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导体球涂覆各向异性铁氧体介质电磁散射的解析解 耿友林 吴信宝 官伯然(杭州电子科技大学 天线与微波研究所 杭州 310018) (上海微波技术研究所 上海 200063) 摘 要 该文用球矢量波函数对各向异性铁氧体介质涂覆导体球的电磁散射解析解开展研究。各向异性铁氧体介质中电磁场的球矢量波函数解可表示成第一、二、三、四类球矢量波函数之和。根据球Bessel函数的性质，可以得出导体球涂覆各向异性铁氧体介质的球矢量波函数解析解。应用铁氧体与自由空间分界面上电磁场切向连续和在导体球面上切向电场等于零的边界条件以及球矢量波函数切向正交性质，可分别得出铁氧体介质中电磁场和散射场的展开系数。给出了平面波入射情况下的数值计算结果。该文的结果可应用于有关微波器件、天线以及目标特征的分析和计算。 关键词 球矢量波函数，铁氧体，各向异性 中图分类号：TM277 文献标识码：A 文章编号：1009-5896(2006)09-1740-04 The Analytical Solution to the Electromagnetic Scattering by an Anisotropic Ferrite-Coated Conducting Sphere Geng You-lin Wu Xin-bao Guan Bo-ran(The Institute of Antenna and Microwaves, Hangzhou Dianzi University, Hangzhou 310018, China) (Shanghai Research Institute of Microwave Technology, Shanghai 200063, China) Abstract The analytical solution of electromagnetic scattering to an anisotropic ferrite-coated conducting sphere by a plane wave is discussed in this paper. It is well known that the electromagnetic fields in three-dimensional ferrite medium and free space can be expressed into spherical vector wave functions of the first, second, third and fourth in ferrite anisotropic medium and isotropic medium. Applying the continue boundary condition of tangential component of electromagnetic fields in the interface between the ferrite anisotropic medium and free space, and the tangential electric field vanishing in the interface of the conducting sphere, the expansion coefficients of electromagnetic fields with spherical vector wave function in ferrite medium and the scattering fields in free space can be derived. The theoretic analysis and numerical result show that when the radius of conducting sphere approaches zero, the present method can be reduced to that of the homogeneous ferrite anisotropic medium. The present method can be applied to the analyses of related microwave devices, antennas and the character of radar targets. Key words Spherical vector wave function, Ferrite medium, Anisotropy 1 preface Currently, the ferrite in the microwave and millimeter-wave devices (such as circulators, isolators, switches and phase shifters) has been widely used. Its material properties can be with the applied field strength and direction of the magnetic field changes not only the unique nature of the ferromagnetic material used in making microwave integrated circuits, but also used in antenna technology. When not considering the role of the external magnetic field, the ferrite can be degraded as a general isotropic media. In this case, Mie theory can be used for conducting sphere coated ferrite isotropic scattering media research 1; and when taking into account the anisotropic ferrite medium, according to the literature to our knowledge see, only two-dimensional column coated with anisotropic ferrite medium to carry out a study 2; In three-dimensional case, only the uniform ferrite media interaction with the electromagnetic field research carried out; which were based on differential equation finite-difference time-domain (FDTD) method 3,4 ;and integral equations method of moments combined with a total of conjugate gradient and Fourier transform (MOM-CG-FFT) and other numerical methods 5, there are vector based on the intrinsic and the analytical solution of Fourier transform method 6,7 .As in 1 above, we have not found the electromagnetic field and anisotropic ferrite coated conducting sphere interaction theory and numerical results.7 based on Fourier transform and expand the plane wave factor, based on the ball with the intrinsic anisotropy vector wave function of the uniform ferrite media to carry out a theoretical study. The numerical results with the MOM-CG-FFT calculation results are compared, given the electrically large electromagnetic scattering anisotropy ferrite sphere the numerical results. In this paper, homogeneous ferrite anisotropic media intrinsic ball vector wave function on the basis of the analytical solution 7, the nature of the use of Bessel functions, gives the case of plane wave incident with wave vector expressed as a function of the ball ferrite media coated conductor ball ball intrinsic analytic expression vector wave functions and the scattering vector wave function of the analytical expressions.The use of electromagnetic field boundary conditions and cut the ball to the continuous wave function vector orthogonal nature of the tangential component can be drawn from the ball to start the electromagnetic field vector wave function expansion coefficients. Theoretical analysis and numerical results show that when the conductor radius trend style at 0:00, the text of the results can be degraded to a uniform anisotropic ferrite ball situation. This article also gives electrically large numerical results. This result can be applied to microwave devices, antennas, and other aspects of stealth targets .2 Theoretical formula This paper studied the problem shown in Figure 1, conducting sphere coated with anisotropic ferrite medium is located in free space, divided into three regions, namely the region 0, free space; region 1, anisotropic ferrite media and zone 2, conducting sphere. The selected coordinate system shown in Figure 1, the origin of rectangular coordinates of the center of the sphere in the conducting sphere, located outside the role of ferrite magnetic field direction along the positive z-axis, anisotropic ferrite spherical shell of external radius are a2 and a1, the incident plane wave incident along the z-axis direction, the magnitude of an electric field, the polarization direction along the positive x-axis direction.Anisotropic ferrite dielectric parameters 0 (vacuum permittivity), the corresponding permeability , can be expressed as 3-7.1, 2, 3 is added by the external magnetic field, electromagnetic field frequency and other parameters of the function, the specific expression can be found in the literature 2-4. This article does not involve a specific function expression, it is not given a detailed analysis of its expressionBy the Maxwell equations known anisotropic ferrite passive media (area 1) differential equation satisfied by the magnetic field is.In the type (3a) and (3b) in, n and n is from 0 to + , the sum, m-n to n from the sum, in spherical coordinates, k is the direction vector is (,k,k), and the r vector in the direction of (, ).M(l)mn(l,k) ,N(l)mn(l,k)and L(l)mn(l,k)l like the ball for the first vector wave functions, which are 6-8.zn(l)ball for the first-order Bessel function of type l, respectively, jn(x), yn(x) ,hn1(x) and hn2(x)(l=1,2,3,4 ), the associated Legendre function, pnm(x) ,kq(q = 1,2) anisotropic ferrite medium is the intrinsic value, and the ball is ferrite media using the electromagnetic vector wave function expansion coefficients, in 7 has given due to space limitations, this article will not give its detailed expressions, such as 7, the electromagnetic field expansion coefficients, Fmnql(q=1,2;l=1,2)as the unknown quantity, by the boundary conditionsElectromagnetic fields from anisotropic ferrite medium the expression (3a), (3b) can be seen, when the conductor radius of the sphere tends to zero, while at the origin of the electromagnetic field is finite we can see, at this time should be equal to 0, that is now degraded to a uniform ferrite sphere in the form of electromagnetic scattering, with the literature 7 the same result. (2) mnqF From the literature 7,8, the time of the incident wave can be expanded in the following form (superscript i said) Amn5 ,Bmn5is the scattered field expansion coefficients for the unknown quantity. This, we get a conducting sphere coated with anisotropic ferrite medium ball vector of the intrinsic wave function of the analytical expressions. Here we use the boundary conditions and spherical vector wave function of the orthogonal cutting conditions, find the electromagnetic field and the anisotropic ferrite medium scattered field expansion coefficients. At r = a1 and r = a2 have the following boundary conditions Anisotropic ferrite medium, respectively, the electromagnetic field, electromagnetic field of incident electromagnetic fields ball and scattering wave function of the expression vector into the above two equations, using the ball vector wave function orthogonality of the tangential component, and then through a series of mathematical transformations, available ferrite media eigenfunction expansion coefficients satisfy the following algebraic equations: By (9) can be obtained conducting sphere coated with anisotropic ferrite medium ferrite media using the intrinsic ball that the electromagnetic field vector wave function expansion coefficients, then the scattering can be derived scattering coefficient, which may eventually come anisotropic ferrite medium coated conducting sphere of radar cross section.3 Numerical results and discussion In this section, we first derive a formula for carrying out the numerical calculation, the main numerical calculation of the plane wave incident case, the conducting sphere coated with anisotropic ferrite media radar cross section.To test this theory and the accuracy of numerical results, we calculated the radius of the ball when the conductor tends to 0, that is to take the ball of electric conductor size k0a2=0.0001.the calculation results shown in Figure 2, this time k0a2=0.2, ferrite magnetic parameters obtained 1=50 , 2=i0 and 3=70, the solid line is the result of this calculation, the literature 7 results in Figure 2, then in equation (9) n and n take the maximum value of 4. It can be seen by the figure, when the conductor radius tends to 0, the coated anisotropic ferrite medium results and even ferrite sphere are in good results, this paper describes the theoretical model and the Fortran computer program written in the correctness of .On this basis, this paper calculated the size and medium-power ferrite electrically large dielectric coated metal ball of the radar cross section, specific numerical calculation shown in Figure 3, Figure 4. These results have not been reported in the literature. Figure 3, the specific numerical parameters are as follows: the outer ferrite medium sizes are within the power: k0a1 =2and k0a2 =1.6,the magnetic parameters 1=（2+0.2i)0 ,1=i0 and 1=（4+0.4i)0 .At this point type (9) n and n take the maximum value of 12, The solid line is the electric field (E) surface, while the dotted line for the magnetic field (H) surface. Figure 4, the specific numerical parameters are as follows: the outer ferrite medium sizes are within the power: k0a1 = 3.6and k0a2 = 2, the magnetic parameters 1=（4+0.2i)0 , 200.5i = and 1=5i0 2=（3+0.1i)0 . At this point type (9) n and n take the maximum value of 20, The solid line is the electric field (E) surface, while the dotted line for the magnetic field (H) surface. By these three graphs the results can be drawn the following conclusions: (1) In conducting sphere radius tends to 0, the numerical results calculated with the uniform ferrite sphere results in good agreement; (2) numerical calculation of the number of items to take with the size of the increase of power, radar cross section derived from the width of the upper and lower shock also will increase. All of the computing time on the P-4 computer in less than 2 min; (3) forward (scattering angle is 0) of the radar cross section is always larger than the backward (scattering angle 180) of the radar cross section; (4) With the electric size increases, E and H are the oscillation phenomenon, the greater t