Introduction

With the advancement of aerospace technology, wireless communication has emerged as a promising technology for space vehicles1,2,3,4,5. In comparison to traditional wired cable networks6,7,8, wireless communication offers significant advantages in terms of system layout and design within the cabin, leading to reduced design cycles and costs9,10. Moreover, wireless communication enables high-speed data transmission, large capacity handling, and low latency, thereby enhancing data interaction performance among various onboard equipment11,12.

As an almost completely closed metal shell, aircraft presents complicated transmission property for the EM wave within each compartment, since there are multiple refractions and reflections. Additionally, the structural stability and functional partition between compartments lead to electromagnetic disconnection13,14,15. However, if there is an aperture in the partition, it allows for electromagnetic coupling between compartments, providing the possibility of wireless communication within the aircraft.

The topic of EM shielding and coupling has been extensively discussed in the past decade16,17,18,19. For example, Reference 16 introduces an efficient time-domain mixed potential integral equation (TD-MPIE) technique for analyzing the transient electromagnetic field penetrating a perforated rectangular metallic enclosure loaded with conducting objects. Reference 17 presents a formulation for evaluating the magnetic shielding effectiveness of an infinite PEC planar screen with a circular aperture, which is applicable to the case of a current loop source with a finite radius coaxial with the aperture. Additionally, Reference 18 develops an analytical formulation to evaluate the shielding effectiveness (SE) of two coplanar rectangular metallic enclosures with a circular aperture excited by an internal electric dipole source. Reference 19 investigates the shielding performance and inner wideband pulse responses of metallic rectangular multistage cascaded enclosures with multiple rectangular apertures when illuminated by an electromagnetic pulse (EMP). It is worth noting that while traditional aperture coupling or shielding theory mainly focuses on cases involving metal cavities and free space or waveguides, further investigation is needed regarding EM coupling between two closed metal compartments through an aperture.

In this paper, we consider a simplified model of the aircraft cabin, which consists of a metallic cylindrical cavity containing two independent compartments. A source is located in compartment 1, and a small aperture is introduced in the partition to allow for EM coupling between two compartments. It is demonstrated that the effect of the aperture in the partition can be described by an electric dipole and a magnetic dipole. The field distribution in compartment 2 is obtained by analyzing the field excited by the equivalent dipoles of the small aperture, while the field in compartment 1 consists of both standing waves in the original compartment and fields created by the dipole. Specifically, analytical equations for electric and magnetic fields in both compartments are derived, which depend on frequency and spatial parameters. Furthermore, we analyze the field distribution characteristics in both compartments using HFSS software. A great agreement between simulation results and theoretical values can be observed, validating our analytic analysis presented in this paper.

Theoretical model and analysis

Considering aerodynamics, the classic shape of the aircraft cabin shell is generally designed as a cylindrical shape, as shown in Fig. 1. There are also separate compartments inside the cabin. The two adjacent compartments are physically separated by metal partitions, with small holes introduced for electromagnetic coupling.

Fig. 1
figure 1

The aircraft model and the interior view of its separate compartments.

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To simplify the theoretical derivation of the field distribution within the aircraft cabin, we propose the use of a cylindrical metal cavity to represent the aircraft cabin. As illustrated in Fig. 2, the aircraft is comprised of two compartments with lengths \({l_1}\)and \({l_2}\), while the radius of the cavity is R. Additionally, a circular aperture with radius \({r_0}\) is cut into the metal partition between these two compartments.

Fig. 2
figure 2

Simplified model of the aircraft cabin.

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Obviously, when two compartments are totally physical isolated, the electromagnetic (EM) waves in compartment 1 cannot reach compartment 2 due to the shielding effect of metal walls. However, introducing a small aperture in the partition between the two compartments allows for coupling of EM waves from compartment 1 to compartment 2. Figure 3 illustrates the transmission properties of EM waves between point 1 and point 2 in two adjacent compartments based on HFSS, with varying aperture sizes of 1 mm, 3 mm and 9 mm, respectively.

Fig. 3
figure 3

Transmission curves between two points located in two separate compartments with an aperture on the partition, the radius of the aperture is 1 mm, 3 mm and 9 mm respectively.

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It is noted that the transmission coefficient of EM waves between two points is approximately − 140 dB for an aperture with a radius 1 mm. With an increase in the aperture’s radius to 3 mm and 9 mm, the transmission coefficient between these two points improves to − 90 dB and − 25dB, respectively, as depicted by the red dot and black solid lines in Fig. 3. To explain this phenomenon, we further theoretically analyze the relationship between the aperture and electromagnetic transmission characteristics.

Equivalent model of the aperture

According to the resonator theory, EM waves are considered as standing waves in the metal closed compartment 1. On the metal partition, only the vertical component of the electric field and the parallel component of the magnetic field exist, as shown in Fig. 4a,d. There is no EM coupling between two compartments. However, if a small aperture is cut in the partition, it will allow for electromagnetic field coupling from compartment 1 into compartment 2 through this small hole, as indicated in Fig. 4b,e, respectively.

On the other hand, when an electric dipole \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}\) placed perpendicular to the metal partition without a hole in Fig. 4c, it presents the same electric field distribution as in Fig. 3b. Thus, the coupling effect of the normal electric field through the small aperture can be represented by two oppositely directed electric dipoles \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}\), which are normal to the metal partition. The magnitude of electric dipole is proportional to the normal electric field and satisfies20,21

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}=\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n} {\varepsilon _0}{\alpha _e}{E_n}\delta (x - {x_0})\delta (y - {y_0})\delta (z - {z_0})$$
(1)

where \({\varepsilon _0}\) is the permittivity of free-space, \({\alpha _e}\) is defined as the electric polarization coefficient of the aperture and depends on the size and shape of the aperture, \({E_n}\) is the normal component of the electric field on the partition, \(\left( {{x_0},{y_0},{z_0}} \right)\) are the coordinates of the center of the aperture.

Similarly, Fig. 4d,e depict the distribution of tangential magnetic field on the metal partition with and without a small aperture. Figure 4f illustrates the magnetic distribution of two oppositely directed magnetic dipoles \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _m}\) that are parallel to the partition, sharing similar characteristics as Fig. 4e. Therefore, the coupling effect for the parallel magnetic field through the small aperture can be represented by two oppositely directed magnetic dipoles \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _m}\).

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _m}= - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {t} {\alpha _m}{H_t}\delta (x - {x_0})\delta (y - {y_0})\delta (z - {z_0})$$
(2)

\({\alpha _m}\) is the magnetic polarization coefficient of the aperture, \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {t}\) is the tangential component of the partition, and \({H_t}\) is the tangential component of the magnetic field on the partition.

Fig. 4
figure 4

Schematic diagram of the equivalent process for a small aperture in the metal partition as electric and magnetic dipoles.

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The polarization coefficients for circular aperture are22

$${\alpha _e}=2r_{0}^{3}/3, {\alpha _m}=4r_{0}^{3}/3$$
(3)

Field distribution in the two compartments

Since the impact of aperture in the partition on the EM properties is comparable to that of an electric dipole and a magnetic dipole, this section will delve into a further analysis of the EM field distribution in compartments 1 and 2.

When there is no aperture, the EM field in compartment 1 is independent and exhibits a standing wave property when excited. However, if an aperture is introduced in the partition, the EM waves in compartment 1 will be coupled to compartment 2 through the small aperture, as shown in Fig. 5. Based on the analysis in the previous section, the EM field in compartment 1 consists of two parts, one part is the original field \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _0}\) within the closed compartment 1, and the other part comprises the field \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _e}\)、\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _m}\) generated by equivalent electric and magnetic dipoles along the partition.

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _1}={\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _0}+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} ^{\prime}={\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _0}+{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} ^{\prime}_{{P_{\text{e}}}}}+{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} ^{\prime}_{{P_m}}}$$
(4)

While the EM field in compartment 2 is introduced by the equivalent electric and magnetic dipoles as

$${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _2}={\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _{{P_{\text{e}}}}}+{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _{{P_m}}}$$
(5)
Fig. 5
figure 5

(a) Two compartments with a small aperture in the partition, (b) The equivalent model represents the aperture as a combination of electric and magnetic dipoles.

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It is important to highlight that both of these compartments function serve as closed resonators, and the coupling effect differs from the traditional configuration where coupling occurs either between a half-space and a resonator or between a waveguide and a resonator. Specifically, we will use the initial field in compartment 1 without an aperture in the partition, i.e. \(T{E_{mnp}}\) mode as an example. The corresponding components of the EM fields in the cylindrical coordinate system \(\left( {\rho ,\varphi ,z} \right)\) are expressed as follows23

$$\begin{gathered} {E_\rho }= \pm j\frac{{m{\eta _1}{K_1}}}{{K_{{1c}}^{2}\rho }}{H_0}{J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {E_\varphi }=j\frac{{{\eta _1}{K_1}}}{{{K_{1c}}}}{H_0}J_{{_{m}}}^{\prime }({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {E_z}=0 \\ {H_\rho }=\frac{{p\pi }}{{{K_{1c}}{l_1}}}{H_0}J_{{_{m}}}^{\prime }({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {H_\varphi }= \mp \frac{{mp\pi }}{{K_{{1c}}^{2}\rho {l_1}}}{H_0}{J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {H_z}={H_0}{J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ \end{gathered}$$
(6)

\({H_0}\) is the amplitude of the magnetic field, \({J_m}\) and \({J^{\prime}_m}\) are the mth order Bessel function and its derivatives, respectively. \({\eta _1}\), \({K_1}\), \({K_{1c}}\)are the wave impedance, wave number and its tangential components.

$$K_{1}^{2}=K_{{1c}}^{2}+{\left( {\frac{{p\pi }}{{{l_1}}}} \right)^2}={\left( {\frac{{{u_{mn}}}}{R}} \right)^2}+{\left( {\frac{{p\pi }}{{{l_1}}}} \right)^2},{\eta _1}=\sqrt {\frac{{{\mu _1}}}{{{\varepsilon _1}}}}$$
(7)

where \({u_{mn}}\) is the nth root of \({J^{\prime}_m}\), \({\mu _1}\)and \({\varepsilon _1}\) are the permeability and permittivity of the dielectric in compartment 1.

The fields produced by the equivalent electric and magnetic dipoles of the aperture in compartment 1, scatter once again within a closed cabin, and exhibit the same standing wave pattern:

$$\begin{gathered} {{E^{\prime}}_\rho }= \pm \left( {A_{{mnp}}^{ - }+B_{{mnp}}^{ - }} \right){J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {{E^{\prime}}_\varphi }=\left( {A_{{mnp}}^{ - }+B_{{mnp}}^{ - }} \right)J_{{_{m}}}^{\prime }({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {{E^{\prime}}_z}=0 \\ {{H^{\prime}}_\rho }=\left( {A_{{mnp}}^{ - }+B_{{mnp}}^{ - }} \right)J_{{_{m}}}^{\prime }({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {{H^{\prime}}_\varphi }= \mp \left( {A_{{mnp}}^{ - }+B_{{mnp}}^{ - }} \right){J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ {{H^{\prime}}_z}=\left( {A_{{mnp}}^{ - }+B_{{mnp}}^{ - }} \right){J_m}({K_{1c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_1}}}z} \right) \\ \end{gathered}$$
(8)

Similarly, the field distribution in compartment 2 is

$$\begin{gathered} {E_\rho }= \pm \left( {A_{{mnp}}^{+}+B_{{mnp}}^{+}} \right){J_m}({K_{2c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_2}}}z} \right) \\ {E_\varphi }=\left( {A_{{mnp}}^{+}+B_{{mnp}}^{+}} \right)J_{{_{m}}}^{\prime }({K_{2c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_2}}}z} \right) \\ {E_{2z}}=0 \\ {H_\rho }=\left( {A_{{mnp}}^{+}+B_{{mnp}}^{+}} \right)J_{{_{m}}}^{\prime }({K_{2c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_2}}}z} \right) \\ {H_\varphi }= \mp \left( {A_{{mnp}}^{+}+B_{{mnp}}^{+}} \right){J_m}({K_{2c}}\rho )\begin{array}{*{20}{c}} {\sin m\varphi } \\ {\cos m\varphi } \end{array}\cos \left( {\frac{{p\pi }}{{{l_2}}}z} \right) \\ {H_z}=\left( {A_{{mnp}}^{+}+B_{{mnp}}^{+}} \right){J_m}({K_{2c}}\rho )\begin{array}{*{20}{c}} {\cos m\varphi } \\ {\sin m\varphi } \end{array}\sin \left( {\frac{{p\pi }}{{{l_2}}}z} \right) \\ \end{gathered}$$
(9)

where \({K_{2c}}={u_{mn}}/R\), \(A_{{mnp}}^{+}\), \(A_{{mnp}}^{ - }\) are the excitation coefficients of the forward (along the positive z direction) and backward waves generated by the equivalent electric dipole \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}\), which can be obtained by

$$A_{{mnp}}^{ \pm }= \pm \frac{1}{{{P_{mnp}}}}\int_{V} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{mnp}}} \cdot j\omega {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}dv= \pm \frac{1}{{{P_{mnp}}}}\int_{V} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {t} {e_t} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} {e_z})} \cdot j\omega {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _e}dv$$
(10)

Similarly, \(B_{{mnp}}^{+}\)、\(B_{{mnp}}^{ - }\) are the excitation coefficients of the forward and backward waves generated by the equivalent magnetic dipole:

$$B_{{mnp}}^{ \pm }= \pm \frac{1}{{{P_{mnp}}}}\int_{V} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{mnp}}} \cdot j\omega {\mu _0}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _m}dv= \pm \frac{1}{{{P_{mnp}}}}\int_{V} {(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {t} {h_t} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} {h_z})} \cdot j\omega {\mu _0}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} _m}dv$$
(11)

where \({e_t}\) (\({h_t}\))、\({e_z}\) (\({h_z}\)) are the tangential and normal components of the electric field \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _{mnp}}\) (\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} _{mnp}}\)), \({\mu _0}\) is the permeability of vacuum. The coefficient \({P_{mnp}}\) is

$${P_{mnp}}=2\int_{S} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{mnp}} \times {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{mnp}} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} ds}$$
(12)

So far, we have theoretically derived the distribution of the EM field in two compartments, which depends on the radius of the apertures, the spatial position of the observation point, as well as the resonant modes in a fixed cabin.

Numerical analysis

To validate the theoretical analysis, we will use the model illustrated in Fig. 2 as a demonstration. In this model, the length of the cavity is denoted by \({l_1}={l_2}=400\;mm\), the radius inside the cavity is \(R=60\;mm\), and the orientation of the cavity is along the z-direction. Additionally, a circular aperture with a radius of \({r_0}=10\;mm\) is carved into the wall of both compartments.

EM field of resonant modes as a function of spatial positions

First, we begin with the simple case, assuming only the fundamental mode \(T{E_{111}}\) in compartment 1, which couples to compartment 2 through the small aperture. The field distribution within compartments 1 and 2 can be obtained from Eqs. (4)–(12), wherein \(A_{{111}}^{+}=A_{{111}}^{ - }=0\). Therefore, there are only equivalent fields generated by equivalent magnetic dipoles in both compartments.

$$\begin{aligned} & B_{{111}}^{+}=\frac{1}{{{P_{111}}}}\int_{V} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{111}} \cdot j\omega {\mu _0}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} }_m}} dv \\ & \quad =\frac{{ - 1}}{{{P_{111}}}}\int_{V} ( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\rho } J_{1}^{\prime }({K_{2c}}\rho )\sin \varphi \cos (\frac{\pi }{{{l_1}}}z)+\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\varphi } {J_1}({K_{2c}}\rho )\cos \varphi \cos (\frac{\pi }{{{l_1}}}z) \\ &\quad - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} {J_1}({K_{2c}}\rho )\sin \varphi \sin (\frac{\pi }{{{l_1}}}z)) \cdot ( - j\omega {\mu _0}{\alpha _m}(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\rho } \frac{\pi }{{{K_{2c}}{l_1}}}{H_0}J_{1}^{\prime }({K_{2c}}\rho )\sin \varphi \cos (\frac{\pi }{{{l_1}}}z) \\ &\quad +\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\varphi } \frac{\pi }{{K_{{2c}}^{2}\rho {l_1}}}{H_0}{J_1}({K_{2c}}\rho )\cos \varphi \cos (\frac{\pi }{{{l_1}}}z))\delta (\rho )\delta (\varphi - \frac{\pi }{2})\delta (z - {l_1}))dv \\ & \quad =\frac{{j\omega {\mu _0}{\alpha _m}}}{{{P_{111}}}}\frac{\pi }{{{K_{2c}}{l_1}}}{H_0}J{_{1}^{\prime 2}}(0) \\ \end{aligned}$$
(13a)
$$B_{{111}}^{ - }=\frac{1}{{{P_{111}}}}\int_{V} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{111}} \cdot j\omega {\mu _0}{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {P} }_m}} dv= - \frac{{j\omega {\mu _0}{\alpha _m}}}{{{P_{111}}}}\frac{\pi }{{{K_{1c}}{l_1}}}{H_0}J{_{1}^{\prime 2 }}(0)$$
(13b)

In Eq. (13), the coefficient \({P_{111}}\) can be derived through

$${P_{111}}=2\int_{S} {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{111}} \times {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{111}} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {z} ds} =\frac{{\pi {R^2} \cdot \sqrt {K_{1}^{2} - {{\left( {{u_{11}}/R} \right)}^2}} }}{{{K_1} \cdot {\eta _1}}}$$
(14)

.

Fig. 6
figure 6

Electric field with and without an aperture: (a) compartment 1 (b) compartment 2.

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Figure 6 illustrates the comparison between the theoretical and simulation data of the electric field in two compartments, when only considering the fundamental mode \(T{E_{111}}\) in compartment 1. A significant agreement can be observed, which confirms the accuracy of the analytic expressions for the EM field in previous sections.

Furthermore, let us consider the multimode case in compartment 1, e.g., \(T{E_{111}}\), \(T{E_{112}}\), \(T{E_{113}}\), \(T{M_{010}}\), \(T{M_{011}}\), the electric field distribution in compartment 1 and 2 are compared in Fig. 7, the solid/dashed lines are the theoretical data while the symbolic lines are the simulation ones. The good agreement between these two curves verifies once again the validity of theoretical formula for the EM field.

Fig. 7
figure 7

Electric field of the first five modes with and without an aperture: (a) compartment 1 (b) compartment 2.

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EM Field of resonant modes as a function of frequency

Moreover, considering all the modes in compartment 1, the EM field coupling to compartment 2 through the small aperture can be obtained by superposition of all modes, i.e.

$$\begin{gathered} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} =\sum\limits_{{m=0,n=1,p=1}}^{\infty } {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_{mnp}}}}} +\sum\limits_{{m=0,n=1,p=0}}^{\infty } {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_{mnp}}}}} \hfill \\ ={{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_{011}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_{111}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_{211}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_{010}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_{110}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_{111}}}}+... \hfill \\ =\sum\limits_{{m=0,n=1,p=1}}^{\infty } {({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_\rho }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_\varphi }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{E_z}}})} +\sum\limits_{{m=0,n=1,p=0}}^{\infty } {({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_\rho }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_\varphi }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} }_{T{M_z}}})} \hfill \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} =\sum\limits_{{m=0,n=1,p=1}}^{\infty } {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_{mnp}}}}} +\sum\limits_{{m=0,n=1,p=0}}^{\infty } {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_{mnp}}}}} \hfill \\ ={{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_{011}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_{111}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_{211}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_{010}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_{110}}}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_{111}}}}+... \hfill \\ {\text{ }}=\sum\limits_{{m=0,n=1,p=1}}^{\infty } {({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_\rho }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_\varphi }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{E_z}}})} +\sum\limits_{{m=0,n=1,p=0}}^{\infty } {({{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_\rho }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_\varphi }}}+{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} }_{T{M_z}}})} \hfill \\ \end{gathered}$$
(15)

\({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _{T{E_{mnp}}}}\), \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {E} _{T{M_{mnp}}}}\), \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} _{T{E_{mnp}}}}\)and \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {H} _{T{M_{mnp}}}}\) are the corresponding electric fields and magnetic fields for each mode in the resonant cavity. We are aware that the location of the peak magnitude and resonant frequency of the EM field for each mode can vary. Therefore, the total EM field in the cavity is not only dependent on spatial location (as shown in Figs. 6 and 7), but also on frequency.

Figure 8 illustrates the normalized distribution of all the resonant fields from 1 GHz to 3Ghz at a fixed point, i.e. \({p_2}\left( {0,0,620mm} \right)\), specifically in compartment 2. The blue stars and red triangles represent the \(T{E_{mnp}}\)and \(T{M_{mnp}}\)respectively. It is evident that only specific resonant fields exist at certain frequencies, each with varying amplitudes. In other words, the EM field has discrete distribution characteristics as anticipated from the aforementioned analytic expressions.

Fig. 8
figure 8

Normalized electric field for all the resonant modes in 1 ~ 3 GHz at a specific location \({p_2}\left( {0,0,620mm} \right)\) within compartment 2.

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Fig. 9
figure 9

The distribution of electric field and the transmission curve versus frequency in compartment 2.

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Furthermore, we aim to establish the relationship between the distribution of the electric field and transmission curves within the same frequency band, as shown in Fig. 9. The blue solid line represents the transmission curve from a radiator at \({p_1}\left( {0,0,20\;mm} \right)\) located in compartment 1 to a receiver at \({p_2}\left( {0,0,620\;mm} \right)\) situated in compartment 2. It is evident that the electric field of each resonant modes aligns with the extreme values of the transmission curve at their respective resonant frequencies. Specifically, it can be observed that the amplitude of the electric field is directly proportional to the corresponding transmission coefficient, indicating a positive correlation between the amplitude of resonant field and the transmission curve.

EM Field of resonant modes as a function of aperture radius

As shown in Eqs. (6)–(12), the electric field is also influenced by the radius of the aperture. We consider apertures with radii of 1 mm, 3 mm, 5 mm, 7 mm, 9 mm, 10 mm. The corresponding electric field for both the first mode and the first five modes are depicted in Fig. 10. Apparently, the amplitude of the electric field increases overall as the aperture’s radius increase, which indicates that a larger aperture allows for more electromagnetic coupling between compartments 1 and 2. This observation aligns with the trend of increasing transmission coefficient with an increase in aperture radius shown in Fig. 3.

Fig. 10
figure 10

Distribution of electric field at compartment 2 along the aperture with different sizes: (a) \(TE_{{111}}\); (b) combination of \(TE_{{111}}\), \(TE_{{112}}\), \(TE_{{113}}\), \(TE_{{010}}\), \(TE_{{011}}\).

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Furthermore, it is widely recognized that the characteristics of EM transmission between the radiator and receiver play a crucial role in the selection of frequency bands and wireless communication system design. Specifically, the transmission coefficient is defined as

$${S_{21}}=20*\log (E/{E_0})$$
(16)

.

where E and \({E_0}\) are the amplitude of electric field at the receiver and radiator, respectively. Since the amplitude of electric field is positively correlated with the cubic radius of the aperture’s radius as shown in Eqs. (1), (3) and (9), it follows that the transmission coefficient in the compartment 2 is also positively correlated with the \(\log (r_{0}^{3})\). Figure 11a illustrates the relative increments for both the transmission coefficient (blue solid line) and electric field (red dash line with triangle symbols) versus the aperture’s radius, using the transmission coefficient and electric field at \({r_0}=3 \; mm\) as a baseline. The two curves demonstrate a good agreement and further validate our previous theoretical analysis.

Fig. 11
figure 11

(a) Relative increase for both and amplitude of the electric field as a function of aperture radius. (b) Transmission curves along with r0.

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Discussion

Considering the electromagnetic distribution characteristics within a real aircraft cabin, this phenomenon is primarily governed by two key factors. First, the overall structure of the aircraft cabin establishes the boundary conditions for the electromagnetic field, and thus determines the nature of the field distribution inside the cabin, playing a significant constraining influence. Second, the layout of various metallic electrical devices within the cabin plays a critical role. During the propagation of electromagnetic waves within the cabin, these waves interact with the enclosures of the metallic electrical devices, resulting in reflection, transmission, and diffraction. This interaction modifies the boundary conditions during wave propagation, thereby inducing perturbations in the transmission of electromagnetic waves, the EM distribution in the cabin is more complex and difficult to obtain an accurate analytical solution.

Here we introduce several metallic electrical devices on the cabin wall to simulate a real cabin environment. To balance the weight distribution within the cabin, these devices are typically installed and concentrated in specific areas, the height and width of each area are denoted by \({h_i}\) and \({w_i}\), respectively, as shown in Fig. 12a.

Fig. 12
figure 12

(a) Cabin with various metallic electrical devices on the wall; Transmission curves between two points \({p_1}\) and\({p_2}\) in the real cabin (lower right corner is a cross-sectional view of the cabin) and its equivalent cabin (b) four strip-shaped metal areas (\({h_i} \times {w_i}\)) are 5 mm × 100 mm, 8 mm × 100 mm, 7 mm × 100 mm, 6 mm × 100 mm; (c) four strip-shaped metal areas with thickness 10 mm × 80 mm, 12 mm × 150 mm, 6 mm × 50 mm, 4 mm × 120 mm and 30 degree arc non-metallic cables area.

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These devices exhibit a diverse range of structures and spatial distributions, which have a direct impact on the effective radius of the cabin. As previously demonstrated, the electromagnetic transmission characteristics are primarily determined by the dimensions of the resonator. Therefore, we introduce an empty-loaded cabin with an effective radius to simplify the complex analysis required for assessing the actual cabin. The equivalent radius of the real cabin is formally defined as

$${R_{eq}}=R - \sum\limits_{i} {({h_i} \cdot {w_i})} /\sum\limits_{i} {{w_i}}$$
(17)

.

The transmission curve of cabin equipped with four distinct device areas is compared to that of a simple empty-loaded cabin with an equivalent radius under two different scenarios. The cabin has a radius of 60 mm and a total length of 400 mm, the dimensions (\({h_i} \times {w_i}\)) of the four strip-shaped areas along the axial direction are as follows: 5 mm × 100 mm, 8 mm × 100 mm, 7 mm × 100 mm, and 6 mm × 100 mm for Fig. 12b. Figure 12c shows the case with four strip-shaped areas measuring 10 mm × 80 mm, 12 mm × 150 mm, 6 mm × 50 mm, and 4 mm × 120 mm and 30-degree arc non-metallic cables area. The transmission properties between two points \({p_1}\) and \({p_2}\)are obtained based on EM simulation. The two points are located on the central axis, with distances of 20 mm and 80 mm, respectively, from the adjacent end face. The solid one represents the cabin with metallic electrical devices on its walls, while the dashed line corresponds to the equivalent cabin. These two curves exhibit a similar trend and are in good agreement with each other. This indicates that the simplified equivalent empty-loaded model can be effectively employed to predict the electromagnetic (EM) properties within complex cabin environments. Although this modeling focuses on the equivalent model analysis of electromagnetic transmission characteristics within a single compartment containing devices, the method can be extended to analyze coupling between two compartments. The equivalent model enables rapid analysis of appropriate frequency bands for wireless communication.

Conclusion

In this paper, we propose the introduction of an aperture in the partition to facilitate electromagnetic(EM) coupling between two separated compartments of an aircraft. The effect of the aperture is equivalent to that of an electric dipole and a magnetic dipole, which are perpendicular and parallel to the partition, respectively. Analytical expressions for the EM field in both compartments are derived by considering them as metal resonators. The resonant modes and frequency are primarily determined by the size of each compartment, while the amplitude of the EM field depends on the size of the aperture and the spatial position. The distribution of EM field versus frequency exhibits obvious discrete and discontinuous characteristics. Furthermore, we simulate the distribution of EM waves in both compartments using Ansys HFSS, including single mode and multimode scenarios. The electric fields extracted from these simulations show good agreement with the theoretical results. Additionally, we discuss how transmission coefficient relates to EM field distribution, and a high degree of consistency between them can be observed, indicating that one can use EM distribution to predict transmission properties between a radiator and a receiver within a cabin. Furthermore, our approach in cabin with several electrical devices on the wall is examined. An equivalent radius of the aircraft is defined and theoretically validated to account for the impact of diverse electronic devices on the transmission characteristics within the cabin. The analysis presented in this paper provides a theoretical basis for understanding electromagnetic coupling between separated compartments within an aircraft through small apertures, which is important for further research into wireless communication within aircrafts.