by (5) [17]: (4) Z0=EH=μ0ω0 {Z_0} = {{\left| {\boldsymbol{E}} \right|} \over {\left| {\boldsymbol{H}} \right|}} = \sqrt {{{{\mu _0}} \over {{\omega _0}}}} (5) ZS=Zm⋅ tan (γm⋅dm) {Z_s} = {Z_m} \cdot \;\tan \;({\gamma _m} \cdot {d_m}) where Zm=μr/εr {Z_m} = \sqrt {{\mu _r}/{\varepsilon _r}} is the intrinsic impedance of the wave-absorbing material, γm=jωμrεr {\gamma _m} = j\omega \sqrt {{\mu _r}{\varepsilon _r}} is the propagation constant of the wave-absorbing material, and d_m is the thickness of the wave-absorbing coating layer. One portion of the wave is reflected back, while the other portion enters the material. Through the combined effects of dielectric loss and magnetic loss, the absorbing material converts the electromagnetic wave into thermal energy or other forms of energy, thereby attenuating the electromagnetic wave.

The BLT equation can describe the equivalent transmission characteristics of electromagnetic waves on transmission lines, including the propagation and absorption properties when electromagnetic waves enter the cavity interior. When establishing this model, the external electric field strength is converted into an equivalent potential (voltage) based on the principle of field-circuit equivalence. If an external electromagnetic wave irradiates the cavity aperture, the tangential electric field component Etan inc(z) {\boldsymbol{E}}_{\tan \;}^{inc}(z) across the aperture serves as the key field variable. By introducing the effective aperture height h_eff, the tangential electric field is directly equivalent to a distributed voltage source V_s(z) on the transmission line [18]: (6) Vs(z)=Etan inc(z)⋅heff {{\boldsymbol{V}}_S}(z) = {\boldsymbol{E}}_{\tan \;}^{inc}(z) \cdot {h_{eff}}

To match the wave modes of the transmission line, a mode-matching integration is performed on this distributed source, yielding lumped equivalent wave sources V0+ V_0^ + and V0− V_0^ - that can be injected into the BLT equation network model [19]: (7) V0±=12∫Vs(z)e∓jβzdz V_0^ \pm = {1 \over 2}\int {{{\boldsymbol{V}}_s}(z){e^{\mp j\beta z}}{\rm{d}}z}

The voltage values at each node of the network are then obtained, completing the analysis of the shielding characteristics of the cavity. Based on the aforementioned electromagnetic energy flow relationships, the signal flow graph of the cavity model constructed is shown in Fig. 1. It demonstrates the transmission paths and energy conversion processes of electromagnetic waves within the cavity structure.

Fig. 1.

Signal flow diagram of cavity model.

Here Vi,iinc {\boldsymbol{V}}_{i,i}^{{\rm{inc}}} and Vi,iref {\boldsymbol{V}}_{i,i}^{{\rm{ref}}} represent the incident and reflected voltages, respectively, at node J_i on the transmission channel Tube; Vi,jinc {\boldsymbol{V}}_{i,j}^{{\rm{inc}}} and Vi,jref {\boldsymbol{V}}_{i,j}^{{\rm{ref}}} represent the incident and reflected voltages, respectively, at node J_j on the transmission channel Tube. According to Fig. 1, the transmission equation, scattering equation, and voltage equation are expressed in (8), (9) and (10), respectively. As a result, the BLT equation can be obtained in (10) [13]. (8) Vi,jref=PVi,jinc−V {\boldsymbol{V}}_{i,j}^{{\rm{ref}}} = {\boldsymbol{PV}}_{i,j}^{{\rm{inc}}} - {\boldsymbol{V}} (9) Vi,jref=SVi,jinc {\boldsymbol{V}}_{i,j}^{{\rm{ref}}} = {\boldsymbol{SV}}_{i,j}^{{\rm{inc}}} (10) Vi,j=Vi,jinc+Vi,jref {{\boldsymbol{V}}_{i,j}} = {\boldsymbol{V}}_{i,j}^{{\rm{inc}}} + {\boldsymbol{V}}_{i,j}^{{\rm{ref}}} (11) Vi,j=(S+E)(P−S)−1V {{\boldsymbol{V}}_{i,j}} = ({\boldsymbol{S}} + {\boldsymbol{E}}){({\boldsymbol{P}} - {\boldsymbol{S}})^{- 1}}{\boldsymbol{V}} where V_i,j represents the total voltage vector of the node; S represents the node scattering matrix, E represents the identity matrix; P represents the transmission matrix of the transmission channel, V represents the equivalent voltage source vector of the node, the unit vector composed of an equivalent voltage source V₀, i.e. V = [V₀ 0 0 0 ...0]^T.

Results should be briefly summarized, and the authors’ main scientific contributions should be demonstrated.

In this study, the rectangular open-hole shielding cavity is taken as an example for SE analysis, and its schematic diagram is shown in Fig. 2. Here, absorbing materials are selected based on their electromagnetic parameters, such as the relative dielectric constant ɛ_r and relative magnetic permeability μ_r [13]. The rectangular hole gap has length l₁ and width w₁, located in the center of the front wall in the cavity. The direction of the interference electromagnetic wave electric field is along the y axis, and the propagation direction is along the −z axis, that is, vertical incidence and vertical polarization. The distance between the observation point P₂ and the aperture center is d₂. The distance between the observation point P₁ and the aperture center is d₁; the SE of the rectangular shielding cavity is obtained by calculating the voltage at the observation point P₁.

Fig. 2.

Schematic diagram of rectangular cavity model structure (unit: mm).

The schematic diagram of a cavity coated with and without absorbing material is shown in Fig. 3, where the gray area indicates no absorbing material coated, and the blue area shows the inner wall of the cavity coated with absorbing material.

Fig. 3.

Coating schematic diagram in rectangular shielding cavities with openings.

When coating absorbing materials on the inner wall of the cavity and establishing the equivalent circuit, the aperture impedance is equivalent to Z_ap, which can be expressed as the parallel connection of Z_ap1 and Z_ap2. The equivalent impedance of the aperture is Z_ap1 (Z_ap1 = 1/Y_ap1). When using coated absorbing materials, the equivalent impedance of the absorbing material at the aperture is Z_ap2 (Z_ap2 = 1/Y_ap2). Z_gmn1 is the characteristic impedance of the rectangular cavity, and Z_gmn2 is the characteristic impedance of the coated absorbing material waveguide. The impedance at the inner wall of the rectangular cavity is equivalent to the parallel connection of Z_gmn1 and Z_gmn2. The equivalent circuit of anti-interference using the coating material against electromagnetic wave irradiation is therefore established, as shown in Fig. 4.

Fig. 4.

Equivalent circuit of cavity inner wall.

According to the transmission line equivalent circuit [20], voltage source V₀ (V₀ = 1 V) represents the external incident plane wave, and Z₀ (Z₀ = 377 Ω) is the characteristic impedance of free space; the external space transmission coefficient k₀ = 2π/λ, where λ is the interference electromagnetic wavelength.

The equivalent impedance Z_ap1 of the aperture is expressed as (12) Zap1=j12l1aZostan k0l12 {Z_{{\rm{ap}}1}} = j{1 \over 2}{{{l_1}} \over a}{Z_{{\rm{os}}}}\tan \;\left({{{{k_0}{l_1}} \over 2}} \right) where Z_os is the equivalent impedance of the perforated cavity wall.

Z_os is expressed as (13) Zos=120π2ln21+1−(we/b)241−1−(we/b)24−1 {Z_{{\rm{os}}}} = 120{\pi ^2}{\left[ {\ln \left({2{{1 + \root 4 \of {1 - {{({w_e}/b)}^2}}} \over {1 - \root 4 \of {1 - {{({w_e}/b)}^2}}}}} \right)} \right]^{- 1}} where w_e is the effective width of the coplanar transmission line, and w_e = l₁−(5t/4π)[1+ln(4πl₁/t)]. The transmission coefficient k_g of the cavity coated with absorbing materials is 2π(ɛ_rμ_r)_1/2/λ. The wall impedance Z_ap2 in the cavity is used to characterize the absorption capability against an incident electromagnetic wave, expressed in (14). (14) Zap2=ln21+1−(we/b)241−1−(we/b)24−1⋅j120π2l1tan πl1λεrμr2a {Z_{{\rm{ap}}2}} = {\left[ {\ln \left({2{{1 + \root 4 \of {1 - {{({w_e}/b)}^2}}} \over {1 - \root 4 \of {1 - {{({w_e}/b)}^2}}}}} \right)} \right]^{- 1}} \cdot {{j120{\pi ^2}{l_1}\tan \;\left({{{\pi {l_1}} \over \lambda}\sqrt {{\varepsilon _r}{\mu _r}}} \right)} \over {2a}} where ɛ_r and μ_r represent the relative dielectric constant and relative permeability of the absorbing material filled with the cavity, respectively. Note that the influence of the hole seam position on the impedance of the hole array is ignored.

The impedance and transmission coefficients in the shielding cavity are equivalent to Z_gmn1 and k_gmn1 in the rectangular waveguide, respectively. (15) Zgmn1=Z0/1+(mλ2a)2−(nλ2b)2 {Z_{{\rm{gmn}}1}} = {Z_0}/\sqrt {1 + {{({{m\lambda} \over {2a}})}^2} - {{({{n\lambda} \over {2b}})}^2}} (16) kgmn1=k01−(mλ2a)2−(nλ2b)2 {k_{{\rm{gmn}}1}} = {k_0}\sqrt {1 - {{({{m\lambda} \over {2a}})}^2} - {{({{n\lambda} \over {2b}})}^2}}

In (15) and (16), the values of m and n are the number of standing waves along the x and y axes. The characteristic impedance Z_gmn2 of a waveguide coated with absorbing material is expressed as (17) Zgmn2=2da⋅Z01−(fc/f)2⋅k1γ {Z_{{\rm{gmn}}2}} = {{2d} \over a} \cdot {{{Z_0}} \over {\sqrt {1 - {{({f_c}/f)}^2}}}} \cdot {{{k_1}} \over \gamma} where a and d are the length and width dimensions of the cavity in the model shown in Fig. 2, respectively, and f_c = c/2d is the cutoff frequency of the dominant mode (TE₁₀) of the rectangular waveguide [13], with c as the speed of light in vacuum and as the operating frequency γ is the complex propagation constant of the rectangular waveguide after coating with absorbing material. This is due to the introduction of the complex impedance boundary condition resulting from the inner walls coated with absorbing material. The complex propagation constant γ must be determined by solving the eigenvalue equation. When the material loss is relatively small, the perturbation method can be adopted for efficient solution [15], and the expression of γ can be derived as shown in (18). (18) γ=kc2−k12+j2k1aZ0⋅ZS=α−jβ \gamma = \sqrt {k_{\rm{c}}^2 - k_1^2 + j{{2{k_1}} \over {a{Z_0}}} \cdot {Z_{\rm{S}}}} = \alpha - j\beta where k_c = π/d is the cutoff wavenumber, α and β are the real and imaginary parts of γ, respectively [21]. Substituting (17) into (18) yields (19). (19) Zgmn2=2da⋅Z01−(fc/f)2⋅k1α−jβ {Z_{{\rm{gmn}}2}} = {{2d} \over a} \cdot {{{Z_0}} \over {\sqrt {1 - {{({f_c}/f)}^2}}}} \cdot {{{k_1}} \over {\alpha - j\beta}}

The equivalent impedance Z_ap of the slot in Fig. 4 is shown in (20), and the total impedance Z_gmn of the rectangular waveguide is shown in (21). (20) Zap=Zap1//Zap2==ln21+1−(we/b)241−1−(we/b)24−1⋅j3π2l1a⋅tan πl1λtan πl1λεrμrtan πl1λ+tan πl1λεrμr \eqalign{& {Z_{{\rm{ap}}}} = {Z_{{\rm{ap}}1}}//{Z_{{\rm{ap}}2}} = \cr & = {\left[ {\ln \left({2{{1 + \root 4 \of {1 - {{({w_e}/b)}^2}}} \over {1 - \root 4 \of {1 - {{({w_e}/b)}^2}}}}} \right)} \right]^{- 1}} \cdot {{j3{\pi ^2}{l_1}} \over a} \cdot {{\tan \;\left({{{\pi {l_1}} \over \lambda}} \right)\tan \;\left({{{\pi {l_1}} \over \lambda}\sqrt {{{{\varepsilon _r}} \over {{\mu _r}}}}} \right)} \over {\tan \;\left({{{\pi {l_1}} \over \lambda}} \right) + \tan \;\left({{{\pi {l_1}} \over \lambda}\sqrt {{{{\varepsilon _r}} \over {{\mu _r}}}}} \right)}} \cr} (21) Zgmn=Zgmn1//Zgmn2==dλZ02a(α−jβ)1−fcf2+dλZ01−mλ2a−nλ2b2 \eqalign{& {Z_{{\rm{gmn}}}} = {Z_{{\rm{gmn}}1}}//{Z_{{\rm{gmn}}2}} = \cr & = {{d\lambda Z_0^2} \over {a(\alpha - j\beta)\sqrt {1 - {{\left({{{{f_c}} \over f}} \right)}^2} + d\lambda {Z_0}\sqrt {1 - \left({{{m\lambda} \over {2a}}} \right) - {{\left({{{n\lambda} \over {2b}}} \right)}^2}}}}} \cr}

According to the electromagnetic topology theory [17], the signal flow diagram of the cavity using the coated suction material is established as shown in Fig. 5. Here, Tube1 represents free space, and Tube2 and Tube3 are shielded cavity, where J₁ is the cavity in vitro observation point P₂, W_s is the interference electromagnetic wave equivalent voltage source, J₂ is the hole seam node, J₃ is the intra-cavity observation point P₁, and J₄ is the terminal node of the cavity.

Fig. 5.

Signal-flow graph of cavity coated with absorbing material.

The BLT equation, formed from the inner wall using a coated absorbing material cavity, is established based on Fig. 4 and Fig. 5. The scattering matrix S can be obtained from the impedance, S = diag [ρ₁ρ₂ρ₃ρ₄]. ρ₁ indicates the node J₁ dissipation coefficient, and ρ₁ = 0, ρ₄ is the node J₄ dissipation coefficient, and ρ₄ = −1. ρ_i (i = 2,3,4) represents the node J_j of the scattering matrix, expressed by (22). (22) ρ2=Y0−Ygmn−YapY0+Ygmn+Yap2YgmnY0+Ygmn+Yap2Y0Y0+Ygmn+YapYgmn−Yap−Y0Y0+Ygmn+Yapρ3=0110 \left\{{\matrix{{{\rho _2} = \left[ {\matrix{{{{{Y_0} - {Y_{{\rm{gmn}}}} - {Y_{{\rm{ap}}}}} \over {{Y_0} + {Y_{{\rm{gmn}}}} + {Y_{{\rm{ap}}}}}}} & {{{2{Y_{{\rm{gmn}}}}} \over {{Y_0} + {Y_{{\rm{gmn}}}} + {Y_{{\rm{ap}}}}}}} \cr {{{2{Y_0}} \over {{Y_0} + {Y_{{\rm{gmn}}}} + {Y_{{\rm{ap}}}}}}} & {{{{Y_{{\rm{gmn}}}} - {Y_{{\rm{ap}}}} - {Y_0}} \over {{Y_0} + {Y_{{\rm{gmn}}}} + {Y_{{\rm{ap}}}}}}} \cr}} \right]} \hfill \cr {{\rho _3} = \left[ {\matrix{0 & 1 \cr 1 & 0 \cr}} \right]} \hfill \cr}} \right. where Y₀( = 1/Z₀) represents the free space admittance, and Y_gmn( = 1/Z_gmn) represents a rectangular waveguide guide.

The transmission coefficient comprises the free-space transmission coefficient k₀ and the transmission coefficient k_g after the cavity is coated with an absorbing material. The transmission matrix P can be derived from the transmission coefficient, P = diag [T₁T₂T₃], where T_i (i = 1,2,3) is the transmission matrix of each segment transmission channel, as expressed in (23). (23) T1=0eik0d2eik0d20T2=0eikgd1eikgd10T3=0eikg(d−d1)eikg(d−d1)0 \left\{{\matrix{{{{\boldsymbol{T}}_1} = \left[ {\matrix{0 & {{e^{i{k_0}{d_2}}}} \cr {{e^{i{k_0}{d_2}}}} & 0 \cr}} \right]} \hfill \cr {{{\boldsymbol{T}}_2} = \left[ {\matrix{0 & {{e^{i{k_g}{d_1}}}} \cr {{e^{i{k_{\rm{g}}}{d_1}}}} & 0 \cr}} \right]} \hfill \cr {{{\boldsymbol{T}}_3} = \left[ {\matrix{0 & {{e^{i{k_g}(d - {d_1})}}} \cr {{e^{i{k_g}(d - {d_1})}}} & 0 \cr}} \right]} \hfill \cr}} \right.

Substitute the scattering matrix S (19) and the transmission matrix P (20) into (10) to obtain the total voltage V_3,3 at observation point P₁, and thus obtain the SE of the cavity, as shown in (24). (24) SE=20lg(V0/V3,3) SE = 20\lg ({V_0}/{V_{3,3}})

Therefore, the basic process flow diagram of the BLT equation modeling principle for the SE of the cavity coated with absorbing materials on the inner wall is shown in Fig. 6.

Fig. 6.

Flowchart of BLT equation modeling.

In this study, the BLT-based equation analytical model has been successfully established to accurately calculate the SE of absorbing materials. It verifies that the results obtained using the BLT equation and CST simulation are very close, with an error of less than 8 dB. More importantly, the BLT equation model calculates the SE and resonant frequency more quickly than CST. In conclusion, the BLT equation can rapidly complete single SE calculations, enabling quick optimization of multiple variables, such as material thickness and dielectric parameters. This makes it particularly suitable for preliminary rapid parameter scanning and resonant risk assessment. While CST simulations are more time-consuming, they can accurately capture the field distribution characteristics of higher-order resonant modes, such as TE₁₀₃, providing critical evidence for understanding electromagnetic energy coupling paths and an effective means for final design validation and mechanism investigation.

It also confirms that the inner wall of the cavity, coated with a graphene, SiC, and Ni/rGO composite material, has no effect on the frequency at the TE₁₀₁ and TE₁₀₂ resonant modes; however, an effect occurs at the higher-order resonant TE₁₀₃ modes. However, the coated thickness using PTFE does not change the resonant frequency of the cavity. Moreover, the 2 mm thickness of the cavity wall coated with graphene, SiC, or Ni/rGO composite materials is found to achieve the best electromagnetic shielding performance. On the other hand, a 1 mm thick coating using PTFE can achieve the best SE in the cavity wall.