The Terminal Responses of the Two-Wire Line in Multiaperture Cavities Based on Electromagnetic Topology and Method of Moments

Abstract

A simulation technique based on electromagnetic topology (EMT) theory is proposed for analyzing electromagnetic interference (EMI) coupling through apertures onto the two-transmission line enclosed within metallic structures. The electromagnetic interactions between apertures and the external-internal interactions were treated through the topological decomposition and the multistep iterative method. Then, the load responses of the two-wire transmission line are resolved by the the Baum-Liu-Tesche (BLT) equation. The simulation results both without and with the electromagnetic interaction are presented for the frequency range from 100 MHz to 3 GHz. These numerical results obtained by two methods imply that the electromagnetic interaction cannot be simply ignored, especially for the frequency range up to 1 GHz.

1. Introduction

While considering analysis and design of EMC, it is important to protect electronic circuits/components from electromagnetic effects due to external illumination. For this purpose, electronic circuits/components are often shielded inside metallic cavities. However, these metallic surfaces are not perfect. On a metallic cavity, there may be apertures that become sources of the EMI problem. In most cases the cavity and apertures are usually rectangular, which has led to a number of attempts to solve the coupling problem of the rectangular cavity with a series of rectangular apertures when illuminated by a plane wave (see [1–3]). So there would be interference between the fields entering through the apertures and the circuitry inside the cavities. A natural problem is a comprehensive analysis of the coupling mechanism for such structures under external EMI illumination. In this paper, electromagnetic interaction problems can be simulated through codes based on electromagnetic topology. An important element of topological analysis is the determination of a mechanism which represents the external-internal coupling through a small aperture and the subsequent propagation process. By combining methodologies suggested earlier (see [4]), an equivalent source can be created to relate the electromagnetic coupling at the exterior surface and the transfer function through the free space by assuming an imaginary transmission line as a source of the aperture radiation (see [5, 6]), but this method only adapts to the case where the dimension of the aperture is electrically small when compared to the wavelength. Moreover most papers in the literature neglect the electromagnetic interaction among apertures (see [3, 7]), which is not reasonable.

This paper provides a novel numerical technique for the electromagnetic coupling. More precisely, we first use the EMT theory and the multistep iteration to deal with problems of coupling between apertures and the external-internal interaction. Then, we employ dyadic Green's functions and the method of moments to determine the electromagnetic coupling fields inside the cavity. Finally, we apply the BLT equation to resolve the load response of the two-wire transmission line. The simulation result shows the effect of the external coupling fields on the two-wire line current depends on both the cavity with multiaperture and the electromagnetic interaction.

2. The Topological Decomposition of Complicated Electrical Systems

Electromagnetic interaction problems on very large and complex system, such as an aircraft, can be simulated through codes based on the EMT. The most important aspect of the EMT is the assumption that volumes can be decomposed into subvolumes that can be interacted with each other through apertures (see [8]). To analyze the interaction processes for the electromagnetic coupling by the EMT, we need to establish the topological structure model and the topological diagram for the system configuration. First, we consider a rectangular cavity with 𝑅 energy penetration paths (multiaperture) illuminated by a harmonic plane wave, see Figure 1. Figure 2 shows the topological structure model associated to Figure 1. 𝑉_𝑖 denote these subvolumes, where “𝑖” indicates the hierarchical order of the volume. 𝑆_𝑖,𝑗 is a surface separating volumes 𝑉_𝑖 from 𝑉_𝑗. Figure 3 shows the topological diagram. To analyze the field coupling phenomenon in the cavity, we introduce a “field coupling junction” 𝐽₃, which takes into account the internal EM field coupling to the two-wire line. Moreover, by reciprocity, this junction also provides effects of the EM field radiation by the two-wire line (see [9]). 𝑊_𝑖(0) is the outgoing wave and 𝑊_𝑠 is the network cable coupling source. Signals on an entire transmission line network is expressed through the BLT (Baum-Liu-Tesche) equation which is the multiconductor transmission line (MTL) network composed of the outgoing wave supervector [𝑊(0)] and the source wave supervector [𝑊_𝑠] (see [10]).

Figure 1 

The rectangular cavity with 𝑅 rectangular apertures.

Figure 2 

The system topological structure model.

Figure 3 

The topology diagram of Figure 2.

3. The Multistep Iterative Method

Based on the topology decomposition, we deal with problems of the electromagnetic interaction among apertures and the external-internal interaction using the multistep iterative method (see [11]). We assume 𝑅 apertures on the same wall located on the plane where 𝑧=0. 𝑅 is the number of apertures.

Step 1. Find the zeroth-order approximation for 𝑅-aperture magnetic currents 𝑀₀ and the corresponding coupling electromagnetic fields 𝐸₀, 𝐻₀ in a cavity due to 𝑀₀.

Theoretically, a plane wave excitation represents the simplest electromagnetic source and therefore is particularly suitable to test numerical techniques. In this paper, without lose of practical applications, we will use a harmonic plane wave with a wide frequency spectrum to simplify the problem under consideration. Consider a rectangular cavity with 𝑅 rectangular apertures illuminated by a harmonic plane wave shown in Figure 1. This field is described by angles of incidence 𝜓 and 𝜙, as well as a polarization angles𝛼, which defines the E-field vector direction with respect to the vertical plane of the incidence. It is given as𝐻^𝑖=𝐻_0𝑥𝑥̂+𝐻_0𝑦𝑦̂+𝐻_0𝑧𝑧̂𝑒^−𝑗𝑘0𝑘̂^𝑖_⋅𝑟,𝐸^𝑖=𝑍₀𝐻^𝑖×𝑘̂^𝑖,(3.1) where 𝑟=𝑥𝑥̂+𝑦𝑦̂+𝑧𝑧̂ refers to the location of the field in the free space,𝑘̂^𝑖=−(sin𝜃cos𝜙𝑥̂+sin𝜃sin𝜙𝑦̂+cos𝜃𝑧)̂,(3.2)𝑘₀=𝜔√𝜇₀𝜀₀ is the free space wave number, and 𝑍₀ is the free space intrinsic impedance. For a unity amplitude electric field, the coefficients 𝐻_0𝑥, 𝐻_0𝑦, 𝐻_0𝑧 are given by𝐻_0𝑥=1𝑍₀𝐻(sin𝛼cos𝜃cos𝜙+cos𝛼sin𝜙),(3.3)_0𝑦=1𝑍₀𝐻(sin𝛼cos𝜃sin𝜙−cos𝛼cos𝜙),(3.4)_0𝑧=−1𝑍₀sin𝛼sin𝜃.(3.5)

We introduce equivalent magnetic currents 𝑀₀ on 𝑅 apertures as𝑀₀=^𝑅_𝑟=1𝑀_0𝑟=^𝑅_𝑟=1𝑀_0𝑟𝑥(𝑥,𝑦)𝑥̂+𝑀_0𝑟𝑦(𝑥,𝑦)𝑦̂,(3.6) where 𝑀_0𝑟 is an equivalent magnetic current on the 𝑟th aperture.

The equivalent magnetic current components now are expanded as𝑀_0𝑟𝑥(𝑥,𝑦)=^𝑃𝑟−1_𝑄𝑝=1^𝑟_𝑞=1𝑀𝑀_{0𝑟𝑥𝑝𝑞}Ψ_𝑟𝑝𝑞(𝑥,𝑦),_0𝑟𝑦(𝑥,𝑦)=^𝑃𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}Φ_𝑟𝑝𝑞(𝑥,𝑦),(3.7) whereΨΦ_𝑟𝑝𝑞(𝑥,𝑦)=𝑇_𝑝𝑥−𝑥_𝑎𝑟𝑃_𝑞𝑦−𝑦_𝑏𝑟,(3.8)_𝑟𝑝𝑞(𝑥,𝑦)=𝑃_𝑝𝑥−𝑥_𝑎𝑟𝑇_𝑞𝑦−𝑦_𝑏𝑟,(3.9) in which (𝑥_𝑎𝑟,𝑦_𝑏𝑟) is the coordinates of the lower left-hand corner of the 𝑟th aperture, (𝑝,𝑞) are integers used for multiaperture modes, 𝑇_𝑝(𝑡) and 𝑃_𝑞(𝑡) are triangular and pulse functions defined by𝑇_𝑝(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑡−(𝑝−1)Δ𝑡(𝑝+1Δ)𝑡Δ𝑡−𝑡,(𝑝−1)Δ𝑡≤𝑡≤𝑝Δ𝑡,0,Δ𝑡,𝑝||𝑡Δ−𝑡𝑝≤Δ𝑡𝑡||≤≥(Δ𝑝𝑡+1)Δ𝑡,(3.10) for 𝑝=1,2,…,𝑃_𝑟−1, and𝑃_𝑞(𝑡)=10,,(o𝑞th−er1w)iΔse𝑡,≤𝑡≤𝑞Δ𝑡,(3.11) for 𝑞=1,2,…,𝑄_𝑟.

Substituting (3.7) into expression (3.6), we derive equivalent magnetic currents as𝑀₀=^𝑅_𝑟=1⎡⎢⎢⎣^𝑃𝑟−1_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}Ψ_𝑟𝑝𝑞(𝑥,𝑦)𝑥̂+^𝑃𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}Φ_𝑟𝑝𝑞(𝑥,𝑦)𝑦⎤⎥⎥⎦̂.(3.12) To solve the unknown currents 𝑀₀, we use the continuity of tangential magnetic fields across 𝑅 apertures. Then, we have𝑧̂×𝐻^𝑎𝑀₀+𝐻^𝑖=−𝑧̂×𝐻^𝑏𝑀₀,𝑧=0,(3.13) where 𝐻^𝑎 refers to the corresponding exterior scattered magnetic field due to 𝑀₀, and 𝐻^𝑏 refers to the corresponding interior coupling magnetic field due to 𝑀₀.

The exterior scattered field can be expressed as the radiation caused by the equivalent magnetic current 𝑀₀, say,𝐻^𝑎𝑀₀=−𝑗𝑘⁰𝑌₀_∑^𝑅𝑟=1_𝑆𝑟2𝑀₀𝑟^⋅Γ₀𝑟;𝑟^𝑑𝑠^,(3.14) where 𝑆_𝑟 denotes the surface of the 𝑟th aperture and Γ₀(𝑟;𝑟^) is the dyadic Green's function in the free space, which readsΓ₀𝑟;𝑟^=𝐼^{0𝑥0033𝑓}+1||𝑘²⁰∇∇4𝑒𝜋||𝑟−𝑟^{−𝑗𝑘0|𝑟−𝑟′|}||,𝑟−𝑟^||=(𝑥−𝑥^)²+(𝑦−𝑦^)²,(3.15) in which 𝑟 and 𝑟^ represent the locations of both the field and source points on the aperture, respectively.

The interior field can be formulated as the radiation due to 𝑀₀ on 𝑅 apertures. Using the available dyadic Green's functions for the cavity, we can express the interior field as𝐸^𝑏𝑀₀=_{∑𝑅𝑟=1𝑆𝑟}∇×𝐺_𝐻𝑀⋅𝑀₀𝑑𝑠^,(3.16)𝐻^𝑏𝑀₀=𝑗𝜔𝜀_{∑𝑅𝑟=1𝑆𝑟}𝐺_𝐻𝑀⋅𝑀₀𝑑𝑠^,(3.17) where the dyadic Green's function is defined as𝐺_𝐻𝑀=−1𝑘²⁰𝑧𝑧̂𝛿̂𝑅−𝑅^−_𝑚,𝑛22−𝛿_𝑚𝑛⋅𝑎𝑏𝑘^2𝑐𝑘_𝑚𝑛sin𝑘_𝑚𝑛𝑐𝑀_𝑜𝑒(𝑧+𝑐)𝑀^𝑜𝑒(0)+𝑁_𝑒𝑜(𝑧+𝑐)𝑁^𝑒𝑜(0),(3.18) where 𝛿_𝑚𝑛 denotes the Kronecker delta, say, 𝛿_𝑚𝑛=1 for 𝑚 or 𝑛=0, and zero otherwise. Also, the wave functions within representation (3.18) are given by𝑀_𝑜𝑒(𝑧)=−𝑘𝑘_𝑦𝑛s_𝑥𝑚in𝑘_𝑥𝑚𝑥cos𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛𝑧𝑥̂cos𝑘_𝑥𝑚𝑥sin𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛𝑧𝑦,̂𝑁_𝑒𝑜(𝑧)=1𝑘_𝑏−𝑘_𝑚𝑛−𝑘_𝑚𝑛𝑘_𝑥𝑚sin𝑘_𝑥𝑚𝑥cos𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛𝑧𝑥̂+𝑘_2𝑐𝑘_𝑦𝑛cos𝑘_𝑥𝑚𝑥sin𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛𝑧𝑦̂cos𝑘_𝑥𝑚𝑥cos𝑘_𝑦𝑛𝑦sin𝑘_𝑚𝑛𝑧𝑧̂.(3.19) As usual, 𝑘_𝑏=𝑘₀√𝜇_𝑏𝜀_𝑏=𝜔√𝜇𝜀; 𝑘_𝑥𝑚=𝑚𝜋/𝑎, 𝑘_𝑦𝑛=𝑛𝜋/𝑏, 𝑘^2𝑐=𝑘^2𝑥𝑚+𝑘^2𝑦𝑛, and 𝑘_𝑚𝑛=⎧⎪⎨⎪⎩−𝑗𝑘^2𝑐−𝑘²⁰,𝑘²⁰<𝑘^2𝑐,𝑘²⁰−𝑘^2𝑐,𝑘²⁰>𝑘^2𝑐,(3.20) with 𝑚,𝑛 being nonnegative integers excluding 𝑚=𝑛=0.

Substituting (3.18) into expressions (3.16) and (3.17) yields all components of the electromagnetic field inside the cavity as𝐸^𝑏0𝑥(𝑥,𝑦,𝑧)=−^𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛𝑎𝑏sin𝑘_𝑚𝑛𝑐^𝑃𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛}𝐸⋅cos𝑘_𝑥𝑚𝑥sin𝑘_𝑦𝑛𝑦sin𝑘_𝑚𝑛(𝑧+𝑐)_𝑏0𝑦,(𝑥,𝑦,𝑧)=^𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛𝑎𝑏sin𝑘_𝑚𝑛𝑐^𝑃𝑟−1_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}𝐸⋅sin𝑘_𝑥𝑚𝑥cos𝑘_𝑦𝑛𝑦sin𝑘_𝑚𝑛(𝑧+𝑐)_𝑏0𝑧,(𝑥,𝑦,𝑧)=−^𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛⋅⎡⎢⎢⎣𝑘_{𝑃𝑦𝑛𝑟−1}𝑎𝑏𝑘_𝑚𝑛sin𝑘_𝑚𝑛𝑐_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}−𝑘_𝑃𝑥𝑚^𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛}⋅sin𝑘_𝑥𝑚⎤⎥⎥⎦𝐻𝑥sin𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛(𝑧+𝑐)_𝑏0𝑥,(𝑥,𝑦,𝑧)=−𝑗𝑌_𝑏𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛⋅⎡⎢⎢⎣𝑘_2𝑏𝑎𝑏𝑘_𝑚𝑛𝑘_𝑏sin𝑘_𝑚𝑛𝑐−𝑘^2𝑥𝑚^𝑃𝑟−1_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}+−𝑘_𝑥𝑚𝑘_𝑦𝑛^𝑃𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛}⋅sin𝑘_𝑥𝑚⎤⎥⎥⎦𝐻𝑥cos𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛(𝑧+𝑐)_𝑏0𝑦,(𝑥,𝑦,𝑧)=−𝑗𝑌_𝑏𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛⋅⎡⎢⎢⎣−𝑘_𝑥𝑚𝑎𝑏𝑘_𝑚𝑛𝑘_𝑏sin𝑘_𝑚𝑛𝑐𝑘_𝑦𝑛^𝑃𝑟−1_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}+𝑘^2𝑏−𝑘^2𝑦𝑛^𝑃𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛}⋅cos𝑘_𝑥𝑚⎤⎥⎥⎦𝐻𝑥sin𝑘_𝑦𝑛𝑦cos𝑘_𝑚𝑛(𝑧+𝑐)_𝑏0𝑧,(𝑥,𝑦,𝑧)=𝑗𝑌_𝑏𝑅_𝑟=1_𝑚,𝑛22−𝛿_𝑚𝑛⋅⎡⎢⎢⎣𝑘_{𝑃𝑥𝑚𝑟−1}𝑎𝑏𝑘_𝑏sin𝑘_𝑚𝑛𝑐_𝑄𝑝=1^𝑟_𝑞=1𝑀_{0𝑟𝑥𝑝𝑞}𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}+𝑘_𝑃𝑦𝑛^𝑟_𝑄𝑝=1^𝑟−1_𝑞=1𝑀_{0𝑟𝑦𝑝𝑞}𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛}⋅cos𝑘_𝑥𝑚⎤⎥⎥⎦𝑥cos𝑘_𝑦𝑛𝑦sin𝑘_𝑚𝑛(𝑧+𝑐),(3.21) where 𝑌_𝑏=1/𝑍_𝑏=(1/𝑍₀)√𝜀_𝑏/𝜇_𝑏=√𝜀/𝜇, 𝑘_𝑏=𝑘₀√𝜇_𝑏𝜀_𝑏=𝜔√𝜇𝜀, and𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}=8sin²𝑘_𝑥𝑚Δ𝑥_𝑟/2sin𝑘_𝑦𝑛Δ𝑦_𝑟/2𝑘^2𝑥𝑚𝑘_𝑦𝑛Δ𝑥_𝑟⋅sin𝑘_𝑥𝑚𝑥_𝑎𝑟+𝑝Δ𝑥_𝑟cos𝑘_𝑦𝑛𝑦^𝑏_𝑟+𝑞−1𝐼2Δ𝑦^𝑟_{𝑟𝑦𝑝𝑞𝑚𝑛},=8sin²𝑘_𝑦𝑛Δ𝑦_𝑟/2sin𝑘_𝑥𝑚Δ𝑥_𝑟/2𝑘^2𝑦𝑛𝑘_𝑥𝑚Δ𝑦_𝑟⋅sin𝑘_𝑦𝑛𝑦_𝑏𝑟+𝑞Δ𝑦_𝑟cos𝑘_𝑥𝑚𝑥^𝑎_𝑟+𝑝−12Δ𝑥^𝑟.(3.22)𝑀_{0𝑟𝑥𝑝𝑞} and 𝑀_{0𝑟𝑦𝑝𝑞} are the unknown constants of the 𝑝𝑞 mode on the 𝑟th aperture, 𝑤_𝑟, 𝑙_𝑟 are the width and length of the 𝑟th aperture, respectively. Δ𝑥_𝑟, Δ𝑦_𝑟 refer to the width and length of the surface element used to discretize the 𝑟th aperture.

Using Galerkin's method (see [12]), the integral equation to be solved for 𝑀_{0𝑟𝑥𝑝𝑞}, 𝑀_{0𝑟𝑦𝑝𝑞} is_∑^𝑅𝑟=1_𝑆𝑟𝑧̂×𝐻^𝑎𝑀₀+𝐻^𝑏𝑀₀⋅𝑊𝑑𝑠^=−_{∑𝑅𝑟=1𝑆𝑟}𝑧̂×𝐻^𝑖⋅𝑊𝑑𝑠^,𝑧=0,(3.23) where 𝑊 is a weighting function, 𝐻^𝑎(𝑀₀) and 𝐻^𝑏(𝑀₀) are given by (3.14) and (3.17), respectively. To discretize (3.23), the corresponding weighing functions are given by𝑊_{𝑟′𝑝′𝑞′}=Φ_{𝑟′𝑝′𝑞′}(𝑥,𝑦)𝑥̂+Ψ_{𝑟′𝑝′𝑞′}(𝑥,𝑦)𝑦.̂(3.24) where Φ_{𝑟′𝑝′𝑞′}(𝑥,𝑦) and Ψ_{𝑟′𝑝′𝑞′}(𝑥,𝑦) are given by (3.8) and (3.9), respectively. Substitution of (3.14) and (3.17) into (3.23) yields a matrix equation:𝑌^𝑎+𝑏𝑀₀=𝐶^inc.(3.25) The expressions of all matrix entries in (3.25) are as follows.

Self-admittance matrix [𝑌^𝑏] is represented by𝑌^𝑏11_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=𝑗𝜔𝜀⁰𝑘^2𝑏_𝑚,𝑛𝜀_𝑚𝜀_𝑛𝑘^2𝑏−𝑘^2𝑥𝑚𝑎𝑏𝑘_𝑚𝑛tan𝑘_𝑚𝑛𝑐𝐼^{𝑟𝑥𝑝𝑞𝑚𝑛}𝑌_𝑏12𝐼_{𝑟′𝑥𝑝′𝑞′𝑚𝑛},_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=𝑗𝜔𝜀⁰𝑘^2𝑏_𝑚,𝑛𝜀_𝑚𝜀_𝑛𝑘_𝑥𝑚𝑘_𝑦𝑛𝑎𝑏𝑘_𝑚𝑛tan𝑘_𝑚𝑛𝑐𝐼^{𝑟𝑦𝑝𝑞𝑚𝑛}𝑌_𝑏21𝐼_{𝑟′𝑥𝑝′𝑞′𝑚𝑛},_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=𝑗𝜔𝜀⁰𝑘^2𝑏_𝑚,𝑛𝜀_𝑚𝜀_𝑛𝑘_𝑥𝑚𝑘_𝑦𝑛𝑎𝑏𝑘_𝑚𝑛tan𝑘_𝑚𝑛𝑐𝐼^{𝑟𝑥𝑝𝑞𝑚𝑛}𝑌_𝑏22𝐼_{𝑟′𝑦𝑝′𝑞′𝑚𝑛},_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=𝑗𝜔𝜀⁰𝑘^2𝑏_𝑚,𝑛𝜀_𝑚𝜀_𝑛𝑘^2𝑏−𝑘^2𝑦𝑛𝑎𝑏𝑘_𝑚𝑛tan𝑘_𝑚𝑛𝑐𝐼^{𝑟𝑦𝑝𝑞𝑚𝑛}𝐼_{𝑟′𝑦𝑝′𝑞′𝑚𝑛},(3.26) where 𝜀_𝑚 is Neumann's numbers, and 𝐼_{𝑟𝑥𝑝𝑞𝑚𝑛}, 𝐼_{𝑟𝑦𝑝𝑞𝑚𝑛} are given by (3.22).

External admittance matrix [𝑌^𝑎] is represented by(𝑌^𝑎11)_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=−𝜔𝜀⁰4𝜋𝑘₀²^+∞−∞𝑘²⁰−𝑘^2𝑥𝑚𝑘_𝑚𝑛(𝑌_𝑎12𝐹_𝑟𝑝𝑞𝐹^{∗𝑟𝑝𝑞}𝑑𝑘_𝑥𝑚𝑑𝑘_𝑦𝑛,)_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=−𝜔𝜀⁰4𝜋𝑘₀²^+∞−∞𝑘_𝑥𝑚𝑘_𝑦𝑛𝑘_𝑚𝑛(𝑌_𝑎21𝐺_𝑟𝑝𝑞𝐹^∗𝑟_{′𝑝′𝑞′}𝑑𝑘_𝑥𝑚𝑑𝑘_𝑦𝑛,)_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=−𝜔𝜀⁰4𝜋𝑘₀²^+∞−∞𝑘_𝑥𝑚𝑘_𝑦𝑛𝑘_𝑚𝑛(𝑌_𝑎22𝐹_𝑟𝑝𝑞𝐺^∗𝑟_{′𝑝′𝑞′}𝑑𝑘_𝑥𝑚𝑑𝑘_𝑦𝑛,)_{𝑟𝑝𝑞𝑟′𝑝′𝑞′}=−𝜔𝜀⁰4𝜋𝑘₀²^+∞−∞𝑘²⁰−𝑘^2𝑥𝑚𝑘_𝑚𝑛𝐺_𝑟𝑝𝑞𝐺^{∗𝑟𝑝𝑞}𝑑𝑘_𝑥𝑚𝑑𝑘_𝑦𝑛,(3.27) where 𝐹_𝑟𝑝𝑞(𝑘_𝑥𝑚,𝑘_𝑦𝑛) and 𝐺_𝑟𝑝𝑞(𝑘_𝑥𝑚,𝑘_𝑦𝑛) are the Fourier transforms of Ψ_𝑟𝑝𝑞(𝑥,𝑦) and Φ_𝑟𝑝𝑞(𝑥,𝑦), respectively. By replacing 𝑘_𝑥𝑚, 𝑘_𝑦𝑛 by −𝑘_𝑥𝑚, −𝑘_𝑦𝑛, the expressions 𝐹^{∗𝑟𝑝𝑞} and 𝐺^{∗𝑟𝑝𝑞} can be obtained from expressions of 𝐹_𝑟𝑝𝑞 and 𝐺_𝑟𝑝𝑞, respectively.

Admittance matrix [𝑌^𝑎+𝑏] is represented by𝑌^𝑎+𝑏=𝑌^𝑎+𝑌^𝑏(3.28)

Excitation vector [𝐶^inc] is represented by𝐶^inc_𝑥_𝑟𝑝𝑞=−2𝐻_0𝑥Δ𝑥_𝑟Δ𝑦_𝑟𝑒^{𝑗𝑘0sin𝜃[𝑝Δ𝑥𝑟+𝑥𝑎𝑟]cos𝜙+[(𝑞−1/2)Δ𝑦𝑟+𝑦𝑏𝑟]sin𝜙}⋅sinc²𝑘⁰Δ𝑥_𝑟sin𝜃cos𝜙2𝑠inc𝑘⁰Δ𝑦_𝑟sin𝜃sin𝜙𝐶_inc𝑦2,_𝑟𝑝𝑞=−2𝐻_0𝑦Δ𝑥_𝑟Δ𝑦_𝑟𝑒^{𝑗𝑘0sin𝜃[(𝑝−1/2)Δ𝑥𝑟+𝑥𝑎𝑟]cos𝜙+[𝑞Δ𝑦𝑟+𝑦𝑏𝑟]sin𝜙}⋅sinc𝑘⁰Δ𝑥_𝑟sin𝜃cos𝜙2sinc²𝑘⁰Δ𝑦_𝑟sin𝜃sin𝜙2,(3.29) where sinc(𝑥)=sin(𝑥)/𝑥, 𝐻_0𝑥 and 𝐻_0𝑦 are given by (3.3) and (3.4), respectively.

The solution of this matrix equation (3.25) yields the coefficients 𝑀_{0𝑟𝑥𝑝𝑞}, 𝑀_{0𝑟𝑦𝑝𝑞} of 𝑅-aperture magnetic currents 𝑀₀. A similar result can be found in [3, 11].

Step 2. Find 𝑅-aperture magnetic current increment Δ𝑀₁ due to 𝐸^𝑏0, 𝐻^𝑏0 and find the corresponding exterior scatter fields Δ𝐸^𝑎1, Δ𝐻^𝑎1 and interior coupling fields Δ𝐸^𝑏1, Δ𝐻^𝑏1 due to Δ𝑀₁.

To solve the unknown magnetic current increment Δ𝑀₁, we require the tangential magnetic field be continuous in the following sense:_∑^𝑅𝑟=1_𝑆𝑟𝑧̂×𝐻^𝑏𝑀₀+𝐻^𝑏Δ𝑀₁⋅𝑊𝑑𝑠=−_{∑𝑅𝑟=1𝑆𝑟}𝑧̂×𝐻^𝑎Δ𝑀₁⋅𝑊𝑑𝑠,(3.30) where𝐻^𝑏Δ𝑀₁=𝑗𝜔𝜀_{∑𝑅𝑟=1𝑆𝑟}𝐺_𝐻𝑀⋅Δ𝑀₁𝑑𝑠^,𝐻^𝑎Δ𝑀₁=−𝑗𝑘⁰𝑌₀_∑^𝑅𝑟=1_𝑆𝑟2Δ𝑀₁𝑟^⋅Γ₀𝑟;𝑟^𝑑𝑠^,(3.31)𝐻^𝑏(𝑀₀) and 𝑊 are given by (3.17) and (3.24), respectively. Substitution of (3.17), (3.24), and (3.31) into (3.30) yields a matrix equation:𝑌^𝑎+𝑏Δ𝑀₁=−𝑌^𝑏𝑀₀.(3.32)

The solution of this matrix equation (3.32) will yield the coefficients Δ𝑀_{1𝑟𝑥𝑝𝑞}, Δ𝑀_{1𝑟𝑦𝑝𝑞} of the magnetic current increment Δ𝑀₁.

Step 3. Find 𝑅-aperture magnetic current increment Δ𝑀₂ due to Δ𝐸^𝑎1, Δ𝐻^𝑎1 and find the corresponding exterior scatter fields Δ𝐸^𝑎2, Δ𝐻^𝑎2 and internal coupling fields Δ𝐸^𝑏2, Δ𝐻^𝑏2 due to Δ𝑀₂.

Similarly,_∑^𝑅𝑟=1_𝑆𝑟𝑧̂×𝐻^𝑏Δ𝑀₂⋅𝑊𝑑𝑠=−_{∑𝑅𝑟=1𝑆𝑟}𝑧̂×𝐻^𝑎Δ𝑀₁+𝐻^𝑎Δ𝑀₂⋅𝑊𝑑𝑠,(3.33) and a matrix equation is obtained as𝑌^𝑎+𝑏Δ𝑀₂=−𝑌^𝑎Δ𝑀₁.(3.34) So the solution of this matrix equation (3.34) will yield the coefficients Δ𝑀_{2𝑟𝑥𝑝𝑞}, Δ𝑀_{2𝑟𝑦𝑝𝑞} of the magnetic current increment Δ𝑀₂.

Step 4. Find 𝑅-aperture magnetic current increment Δ𝑀₃ due to Δ𝐸^𝑏2, Δ𝐻^𝑏2 and find the corresponding exterior scatter fields Δ𝐸^𝑎3, Δ𝐻^𝑎3 and internal coupling fields Δ𝐸^𝑏3, Δ𝐻^𝑏3 due to Δ𝑀₃.

Similarly,_∑^𝑅𝑟=1_𝑆𝑟𝑧̂×𝐻^𝑏Δ𝑀₂+𝐻^𝑏Δ𝑀₃⋅𝑊𝑑𝑠=−_{∑𝑅𝑟=1𝑆𝑟}𝑧̂×𝐻^𝑎Δ𝑀₃⋅𝑊𝑑𝑠,(3.35) and a matrix equation is obtained as:𝑌^𝑎+𝑏Δ𝑀₃=−𝑌^𝑏Δ𝑀₂.(3.36) So the solution of this matrix equation (3.36) will yield the coefficients Δ𝑀_{3𝑟𝑥𝑝𝑞}, Δ𝑀_{3𝑟𝑦𝑝𝑞} of the magnetic current increment Δ𝑀₃, and so on.

Finally, this iterative process approximates the magnetic currents on 𝑅 apertures and the total coupling electromagnetic field inside the cavity as𝑀=𝑀₀+^𝑁iter_𝑖=1Δ𝑀_𝑖,(3.37)𝐸^𝑏=𝐸^𝑏0+^𝑁iter_𝑖=1Δ𝐸^𝑏𝑖,(3.38)𝐻^𝑏=𝐻^𝑏0+^𝑁iter_𝑖=1Δ𝐻^𝑏𝑖.(3.39)

We must point out that the four-step algorithm admits high accuracy for the approximation to the interaction among apertures and the external-internal interaction.

4. The Application of the BLT Equation

The EMT can simulate the response of inner circuits of the electrically large complex system. The BLT equation is the key equation of the EMT to express the signals on the entire transmission line network [13]. Hence we use the BLT equation to solve the terminal responses of the two-wire line in a multiaperture cavity illuminated by a plane wave. We consider the case where a lossless two-wire line is illuminated by a coupling EM field. In order to simplify the notation, let us place the two-wire transmission line in the plane where 𝑦=𝑦₀, which parallels to the 𝑥-axis as shown in Figure 4. We put the reference wire at the plane where 𝑧=𝑧₀, and the other wire at 𝑧=𝑧₀+𝑑 so that there is distance 𝑑 between two wires. We assume that 𝑑 is greater than the wire radius 𝑟_𝑎(𝑑≫𝑟_𝑎). 𝜌_𝑖=(𝑍_𝑖−𝑍_𝑐)/(𝑍_𝑖+𝑍_𝑐)(𝑖=1,2) are the reflection coefficients at each node of the line. 𝐿 is the length of the line. 𝑍₁ and 𝑍₂ are the load impedances at the ends 𝑥=𝑥₀ and 𝑥=𝑥₀+𝐿, respectively, and 𝑍_𝑐=120ln(𝑑/𝑟_𝑎) is the characteristic impedance of the line. The transmission line theory shows that the wave propagation constant 𝛾=𝑗𝑘₀.

Figure 4 

The isolated two-wire line excited by the interior coupling field.

In this paper, we only consider the transmission line mode current. The BLT equation of the load currents and the total voltages at the loads can be expressed in the matrix form as𝐼𝑥⁰𝐼𝑥₀+𝐿=𝑍1_𝑐1−𝜌¹01−0𝜌₂−𝜌¹𝑒^𝛾𝐿𝑉𝑥₀𝑒^𝛾𝐿−𝜌₂⁻¹𝑆𝑆¹₂,𝑉𝑥₀+𝐿=1+𝜌¹01+0𝜌₂−𝜌¹𝑒^𝛾𝐿𝑒^𝛾𝐿−𝜌₂⁻¹𝑆𝑆¹₂.(4.1)

We choose the Taylor formulation (see [14, 15]), since it consists of both voltage and current sources that are appropriate for localized excitations through apertures and the source vector given by𝑆𝑆¹₂=⎡⎢⎢⎢⎢⎣12^{𝑥0𝑥+𝐿}₀−1𝑒^𝛾𝑥𝑉_𝑆(𝑥)+𝑍_𝑐𝐼_𝑆(𝑥)𝑑𝑥2^{𝑥0𝑥+𝐿}₀𝑒^{𝛾(𝐿−𝑥)}𝑉_𝑆(𝑥)−𝑍_𝑐𝐼_𝑆(𝑥)𝑑𝑥⎤⎥⎥⎥⎥⎦,(4.2) in which voltage and current sources 𝑉_𝑠(𝑥) and 𝐼_𝑠(𝑥) are given by𝑉_𝑆(𝑥)=−𝑗𝜔𝜇₀^{𝑧0𝑧+𝑑}₀𝐼𝐻^𝑏𝑦𝑥,𝑦₀,𝑧𝑑𝑧,(4.3)_𝑆(𝑥)=−𝑗𝜔𝐶^{𝑧0𝑧+𝑑}₀𝐸^𝑏𝑧𝑥,𝑦₀,𝑧𝑑𝑧,(4.4) where𝐶=𝜋𝜀ln𝑑/𝑟_𝑎𝑑≫𝑟^𝑎(4.5) Substituting (3.38) into (4.4) and (3.39) into (4.3) then into (4.2) can determine the source vector. Thereafter, using the BLT equation, we obtain the induced voltage and current of a two-wire line at each load in a multiaperture cavity under excitation.

5. Simulation

We present some numerical examples to demonstrate the formulation given in the previous section. We consider the coupling in a cavity with two rectangular apertures on the plane 𝑧=0. There is a two-wire transmission line along the 𝑥 direction in this cavity, see Figure 1. We excite the cavity by a plane wave incident normal to the side of the box on which two apertures resides. The parameters are as follows: the cavity's size 𝑎=0.3 m, 𝑏=0.3 m, 𝑐=0.2 m, two identical apertures' size 𝑙=0.1 m, 𝑤=0.005 m, the location of the first aperture (0.15 m,0.12 m,0 m), and the location of the second aperture (0.15 m,0.18 m,0 m), the wire radius 𝑟_𝑎=0.0003 m, the wire length 𝐿=0.15 m, the wire separation distance 𝑑=0.02 m, the characteristic impedance 𝑍_𝑐≈503Ω, the load resistance 𝑍₁=𝑍₂=50Ω.

The two-wire transmission line can be excited by the interior coupling electromagnetic field. By employing the BLT equation, Figure 5 shows the results of the terminal response at Junction 4 with those parameters. This example indicates that the effect of the external coupling fields on the two-wire line current depends on both the multiaperture cavity and the electromagnetic interaction among apertures. More precisely, the difference between the results with the electromagnetic interaction and those without the electromagnetic interaction is actually not small at all. Therefore, this electromagnetic interaction cannot be simply neglected for the frequency range up to 1 GHz. Moreover, using the BLT equation, we can also conclude that the currents and voltages at arbitrary point on the two-wire line can be derived from the interior coupling EM fields.

Figure 5 

Comparison of induced currents of a two-wire line at load under excitation.

6. Conclusion

In this paper, the field coupling phenomenon between external fields and a two-wire line in the multiaperture cavity is studied. For the issue of the field penetration through apertures, the EMT theory and the multistep iteration are used to deal with these problems. This method can also be used for electromagnetic interaction problems on more complex system. Then, using the Modal Green's Function and the method of moments, the electromagnetic total coupling fields are determined inside a multiaperture cavity. Finally, the load response of the two-wire transmission line could be resolved by the BLT equation. The results with the electromagnetic interaction are largely different from those without the electromagnetic interaction for the low frequency.

Acknowledgments

This work was supported by China Postdoctoral Special Science Foundation (no. 200902662), the National Natural Science Foundation of China (no. 10871231), and Pre-research Foundation of Weapon and Equipment (no. 9140A31020609KG0170).