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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 349361, 14 pages
Research Article

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

Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha, Hunan 410073, China

Received 1 September 2010; Accepted 7 January 2011

Academic Editor: Alois Steindl

Copyright © 2011 Ying Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [13]). 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. 𝑉𝑖 denote these subvolumes, where “𝑖” indicates the hierarchical order of the volume. 𝑆𝑖,𝑗 is a surface separating volumes 𝑉𝑖 from 𝑉𝑗. 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]). 𝑊𝑖(0) is the outgoing wave and 𝑊𝑠 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 [𝑊𝑠] (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 as𝐻𝑖=𝐻0𝑥̂𝑥+𝐻0𝑦̂𝑦+𝐻0𝑧𝑒̂𝑧𝑗𝑘0̂𝑘𝑖𝑟,𝐸𝑖=𝑍0𝐻𝑖×̂𝑘𝑖,(3.1) where 𝑟=𝑥̂𝑥+𝑦̂𝑦+𝑧̂𝑧 refers to the location of the field in the free space,̂𝑘𝑖=(sin𝜃cos𝜙̂𝑥+sin𝜃sin𝜙̂𝑦+cos𝜃̂𝑧),(3.2)𝑘0=𝜔𝜇0𝜀0 is the free space wave number, and 𝑍0 is the free space intrinsic impedance. For a unity amplitude electric field, the coefficients 𝐻0𝑥, 𝐻0𝑦, 𝐻0𝑧 are given by𝐻0𝑥=1𝑍0𝐻(sin𝛼cos𝜃cos𝜙+cos𝛼sin𝜙),(3.3)0𝑦=1𝑍0𝐻(sin𝛼cos𝜃sin𝜙cos𝛼cos𝜙),(3.4)0𝑧1=𝑍0sin𝛼sin𝜃.(3.5)

We introduce equivalent magnetic currents 𝑀0 on 𝑅 apertures as𝑀0=𝑅𝑟=1𝑀0𝑟=𝑅𝑟=1𝑀0𝑟𝑥(𝑥,𝑦)̂𝑥+𝑀0𝑟𝑦,(𝑥,𝑦)̂𝑦(3.6) where 𝑀0𝑟 is an equivalent magnetic current on the 𝑟th aperture.

The equivalent magnetic current components now are expanded as𝑀0𝑟𝑥(𝑥,𝑦)=𝑃𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞Ψ𝑟𝑝𝑞𝑀(𝑥,𝑦),0𝑟𝑦(𝑥,𝑦)=𝑃𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞Φ𝑟𝑝𝑞(𝑥,𝑦),(3.7) whereΨ𝑟𝑝𝑞(𝑥,𝑦)=𝑇𝑝𝑥𝑥𝑎𝑟𝑃𝑞𝑦𝑦𝑏𝑟,Φ(3.8)𝑟𝑝𝑞(𝑥,𝑦)=𝑃𝑝𝑥𝑥𝑎𝑟𝑇𝑞𝑦𝑦𝑏𝑟,(3.9) in which (𝑥𝑎𝑟,𝑦𝑏𝑟) is the coordinates of the lower left-hand corner of the 𝑟th aperture, (𝑝,𝑞) are integers used for multiaperture modes, 𝑇𝑝(𝑡) and 𝑃𝑞(𝑡) are triangular and pulse functions defined by𝑇𝑝(𝑡)=𝑡(𝑝1)Δ𝑡(Δ𝑡,(𝑝1)Δ𝑡𝑡𝑝Δ𝑡,𝑝+1)Δ𝑡𝑡||||Δ𝑡,𝑝Δ𝑡𝑡(𝑝+1)Δ𝑡,0,𝑡𝑝Δ𝑡Δ𝑡(3.10) for 𝑝=1,2,,𝑃𝑟1, and𝑃𝑞(𝑡)=1,(𝑞1)Δ𝑡𝑡𝑞Δ𝑡,0,otherwise,(3.11) for 𝑞=1,2,,𝑄𝑟.

Substituting (3.7) into expression (3.6), we derive equivalent magnetic currents as𝑀0=𝑅𝑟=1𝑃𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞Ψ𝑟𝑝𝑞(𝑥,𝑦)̂𝑥+𝑃𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞Φ𝑟𝑝𝑞.(𝑥,𝑦)̂𝑦(3.12) To solve the unknown currents 𝑀0, we use the continuity of tangential magnetic fields across 𝑅 apertures. Then, we havê𝑧×𝐻𝑎𝑀0+𝐻𝑖=̂𝑧×𝐻𝑏𝑀0,𝑧=0,(3.13) where 𝐻𝑎 refers to the corresponding exterior scattered magnetic field due to 𝑀0, and 𝐻𝑏 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,𝐻𝑎𝑀0=𝑗𝑘0𝑌0𝑅𝑟=1𝑆𝑟2𝑀0𝑟Γ0𝑟;𝑟𝑑𝑠,(3.14) where 𝑆𝑟 denotes the surface of the 𝑟th aperture and Γ0(𝑟;𝑟) is the dyadic Green's function in the free space, which readsΓ0𝑟;𝑟=0𝑥0033𝑓𝐼+1𝑘20𝑒𝑗𝑘0|𝑟𝑟|||4𝜋𝑟𝑟||,||𝑟𝑟||=(𝑥𝑥)2+(𝑦𝑦)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 as𝐸𝑏𝑀0=𝑅𝑟=1𝑆𝑟×𝐺𝐻𝑀𝑀0𝑑𝑠,(3.16)𝐻𝑏𝑀0=𝑗𝜔𝜀𝑅𝑟=1𝑆𝑟𝐺𝐻𝑀𝑀0𝑑𝑠,(3.17) where the dyadic Green's function is defined as𝐺𝐻𝑀1=𝑘20̂𝑧̂𝑧𝛿𝑅𝑅𝑚,𝑛22𝛿𝑚𝑛𝑎𝑏𝑘2𝑐𝑘𝑚𝑛𝑘sin𝑚𝑛𝑐𝑀𝑜𝑒(𝑧+𝑐)𝑀𝑜𝑒(0)+𝑁𝑒𝑜(𝑧+𝑐)𝑁𝑒𝑜,(0)(3.18) where 𝛿𝑚𝑛 denotes the Kronecker delta, say, 𝛿𝑚𝑛=1 for 𝑚 or 𝑛=0, and zero otherwise. Also, the wave functions within representation (3.18) are given by𝑀𝑜𝑒(𝑧)=𝑘𝑦𝑛𝑘sin𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘cos𝑚𝑛𝑧̂𝑥𝑘𝑥𝑚𝑘cos𝑥𝑚𝑥𝑘sin𝑦𝑛𝑦𝑘cos𝑚𝑛𝑧̂𝑦,𝑁𝑒𝑜1(𝑧)=𝑘𝑏𝑘𝑚𝑛𝑘𝑥𝑚𝑘sin𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘cos𝑚𝑛𝑧̂𝑥𝑘𝑚𝑛𝑘𝑦𝑛𝑘cos𝑥𝑚𝑥𝑘sin𝑦𝑛𝑦𝑘cos𝑚𝑛𝑧̂𝑦+𝑘2𝑐𝑘cos𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘sin𝑚𝑛𝑧.̂𝑧(3.19) As usual, 𝑘𝑏=𝑘0𝜇𝑏𝜀𝑏=𝜔𝜇𝜀; 𝑘𝑥𝑚=𝑚𝜋/𝑎, 𝑘𝑦𝑛=𝑛𝜋/𝑏, 𝑘2𝑐=𝑘2𝑥𝑚+𝑘2𝑦𝑛, and 𝑘𝑚𝑛=𝑗𝑘2𝑐𝑘20,𝑘20<𝑘2𝑐,𝑘20𝑘2𝑐,𝑘20>𝑘2𝑐,(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 as𝐸𝑏0𝑥(𝑥,𝑦,𝑧)=𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑘𝑎𝑏sin𝑚𝑛𝑐𝑃𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝑘cos𝑥𝑚𝑥𝑘sin𝑦𝑛𝑦𝑘sin𝑚𝑛,𝐸(𝑧+𝑐)𝑏0𝑦(𝑥,𝑦,𝑧)=𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑘𝑎𝑏sin𝑚𝑛𝑐𝑃𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞𝐼𝑟𝑥𝑝𝑞𝑚𝑛𝑘sin𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘sin𝑚𝑛,𝐸(𝑧+𝑐)𝑏0𝑧(𝑥,𝑦,𝑧)=𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑎𝑏𝑘𝑚𝑛𝑘sin𝑚𝑛𝑐𝑘𝑃𝑦𝑛𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞𝐼𝑟𝑥𝑝𝑞𝑚𝑛𝑘𝑃𝑥𝑚𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝑘sin𝑥𝑚𝑥𝑘sin𝑦𝑛𝑦𝑘cos𝑚𝑛,𝐻(𝑧+𝑐)𝑏0𝑥(𝑥,𝑦,𝑧)=𝑗𝑌𝑏𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑎𝑏𝑘𝑚𝑛𝑘𝑏𝑘sin𝑚𝑛𝑐𝑘2𝑏𝑘2𝑥𝑚𝑃𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞𝐼𝑟𝑥𝑝𝑞𝑚𝑛+𝑘𝑥𝑚𝑘𝑦𝑛𝑃𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝑘sin𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘cos𝑚𝑛,𝐻(𝑧+𝑐)𝑏0𝑦(𝑥,𝑦,𝑧)=𝑗𝑌𝑏𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑎𝑏𝑘𝑚𝑛𝑘𝑏𝑘sin𝑚𝑛𝑐𝑘𝑥𝑚𝑘𝑦𝑛𝑃𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞𝐼𝑟𝑥𝑝𝑞𝑚𝑛+𝑘2𝑏𝑘2𝑦𝑛𝑃𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝑘cos𝑥𝑚𝑥𝑘sin𝑦𝑛𝑦𝑘cos𝑚𝑛(,𝐻𝑧+𝑐)𝑏0𝑧(𝑥,𝑦,𝑧)=𝑗𝑌𝑏𝑅𝑟=1𝑚,𝑛22𝛿𝑚𝑛𝑎𝑏𝑘𝑏𝑘sin𝑚𝑛𝑐𝑘𝑃𝑥𝑚𝑟1𝑄𝑝=1𝑟𝑞=1𝑀0𝑟𝑥𝑝𝑞𝐼𝑟𝑥𝑝𝑞𝑚𝑛+𝑘𝑃𝑦𝑛𝑟𝑄𝑝=1𝑟1𝑞=1𝑀0𝑟𝑦𝑝𝑞𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝑘cos𝑥𝑚𝑥𝑘cos𝑦𝑛𝑦𝑘sin𝑚𝑛,(𝑧+𝑐)(3.21) where 𝑌𝑏=1/𝑍𝑏=(1/𝑍0)𝜀𝑏/𝜇𝑏=𝜀/𝜇, 𝑘𝑏=𝑘0𝜇𝑏𝜀𝑏=𝜔𝜇𝜀, and𝐼𝑟𝑥𝑝𝑞𝑚𝑛=8sin2𝑘𝑥𝑚Δ𝑥𝑟𝑘/2sin𝑦𝑛Δ𝑦𝑟/2𝑘2𝑥𝑚𝑘𝑦𝑛Δ𝑥𝑟𝑘sin𝑥𝑚𝑥𝑎𝑟+𝑝Δ𝑥𝑟𝑘cos𝑦𝑛𝑦𝑏𝑟+1𝑞2Δ𝑦𝑟,𝐼𝑟𝑦𝑝𝑞𝑚𝑛=8sin2𝑘𝑦𝑛Δ𝑦𝑟𝑘/2sin𝑥𝑚Δ𝑥𝑟/2𝑘2𝑦𝑛𝑘𝑥𝑚Δ𝑦𝑟𝑘sin𝑦𝑛𝑦𝑏𝑟+𝑞Δ𝑦𝑟𝑘cos𝑥𝑚𝑥𝑎𝑟+1𝑝2Δ𝑥𝑟.(3.22)𝑀0𝑟𝑥𝑝𝑞 and 𝑀0𝑟𝑦𝑝𝑞 are the unknown constants of the 𝑝𝑞 mode on the 𝑟th aperture, 𝑤𝑟, 𝑙𝑟 are the width and length of the 𝑟th aperture, respectively. Δ𝑥𝑟, Δ𝑦𝑟 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 𝑀0𝑟𝑥𝑝𝑞, 𝑀0𝑟𝑦𝑝𝑞 is𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑎𝑀0+𝐻𝑏𝑀0𝑊𝑑𝑠=𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑖𝑊𝑑𝑠,𝑧=0,(3.23) where 𝑊 is a weighting function, 𝐻𝑎(𝑀0) and 𝐻𝑏(𝑀0) are given by (3.14) and (3.17), respectively. To discretize (3.23), the corresponding weighing functions are given by𝑊𝑟𝑝𝑞=Φ𝑟𝑝𝑞(𝑥,𝑦)̂𝑥+Ψ𝑟𝑝𝑞(𝑥,𝑦)̂𝑦.(3.24) where Φ𝑟𝑝𝑞(𝑥,𝑦) and Ψ𝑟𝑝𝑞(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:𝑌𝑎+𝑏𝑀0=𝐶inc.(3.25) The expressions of all matrix entries in (3.25) are as follows.

Self-admittance matrix [𝑌𝑏] is represented by𝑌𝑏11𝑟𝑝𝑞𝑟𝑝𝑞=𝑗𝜔𝜀0𝑘2𝑏𝑚,𝑛𝜀𝑚𝜀𝑛𝑘2𝑏𝑘2𝑥𝑚𝑎𝑏𝑘𝑚𝑛𝑘tan𝑚𝑛𝑐𝐼𝑟𝑥𝑝𝑞𝑚𝑛𝐼𝑟𝑥𝑝𝑞𝑚𝑛,𝑌𝑏12𝑟𝑝𝑞𝑟𝑝𝑞=𝑗𝜔𝜀0𝑘2𝑏𝑚,𝑛𝜀𝑚𝜀𝑛𝑘𝑥𝑚𝑘𝑦𝑛𝑎𝑏𝑘𝑚𝑛𝑘tan𝑚𝑛𝑐𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝐼𝑟𝑥𝑝𝑞𝑚𝑛,𝑌𝑏21𝑟𝑝𝑞𝑟𝑝𝑞=𝑗𝜔𝜀0𝑘2𝑏𝑚,𝑛𝜀𝑚𝜀𝑛𝑘𝑥𝑚𝑘𝑦𝑛𝑎𝑏𝑘𝑚𝑛𝑘tan𝑚𝑛𝑐𝐼𝑟𝑥𝑝𝑞𝑚𝑛𝐼𝑟𝑦𝑝𝑞𝑚𝑛,𝑌𝑏22𝑟𝑝𝑞𝑟𝑝𝑞=𝑗𝜔𝜀0𝑘2𝑏𝑚,𝑛𝜀𝑚𝜀𝑛𝑘2𝑏𝑘2𝑦𝑛𝑎𝑏𝑘𝑚𝑛𝑘tan𝑚𝑛𝑐𝐼𝑟𝑦𝑝𝑞𝑚𝑛𝐼𝑟𝑦𝑝𝑞𝑚𝑛,(3.26) where 𝜀𝑚 is Neumann's numbers, and 𝐼𝑟𝑥𝑝𝑞𝑚𝑛, 𝐼𝑟𝑦𝑝𝑞𝑚𝑛 are given by (3.22).

External admittance matrix [𝑌𝑎] is represented by(𝑌𝑎11)𝑟𝑝𝑞𝑟𝑝𝑞=𝜔𝜀04𝜋𝑘02+𝑘20𝑘2𝑥𝑚𝑘𝑚𝑛𝐹𝑟𝑝𝑞𝐹𝑟𝑝𝑞𝑑𝑘𝑥𝑚𝑑𝑘𝑦𝑛,(𝑌𝑎12)𝑟𝑝𝑞𝑟𝑝𝑞=𝜔𝜀04𝜋𝑘02+𝑘𝑥𝑚𝑘𝑦𝑛𝑘𝑚𝑛𝐺𝑟𝑝𝑞𝐹𝑟𝑝𝑞𝑑𝑘𝑥𝑚𝑑𝑘𝑦𝑛,(𝑌𝑎21)𝑟𝑝𝑞𝑟𝑝𝑞=𝜔𝜀04𝜋𝑘02+𝑘𝑥𝑚𝑘𝑦𝑛𝑘𝑚𝑛𝐹𝑟𝑝𝑞𝐺𝑟𝑝𝑞𝑑𝑘𝑥𝑚𝑑𝑘𝑦𝑛,(𝑌𝑎22)𝑟𝑝𝑞𝑟𝑝𝑞=𝜔𝜀04𝜋𝑘02+𝑘20𝑘2𝑥𝑚𝑘𝑚𝑛𝐺𝑟𝑝𝑞𝐺𝑟𝑝𝑞𝑑𝑘𝑥𝑚𝑑𝑘𝑦𝑛,(3.27) where 𝐹𝑟𝑝𝑞(𝑘𝑥𝑚,𝑘𝑦𝑛) and 𝐺𝑟𝑝𝑞(𝑘𝑥𝑚,𝑘𝑦𝑛) are the Fourier transforms of Ψ𝑟𝑝𝑞(𝑥,𝑦) and Φ𝑟𝑝𝑞(𝑥,𝑦), respectively. By replacing 𝑘𝑥𝑚, 𝑘𝑦𝑛 by 𝑘𝑥𝑚, 𝑘𝑦𝑛, the expressions 𝐹𝑟𝑝𝑞 and 𝐺𝑟𝑝𝑞 can be obtained from expressions of 𝐹𝑟𝑝𝑞 and 𝐺𝑟𝑝𝑞, respectively.

Admittance matrix [𝑌𝑎+𝑏] is represented by𝑌𝑎+𝑏=𝑌𝑎+𝑌𝑏(3.28)

Excitation vector [𝐶inc] is represented by𝐶inc𝑥𝑟𝑝𝑞=2𝐻0𝑥Δ𝑥𝑟Δ𝑦𝑟𝑒𝑗𝑘0sin𝜃[𝑝Δ𝑥𝑟+𝑥𝑎𝑟]cos𝜙+[(𝑞1/2)Δ𝑦𝑟+𝑦𝑏𝑟]sin𝜙sinc2𝑘0Δ𝑥𝑟sin𝜃cos𝜙2𝑘𝑠inc0Δ𝑦𝑟sin𝜃sin𝜙2,𝐶inc𝑦𝑟𝑝𝑞=2𝐻0𝑦Δ𝑥𝑟Δ𝑦𝑟𝑒𝑗𝑘0sin𝜃[(𝑝1/2)Δ𝑥𝑟+𝑥𝑎𝑟]cos𝜙+[𝑞Δ𝑦𝑟+𝑦𝑏𝑟]sin𝜙𝑘sinc0Δ𝑥𝑟sin𝜃cos𝜙2sinc2𝑘0Δ𝑦𝑟sin𝜃sin𝜙2,(3.29) where sinc(𝑥)=sin(𝑥)/𝑥, 𝐻0𝑥 and 𝐻0𝑦 are given by (3.3) and (3.4), respectively.

The solution of this matrix equation (3.25) yields the coefficients 𝑀0𝑟𝑥𝑝𝑞, 𝑀0𝑟𝑦𝑝𝑞 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 𝐸𝑏0, 𝐻𝑏0 and find the corresponding exterior scatter fields Δ𝐸𝑎1, Δ𝐻𝑎1 and interior coupling fields Δ𝐸𝑏1, Δ𝐻𝑏1 due to Δ𝑀1.

To solve the unknown magnetic current increment Δ𝑀1, we require the tangential magnetic field be continuous in the following sense:𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑏𝑀0+𝐻𝑏Δ𝑀1𝑊𝑑𝑠=𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑎Δ𝑀1𝑊𝑑𝑠,(3.30) where𝐻𝑏Δ𝑀1=𝑗𝜔𝜀𝑅𝑟=1𝑆𝑟𝐺𝐻𝑀Δ𝑀1𝑑𝑠,𝐻𝑎Δ𝑀1=𝑗𝑘0𝑌0𝑅𝑟=1𝑆𝑟2Δ𝑀1𝑟Γ0𝑟;𝑟𝑑𝑠,(3.31)𝐻𝑏(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:𝑌𝑎+𝑏Δ𝑀1𝑌=𝑏𝑀0.(3.32)

The solution of this matrix equation (3.32) will yield the coefficients Δ𝑀1𝑟𝑥𝑝𝑞, Δ𝑀1𝑟𝑦𝑝𝑞 of the magnetic current increment Δ𝑀1.

Step 3. Find 𝑅-aperture magnetic current increment Δ𝑀2 due to Δ𝐸𝑎1, Δ𝐻𝑎1 and find the corresponding exterior scatter fields Δ𝐸𝑎2, Δ𝐻𝑎2 and internal coupling fields Δ𝐸𝑏2, Δ𝐻𝑏2 due to Δ𝑀2.

Similarly,𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑏Δ𝑀2𝑊𝑑𝑠=𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑎Δ𝑀1+𝐻𝑎Δ𝑀2𝑊𝑑𝑠,(3.33) and a matrix equation is obtained as𝑌𝑎+𝑏Δ𝑀2𝑌=𝑎Δ𝑀1.(3.34) So the solution of this matrix equation (3.34) will yield the coefficients Δ𝑀2𝑟𝑥𝑝𝑞, Δ𝑀2𝑟𝑦𝑝𝑞 of the magnetic current increment Δ𝑀2.

Step 4. Find 𝑅-aperture magnetic current increment Δ𝑀3 due to Δ𝐸𝑏2, Δ𝐻𝑏2 and find the corresponding exterior scatter fields Δ𝐸𝑎3, Δ𝐻𝑎3 and internal coupling fields Δ𝐸𝑏3, Δ𝐻𝑏3 due to Δ𝑀3.

Similarly,𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑏Δ𝑀2+𝐻𝑏Δ𝑀3𝑊𝑑𝑠=𝑅𝑟=1𝑆𝑟̂𝑧×𝐻𝑎Δ𝑀3𝑊𝑑𝑠,(3.35) and a matrix equation is obtained as:𝑌𝑎+𝑏Δ𝑀3𝑌=𝑏Δ𝑀2.(3.36) So the solution of this matrix equation (3.36) will yield the coefficients Δ𝑀3𝑟𝑥𝑝𝑞, Δ𝑀3𝑟𝑦𝑝𝑞 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+𝑁iter𝑖=1Δ𝑀𝑖,(3.37)𝐸𝑏=𝐸𝑏0+𝑁iter𝑖=1Δ𝐸𝑏𝑖,(3.38)𝐻𝑏=𝐻𝑏0+𝑁iter𝑖=1Δ𝐻𝑏𝑖.(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 𝑟𝑎(𝑑𝑟𝑎). 𝜌𝑖=(𝑍𝑖𝑍𝑐)/(𝑍𝑖+𝑍𝑐)(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 𝑍𝑐=120ln(𝑑/𝑟𝑎) 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 as𝐼𝑥0𝐼𝑥0=1+𝐿𝑍𝑐1𝜌1001𝜌2𝜌1𝑒𝛾𝐿𝑒𝛾𝐿𝜌21𝑆1𝑆2,𝑉𝑥0𝑉𝑥0=+𝐿1+𝜌1001+𝜌2𝜌1𝑒𝛾𝐿𝑒𝛾𝐿𝜌21𝑆1𝑆2.(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 by𝑆1𝑆2=12𝑥0𝑥+𝐿0𝑒𝛾𝑥𝑉𝑆(𝑥)+𝑍𝑐𝐼𝑆1(𝑥)𝑑𝑥2𝑥0𝑥+𝐿0𝑒𝛾(𝐿𝑥)𝑉𝑆(𝑥)𝑍𝑐𝐼𝑆(𝑥)𝑑𝑥,(4.2) in which voltage and current sources 𝑉𝑠(𝑥) and 𝐼𝑠(𝑥) are given by𝑉𝑆(𝑥)=𝑗𝜔𝜇0𝑧0𝑧+𝑑0𝐻𝑏𝑦𝑥,𝑦0𝐼,𝑧𝑑𝑧,(4.3)𝑆(𝑥)=𝑗𝜔𝐶𝑧0𝑧+𝑑0𝐸𝑏𝑧𝑥,𝑦0,𝑧𝑑𝑧,(4.4) where𝐶=𝜋𝜀ln𝑑/𝑟𝑎𝑑𝑟𝑎(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 𝑟𝑎=0.0003 m, the wire length 𝐿=0.15 m, the wire separation distance 𝑑=0.02 m, the characteristic impedance 𝑍𝑐503Ω, 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.


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).


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