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Analytical theory on electromagnetic shielding effectiveness of infinite conductor plate with periodic aperture array under plane wave illumination

Bai Wan-Xin Li Tian-Le Guo An-Qi Cheng Rui-Qi Jiao Chong-Qing

Analytical theory on electromagnetic shielding effectiveness of infinite conductor plate with periodic aperture array under plane wave illumination

Bai Wan-Xin, Li Tian-Le, Guo An-Qi, Cheng Rui-Qi, Jiao Chong-Qing
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  • Penetration of a plane electromagnetic wave through the apertures on a perfectly conducting flat plate is a classical electromagnetic problem. In some practical applications like electromagnetic shielding, where only the fields far from the apertures are concerned and the aperture sizes are small compared with a wavelength, the role of apertures can be represented by the equivalent electric and magnetic dipoles located in the centers of the apertures. In principle, the penetration field can be expressed as the superposition of the radiation fields of the dipoles. However, the direct superposition leads to a double series with complex form and poor convergence. On the other hand, this problem may also be solved by full wave numerical simulations. Even so, finding analytical solutions is still desirable considering that it is clear in physical significance and easy to implement. In this paper, the analytical formula of the penetration fields are derived for both TE and TM polarization mode with different angles of incidence. The derivation is carried out firstly by averagely distributing each dipole moment within each periodic unit. As a result, the dipole array is replaced with a flat sheet with uniform magnetization and polarization intensity. Then, the equivalent surface current and charge distributions are obtained directly from the polarization intensity. Finally, the penetration fields are treated as the radiation fields of the surface sources. It is shown that the amplitude of the penetration field is proportional to aperture magnetic polarization coefficient and wave frequency, and it is inversely proportional to the area of a periodic unit. In regard to the effect of the incidence angle, the amplitude of the penetration field is proportional to the cosine of the incidence angle for TE polarization. However, for the TM polarization, the relationship is a little complicated due to the coexistence of electric and magnetic dipoles: the field is not rigorously inversely proportional to the cosine of the incidence angle due to the existence of a correction term involving both the polarization coefficient and the sine of the angle. The formula is used to calculate the shielding effectiveness for several different aperture shapes and different incidence angles. The results are in good agreement with those from the full wave simulation software.
      Corresponding author: Jiao Chong-Qing, cqjiao@ncepu.edu.cn
    [1]

    阚勇, 闫丽萍, 赵翔, 周海京, 刘强, 黄卡玛 2016 物理学报 65 030702

    Kan Y, Yan L P, Zhao X, Zhou H J, Liu Q, Huang K M 2016 Acta Phys. Sin. 65 030702

    [2]

    Mcdowell A J, Hubing T H 2014 IEEE Trans. Electromagn. Compat 56 1711

    [3]

    焦重庆, 牛帅, 李琳 2015 电工技术学报 30 1

    Jiao C Q, Niu S, Li L 2015 Transactions of China Electrotechnical Society 30 1

    [4]

    罗静雯, 杜平安, 任丹, 聂宝林 2015 物理学报 64 010701

    Luo J W, Du P A, Ren D, Nie B L 2015 Acta Phys. Sin. 64 010701

    [5]

    段兴跃, 李小康, 程谋森, 李干 2016 物理学报 65 197901

    Duan X Y, Li X K, Cheng M S, Li G 2016 Acta Phys. Sin 65 197901

    [6]

    Nie B L, Du P A, Yu Y T, Shi Z 2011 IEEE Trans. Electromagn. Compat. 53 73

    [7]

    任丹, 杜平安, 聂宝林, 曹钟, 刘文奎 2014 物理学报 63 120701

    Ren D, Du P A, Nie B L, Cao Z, Liu W K 2014 Acta Phys. Sin. 63 120701

    [8]

    焦重庆, 牛帅 2013 物理学报 62 114102

    Jiao C Q, Niu S 2013 Acta Phys. Sin. 62 114102

    [9]

    Zhao Y L, Ma F H, Li X F, Ma J J, Jia K 2018 Chin. Phys. B 27 027302

    [10]

    彭强, 周东方, 侯德亭, 余道杰, 胡涛, 王利萍, 夏蔚 2013 强激光与粒子束 25 2355

    Peng Q, Zhou D F, Hou D T, Yu D J, Hu T, Wang L P, Xia W 2013 High Power Laser and Particle Beams 25 2355

    [11]

    Li B, Dong H, Huang X L, Qiu Y, Tao Q, Zhu J M 2018 Chin. Phys. B 27 020701

    [12]

    毛湘宇, 杜平安, 聂宝林 2009 系统仿真学报 21 7493

    Mao X Y, Du P A, Nie B L 2009 Journal of System Simulation 21 7493

    [13]

    Frikha A, Bensetti M, Duval F, Benjelloun N, Lafon F, Pichon L 2015 IEEE Trans. Magn. 51 1

    [14]

    García-Pérez L G, Lozano-Guerrero A J, Blázquez-Ruiz J M, Valenzuela-Valdés J F, Monzó-Cabrera J, Fayos-Fernández J, Díaz-Morcillo A 2017 IEEE Trans. Electromagn. Compat. 59 789

    [15]

    Benhassine S, Pichon L, Tabbara W 2002 IEEE Trans. Magn. 38 709

    [16]

    Ali S, Weile D, Clupper T 2005 IEEE Trans. Electromagn. Compat. 47 367

    [17]

    Wallyn W, De Zutter D, Rogier H 2002 IEEE Trans. Electromagn. Compat. 44 130

    [18]

    焦重庆, 李顺杰 2016 电工技术学报 31 112

    Jiao C Q, Li S J 2016 Transactions of China Electrotechnical Society 31 112

    [19]

    Robinson M P, Benson T M, Christopoulos C, Dawson J F, Ganley M D, Marvin A C, Porter S J, Thomas D W P 1998 IEEE Trans. Electromagn. Compat. 44 240

    [20]

    焦重庆, 齐磊 2012 物理学报 61 134104

    Jiao C Q, Qi L 2012 Acta Phys. Sin. 61 134104

    [21]

    Otoshi T Y 1972 IEEE Trans. Microwave Theory Tech. 20 235

    [22]

    Hyun S Y, Jung I, Hong I P, Jung C, Kim E J, Yook J G 2016 IEEE Trans. Electromagn. Compat. 58 911

    [23]

    Bethe H A 1944 Phys. Rev. 66 163

    [24]

    Nitsch J B, Tkachenko S V, Potthast S 2012 IEEE Trans. Electromagn. Compat. 54 1252

    [25]

    Tesche F M, Ianoz M V, Karlsson T 1997 EMC Analysis Methods and Computational Models(New York: John Wiley& Sons)pp208—211

    [26]

    Cohn S B 1951 Proc. IRE 39 1416

  • 图 1  平面波垂直入射开孔导体板示意图

    Figure 1.  Conductor plate with holes illuminated by a plane wave with vertical polarization.

    图 2  TE极化平面波入射均匀开孔导体板示意图

    Figure 2.  Conductor plate with holes illuminated by a plane wave of TE polarization.

    图 3  TM极化平面波入射均匀开孔金属板

    Figure 3.  Conductor plate with holes illuminated by a plane wave of TM polarization.

    图 4  d1/d2对屏蔽效能的影响

    Figure 4.  Dependence of the SE on frequency for different d1/d2.

    图 5  板厚度对屏蔽效能的影响

    Figure 5.  Dependence of the SE on frequency for different thicknesses of the plane.

    图 6  圆形孔开孔大小对屏蔽效能的影响

    Figure 6.  Dependence of the SE on frequency for circular apertures with different radius.

    图 7  椭圆开孔示意图

    Figure 7.  Diagram of elliptical opening.

    图 8  椭圆孔开孔大小对屏蔽效能的影响

    Figure 8.  Dependence of SE on frequency of different elliptical apertures with different size.

    图 9  方形、十字形开孔示意图

    Figure 9.  Diagram of square and cross opening.

    图 10  不同形状开孔板的屏蔽效能随频率的变化

    Figure 10.  Dependence of SE on frequency of different elliptical apertures with different size.

    图 11  混合形状开孔周期单元示意图

    Figure 11.  Diagram of the unit cell of mix shape opening.

    图 12  混合形状开孔的屏蔽效能随频率的变化

    Figure 12.  Dependence of SE on frequency of apertures with mix shape.

    图 13  圆形孔屏蔽效能随入射角的变化(r = 0.5 cm)

    Figure 13.  Dependence of SE on angle of incidence of different polarization of circular apertures (r = 0.5 cm).

    图 14  椭圆孔屏蔽效能随入射角的变化(l = 2 cm)

    Figure 14.  Dependence of SE on angle of incidence of different polarization of elliptical apertures (l = 2 cm).

    表 1  常见开孔形状极化系数

    Table 1.  Polarization coefficients of typical opening shapes.

    孔形状${\alpha _{\rm{e}}}$${\alpha _{{\rm{m}}x}}$${\alpha _{{\rm{m}}y}}$
    圆形(r为半径)$\frac{{2{r^3}}}{3}$$\frac{{4{r^3}}}{3}$$\frac{{4{r^3}}}{3}$
    椭圆(l为长轴,沿x方向,w为短轴)$\frac{{\text{π}}}{{24}} \cdot \frac{{{w^2}l}}{{{\rm{E}}(e)}}$$\frac{{\text{π}}}{{24}} \cdot \frac{{{e^2}{l^3}}}{{K(e) - E(e)}}$$\frac{{\text{π}}}{{24}} \cdot \frac{{{e^2}{l^3}}}{{{{(l/w)}^2}E(e) - K(e)}}$
    DownLoad: CSV
  • [1]

    阚勇, 闫丽萍, 赵翔, 周海京, 刘强, 黄卡玛 2016 物理学报 65 030702

    Kan Y, Yan L P, Zhao X, Zhou H J, Liu Q, Huang K M 2016 Acta Phys. Sin. 65 030702

    [2]

    Mcdowell A J, Hubing T H 2014 IEEE Trans. Electromagn. Compat 56 1711

    [3]

    焦重庆, 牛帅, 李琳 2015 电工技术学报 30 1

    Jiao C Q, Niu S, Li L 2015 Transactions of China Electrotechnical Society 30 1

    [4]

    罗静雯, 杜平安, 任丹, 聂宝林 2015 物理学报 64 010701

    Luo J W, Du P A, Ren D, Nie B L 2015 Acta Phys. Sin. 64 010701

    [5]

    段兴跃, 李小康, 程谋森, 李干 2016 物理学报 65 197901

    Duan X Y, Li X K, Cheng M S, Li G 2016 Acta Phys. Sin 65 197901

    [6]

    Nie B L, Du P A, Yu Y T, Shi Z 2011 IEEE Trans. Electromagn. Compat. 53 73

    [7]

    任丹, 杜平安, 聂宝林, 曹钟, 刘文奎 2014 物理学报 63 120701

    Ren D, Du P A, Nie B L, Cao Z, Liu W K 2014 Acta Phys. Sin. 63 120701

    [8]

    焦重庆, 牛帅 2013 物理学报 62 114102

    Jiao C Q, Niu S 2013 Acta Phys. Sin. 62 114102

    [9]

    Zhao Y L, Ma F H, Li X F, Ma J J, Jia K 2018 Chin. Phys. B 27 027302

    [10]

    彭强, 周东方, 侯德亭, 余道杰, 胡涛, 王利萍, 夏蔚 2013 强激光与粒子束 25 2355

    Peng Q, Zhou D F, Hou D T, Yu D J, Hu T, Wang L P, Xia W 2013 High Power Laser and Particle Beams 25 2355

    [11]

    Li B, Dong H, Huang X L, Qiu Y, Tao Q, Zhu J M 2018 Chin. Phys. B 27 020701

    [12]

    毛湘宇, 杜平安, 聂宝林 2009 系统仿真学报 21 7493

    Mao X Y, Du P A, Nie B L 2009 Journal of System Simulation 21 7493

    [13]

    Frikha A, Bensetti M, Duval F, Benjelloun N, Lafon F, Pichon L 2015 IEEE Trans. Magn. 51 1

    [14]

    García-Pérez L G, Lozano-Guerrero A J, Blázquez-Ruiz J M, Valenzuela-Valdés J F, Monzó-Cabrera J, Fayos-Fernández J, Díaz-Morcillo A 2017 IEEE Trans. Electromagn. Compat. 59 789

    [15]

    Benhassine S, Pichon L, Tabbara W 2002 IEEE Trans. Magn. 38 709

    [16]

    Ali S, Weile D, Clupper T 2005 IEEE Trans. Electromagn. Compat. 47 367

    [17]

    Wallyn W, De Zutter D, Rogier H 2002 IEEE Trans. Electromagn. Compat. 44 130

    [18]

    焦重庆, 李顺杰 2016 电工技术学报 31 112

    Jiao C Q, Li S J 2016 Transactions of China Electrotechnical Society 31 112

    [19]

    Robinson M P, Benson T M, Christopoulos C, Dawson J F, Ganley M D, Marvin A C, Porter S J, Thomas D W P 1998 IEEE Trans. Electromagn. Compat. 44 240

    [20]

    焦重庆, 齐磊 2012 物理学报 61 134104

    Jiao C Q, Qi L 2012 Acta Phys. Sin. 61 134104

    [21]

    Otoshi T Y 1972 IEEE Trans. Microwave Theory Tech. 20 235

    [22]

    Hyun S Y, Jung I, Hong I P, Jung C, Kim E J, Yook J G 2016 IEEE Trans. Electromagn. Compat. 58 911

    [23]

    Bethe H A 1944 Phys. Rev. 66 163

    [24]

    Nitsch J B, Tkachenko S V, Potthast S 2012 IEEE Trans. Electromagn. Compat. 54 1252

    [25]

    Tesche F M, Ianoz M V, Karlsson T 1997 EMC Analysis Methods and Computational Models(New York: John Wiley& Sons)pp208—211

    [26]

    Cohn S B 1951 Proc. IRE 39 1416

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  • Received Date:  22 November 2018
  • Accepted Date:  26 February 2019
  • Available Online:  01 May 2019
  • Published Online:  20 May 2019

Analytical theory on electromagnetic shielding effectiveness of infinite conductor plate with periodic aperture array under plane wave illumination

    Corresponding author: Jiao Chong-Qing, cqjiao@ncepu.edu.cn
  • State Key Laboratory of Alternate Elelctrical Power System with Renewable Energy Sources, North China Elelctric Power University, Beijing 102206, China

Abstract: Penetration of a plane electromagnetic wave through the apertures on a perfectly conducting flat plate is a classical electromagnetic problem. In some practical applications like electromagnetic shielding, where only the fields far from the apertures are concerned and the aperture sizes are small compared with a wavelength, the role of apertures can be represented by the equivalent electric and magnetic dipoles located in the centers of the apertures. In principle, the penetration field can be expressed as the superposition of the radiation fields of the dipoles. However, the direct superposition leads to a double series with complex form and poor convergence. On the other hand, this problem may also be solved by full wave numerical simulations. Even so, finding analytical solutions is still desirable considering that it is clear in physical significance and easy to implement. In this paper, the analytical formula of the penetration fields are derived for both TE and TM polarization mode with different angles of incidence. The derivation is carried out firstly by averagely distributing each dipole moment within each periodic unit. As a result, the dipole array is replaced with a flat sheet with uniform magnetization and polarization intensity. Then, the equivalent surface current and charge distributions are obtained directly from the polarization intensity. Finally, the penetration fields are treated as the radiation fields of the surface sources. It is shown that the amplitude of the penetration field is proportional to aperture magnetic polarization coefficient and wave frequency, and it is inversely proportional to the area of a periodic unit. In regard to the effect of the incidence angle, the amplitude of the penetration field is proportional to the cosine of the incidence angle for TE polarization. However, for the TM polarization, the relationship is a little complicated due to the coexistence of electric and magnetic dipoles: the field is not rigorously inversely proportional to the cosine of the incidence angle due to the existence of a correction term involving both the polarization coefficient and the sine of the angle. The formula is used to calculate the shielding effectiveness for several different aperture shapes and different incidence angles. The results are in good agreement with those from the full wave simulation software.

    • 平面波对无限大导体板上开孔阵列的电磁透射属于经典电磁问题, 在电磁屏蔽、频率选择表面等方面有实际应用[1-11]. 理论上, 该问题可以通过数值计算方法求解[12-16], 例如, 文献[16]采用矩量法计算了近场辐射源透过无限大导体板的开孔阵列时的电磁耦合特性. 然而数值方法实施相对复杂, 且耗费CPU和存储资源较多, 特别是在需要实时仿真或反复计算以便进行大量参数优化的场合. 解析方法求解虽然适用范围窄, 但具有唯一性、物理意义清晰、易于计算, 特别是具有可作为数值方法的检验基准等优点. 因此, 对于给定的模型, 得到其解析解不仅具有理论价值, 也具有实际意义[17-20].

      本文结合电磁屏蔽应用实际, 在电磁波波长明显大于孔阵周期单元边长, 且孔阵周期单元边长明显大于开孔尺寸的前提下, 研究透射场的解析表达式. 目前, 对于平面波垂直入射圆形孔阵情况, 已有解析结果[21]. 其中孔阵的影响通过表面阻抗描述, 而表面阻抗需要运用分离变量法和平均场方法求解, 过程复杂. 在电磁波入射角的影响方面, 一般简单对照平面波对均匀平板材料电磁透射的入射角依赖规律, 缺少严格的理论分析. 对于平面波入射金属网情况, 文献[22]进行了分析. 网线的影响同样通过表面阻抗描述, 考虑了平面波入射角的影响. 但是该方法普适性有限, 无法推广到其他形状开孔板的情况. 文献[13]针对开单个孔缝的无限大平面板的磁场屏蔽效能进行了分析. 而在实际应用中, 孔缝通常以阵列形式出现, 该方法无法直接推广至多个孔缝的情况.

      本文首先基于Bethe理论[23], 将透射场表示成偶极子阵列的辐射场. 然后对磁偶极子阵列进行平均化, 即将其偶极矩分布于板上, 由此得到均匀磁化强度. 进而, 将磁化强度用平板两侧的磁化面电流表征, 然后可以容易地求出面电流所产生电磁场的解析表达式. 对于电偶极子阵列(仅存在于TM极化斜入射情况), 通过研究电偶极辐射和磁偶极辐射的对应关系, 从而依据磁偶极阵列辐射场导出电偶极阵列辐射场, 最终通过二者叠加得出总的透射场. 所导出的解析解理论上适用于所有形状磁/电极化系数已知的开孔. 对于诸如椭圆、圆形孔, 其极化系数是已知的. 对于其他形状的开孔, 通过数值计算可以获得极化系数, 如矩形开孔等已有相关结果发表.

    2.   理 论
    • 图 1为平面波垂直入射导体板示意图, 入射磁场${{{H}}_{{i}}} = - {H_i}{{\rm{e}}^{ - {\rm{j}}kz}}{{{e}}_y}$, 入射电场${{{E}}_{{i}}} = - {E_i}{{\rm{e}}^{ - {\rm{j}}kz}}{{{e}}_x}$, 平面波波矢量${{k}} = k{{{e}}_{{z}}}$, 其中, $k = \omega \sqrt {{\mu _0}{\varepsilon _0}} $, 相邻两孔x方向距离为d1, y方向距离为d2, 板厚度为t. 假设坐标原点位于板上表面.

      Figure 1.  Conductor plate with holes illuminated by a plane wave with vertical polarization.

      根据Bethe小孔理论[23], 从开孔耦合到z > 0区域的电磁场, 可借助位于开孔中心的等效偶极矩为${{p}}$, ${{m}}$的电偶和磁偶极子来描述:

      式中${\alpha _{\rm{e}}}$为电极化系数, ${\alpha _{{\rm{m}}x}}$为小孔沿x方向磁极化系数, ${\alpha _{{\rm{m}}y}}$为小孔沿y方向磁极化系数. ${{{E}}_n}$为小孔作短路处理(即小孔不存在或被导体封堵)时开孔面上电场的法向分量, ${{{H}}_{{{tx}}}}$, ${{{H}}_{{{ty}}}}$分别为短路处理下开孔面上切向磁场的x, y分量. 对于垂直入射情况, ${{{H}}_{{{tx}}}} = 0,{{{H}}_{{{ty}}}} = 2{{{H}}_{{i}}}$, ${{{E}}_{{n}}}{\rm{ = }}0$. 此时, 只有磁偶极子作用, z > 0区域的电磁场近似为沿导体板上分布的磁偶极阵列的辐射场. 依据镜像原理[24], 导体板可以去除, 同时磁偶极矩加倍.

      通过平均化的方法, 将每一个磁偶极子的磁偶极矩均匀分布在一个周期单元内, 从而得到板的均匀磁化模型, 其等效磁化强度为

      其中S代表一个周期单元的面积, $\Delta V{\rm{ = }}St$. 对于图 1所示情况, S = d1d2. 所以板上、下表面磁化面电流密度为:

      当导体板不开孔时, 板表面传导面电流密度为

      对比(4)式—(6)式可得

      不难理解, z > 0区域, (6)式激发的电磁场与入射场完全抵消. 进而依据(7)式可得, (4)式和(5)式所述电流产生的磁场分别为:

      其中z1, z2分别是场点到板上表面和下表面的距离. 假设场点P的坐标为$({x_p},{y_p},{z_p})$${z_p} > 0$, 则${z_1} = {z_p}$, ${z_2} = {z_p} - t$. 场点P处透射场的磁场强度为

      由于kt趋近于0, 所以$1 - {{\rm{e}}^{{\rm{j}}kt}} \approx - {\rm{j}}kt$, (10)式可写为

      就此得出, 平面波垂直入射开孔金属板情况下, 屏蔽效能公式为

    • 图 2所示, TE极化平面波入射角为θ, 入射磁场${{{H}}_{{i}}} = ( - {H_i}\cos \theta {{{e}}_{{x}}} + {H_i}\sin \theta {{{e}}_{{z}}}){{\rm{e}}^{ - {\rm{j}}{{k}} \cdot {{r}}}}$, 入射电场${{{E}}_{{i}}} = {E_i}{{\rm{e}}^{ - {\rm{j}}{{k}} \cdot {{r}}}}{{{e}}_{{y}}}$, 波矢量${{k}} = k\sin \theta {{{e}}_{{x}}} + k\cos \theta {{{e}}_{{z}}}$.

      Figure 2.  Conductor plate with holes illuminated by a plane wave of TE polarization.

      根据TE极化斜入射情况, ${{{E}}_{{n}}} = 0,$ ${{{H}}_{{t}}} =$$ - 2{H_i}\cos \theta {{\rm{e}}^{ - \operatorname{j} kx\sin \theta }}{{{e}}_{{x}}}$. 此时, 只有磁偶极子作用. 考虑镜像加倍效应后, 等效磁偶极矩为

      同理, 可得上、下表面磁化面电流密度为

      当导体板不开孔时, 板表面传导面电流密度为

      同理, 通过分别比较(14)式、(15)式和(16)式的比例关系, 可得(14)式和(15)式所述电流在z > 0区域产生的磁场分别为:

      同理, 叠加可得场点P处磁场强度为

      磁场屏蔽效能为

      由电场、磁场和传播方向三者关系可得, 场点P处电场强度为

    • 图 3所示, TM极化平面波入射角为θ, 入射电场${{{E}}_{{i}}} = ( - {E_i}\cos \theta {{{e}}_{{x}}} + {E_i}\sin \theta {{{e}}_{{z}}}){{\rm{e}}^{ - {\rm{j}}{{k}} \cdot {{r}}}}$, 入射磁场${{{H}}_{{i}}} = - {H_i}{{\rm{e}}^{ - {\rm{j}}{{k}} \cdot {{r}}}}{{{e}}_{{y}}}$.

      Figure 3.  Conductor plate with holes illuminated by a plane wave of TM polarization.

      根据TM极化斜入射情况, ${{{E}}_{{n}}} = {E_i}\sin \theta {{\rm{e}}^{ - {\rm{j}}{{k}} \cdot {{r}}}}{{{e}}_{{z}}}$, ${{{H}}_{{t}}} = - 2{H_i}{{\rm{e}}^{ - {\rm{j}}kx\sin \theta }}{{{e}}_{{y}}}$, 故电偶极子、磁偶极子同时作用. 考虑镜像加倍效应后, 等效磁偶极矩、电偶极矩分别为:

      $\nabla \times E = - {\rm{j}}\omega {\mu _0}H$, 可以推出

      (24)式表明, 求得电偶极子、磁偶极子产生的x方向电场, 即可求出相应的y方向磁场. 另一方面, TM极化下, 磁场只有y分量, 因此y方向磁场即代表总磁场.

      令TE极化中磁偶极子激发的z方向磁场、y方向电场分别为${{{H}}_{{z}}}({m_{{\rm{TE}}}})$,${{{E}}_{{y}}}({m_{{\rm{TE}}}})$, TM极化中电偶极子、磁偶极子激发的x方向电场分别为${{{E}}_{{x}}}({p_{{\rm{TM}}}})$,${{{E}}_{{x}}}({m_{{\rm{TM}}}})$. 可以证明(证明过程见附录)

      根据(19)式和(21)式, TE极化下磁偶极子产生的电场EyHz为:

      根据(25)式—(28)式可得:

      由(24)式、(29)式和(30)式得, TM极化下, 电偶极子、磁偶极子产生的磁场分别为:

      由(31)式和(32)式得TM极化下透射场的磁场强度为

      所以屏蔽效能为

      比较(20)式和(34)式看出, TE极化下透射场幅值和入射角的关系比较简单, 即与入射角的余弦成正比, 因此屏蔽效能随入射角的增大而增加;TM极化下, 透射场幅值与入射角的关系相对复杂, 随着入射角的增大, 屏蔽效能有减小的趋势. 当电极化系数相对磁化系数很小时, 附加项可以忽略, 但在电极化系数和磁化系数接近的情况下, 附加项将会对屏蔽效能产生显著影响.

    3.   结果与讨论
    • 对于圆孔和椭圆孔, 其电、磁极化系数有解析公式, 如表 1所列[25].

      孔形状${\alpha _{\rm{e}}}$${\alpha _{{\rm{m}}x}}$${\alpha _{{\rm{m}}y}}$
      圆形(r为半径)$\frac{{2{r^3}}}{3}$$\frac{{4{r^3}}}{3}$$\frac{{4{r^3}}}{3}$
      椭圆(l为长轴,沿x方向,w为短轴)$\frac{{\text{π}}}{{24}} \cdot \frac{{{w^2}l}}{{{\rm{E}}(e)}}$$\frac{{\text{π}}}{{24}} \cdot \frac{{{e^2}{l^3}}}{{K(e) - E(e)}}$$\frac{{\text{π}}}{{24}} \cdot \frac{{{e^2}{l^3}}}{{{{(l/w)}^2}E(e) - K(e)}}$

      Table 1.  Polarization coefficients of typical opening shapes.

      表 1K为第一类完全椭圆积分, E为第二类完全椭圆积分:

      其中, 椭圆离心率$e = \sqrt {1 - {{(w/l)}^2}} $. 对于其他类型孔径, 如矩形、正方形、十字形等, 通过数值计算和实验可以得到极化系数的值[26].

    • 为了验证上述公式的准确性, 本文选定几种情况进行分析. 由(12)式可以看出, 屏蔽效能的大小与周期单元边长比(d1/d2)无关, 为了验证该结论, 在周期单元面积S = 16 cm2, 开孔为半径r = 0.25 cm圆孔阵列的情况下, 选择d1/d2 = 1,4,16进行计算, CST软件全波模拟仿真(微波工作室, 采用unitcell边界)结果如图 4所示. 随着频率的增大, 屏蔽效能减小. 对于d1/d2 = 1和d1/d2 = 4两种情况, 仿真结果几乎一致, 对于d1/d2 = 16, 仿真结果与其余两种情况相差约1 dB. 所以在d1, d2大于最小开孔尺寸的前提下, 屏蔽效能与周期边长比关系不大, 后续分析均取d1 = d2 = 4 cm.

      Figure 4.  Dependence of the SE on frequency for different d1/d2.

      在电磁屏蔽领域, 实际应用中平面板厚度为mm级. 当厚度相对小孔尺寸的比值(以圆孔为例, t/r)增大时, 小孔的波导效应(以圆孔为例, SE波导 ≈ 16t/r)随之增大, 会带来一个附加项的衰减. 图 5展示了开孔半径为1 cm, 厚度t = 0.001 mm和t = 1 mm两种情况下的公式计算结果和全波仿真结果. 根据波导效应的公式, 本文考察的算例中, 波导效应带来的衰减并不大, 两种厚度下的屏蔽效能基本一致. 后续分析中, 取t = 0.001 mm.

      Figure 5.  Dependence of the SE on frequency for different thicknesses of the plane.

      图 6展示了圆形开孔下, 屏蔽效能随频率的变化情况. 可以看出, 随着开孔半径的增大, 屏蔽效能下降. r = 0.25, 0.5 cm时, 公式计算结果和仿真结果相差均约为4 dB, r = 1 cm时, 两者相差约为2 dB, 且频率越高, 一致性越好.

      Figure 6.  Dependence of the SE on frequency for circular apertures with different radius.

      对于图 7所示l/w = 10的椭圆形开孔, 图 8展示了长轴l分别为0.5, 1 和2 cm时屏蔽效能随频率的变化, 其中入射波的电场沿x方向. 对于长轴l = 0.5 cm和l = 1 cm公式计算结果与全波仿真结果一致性较好, 对于长轴l = 2 cm, 两者差值随着频率的增大而增大.

      Figure 7.  Diagram of elliptical opening.

      Figure 8.  Dependence of SE on frequency of different elliptical apertures with different size.

      针对如图 9所示的方形(图 9(a))、十字形(图 9(b))开孔, 文献[26]分别给出了其磁化系数, 将磁化系数代入(12)式可得其屏蔽效能. 图 10展示了两种开孔形状板的屏蔽效能随频率的变化. 对于边长a = 1 cm的方形开孔, 全波仿真结果与公式计算结果一致性较好. 对于十字形开孔(l = 30 cm, w/l = 0.33), 本文比较了入射波频率为0—5 GHz的情况, 在频率低于3 GHz时公式计算结果与全波仿真结果比较一致, 在3 GHz之后, 随着屏蔽效能变差, 两种方法的一致性也稍有下降.

      Figure 9.  Diagram of square and cross opening.

      Figure 10.  Dependence of SE on frequency of different elliptical apertures with different size.

      对于混合形状开孔的情况, 满足各个开孔形状磁化系数相同即可计算其屏蔽效能. 由圆孔、椭圆孔磁化系数的表达式可以得到: r = 0.25 cm的圆孔和l = 24.97 mm, l/w = 10的椭圆孔y方向磁化系数相等, 所以他们在磁场沿y方向电磁波照射下的屏蔽效能可由(12)式得出.  图 11所示为混合形状开孔周期单元示意图, 图 12展示了周期单元如图 11所示的开孔板的屏蔽效能随频率的变化, 公式计算结果和仿真结果比较一致.

      Figure 11.  Diagram of the unit cell of mix shape opening.

      Figure 12.  Dependence of SE on frequency of apertures with mix shape.

    • 取开孔为半径0.5 cm的圆孔, 波频率3 GHz, 图 13给出了屏蔽效能随入射角θ的变化. 对于TE极化入射波, 公式计算结果和全波仿真结果的差值约为4 dB;对于TM极化入射波, 差值约为3 dB. 开孔为l = 2 cm的椭圆, 屏蔽效能随入射角变化情况如图 14所示. 入射波为TE极化时, 公式计算结果与全波仿真的差值约为2 dB, 入射波为TM模时, 公式计算结果和仿真结果差值大部分在5 dB内, 且随着入射角的增大而增加.

      Figure 13.  Dependence of SE on angle of incidence of different polarization of circular apertures (r = 0.5 cm).

      Figure 14.  Dependence of SE on angle of incidence of different polarization of elliptical apertures (l = 2 cm).

      图 14所示的TM极化中, 磁场方向沿y方向, 与椭圆开孔方向垂直, (34)式中电极化系数和磁化系数比较接近, 附加项影响比较显著, 故随着入射角的增大, 屏蔽效能先增大后减小.

    4.   结 论
    • 本文提出了一种求解平面波对无限大导体板上周期孔阵的电磁透射场的解析方法. 该方法适用的前提条件是: 波长大于周期单元尺寸;周期单元尺寸大于开孔尺寸. 该方法首先基于Bethe理论将透射场表示为偶极子阵列的辐射场, 然后通过平均化将偶极阵列近似成均匀磁化/极化板, 进而通过等效磁化面电流求得磁偶极阵列的透射场, 以及通过电偶极和磁偶极的对应关系求得电偶极阵列的透射场, 最后二者叠加得出总的透射场. 依据该方法分别导出了TE和TM两种极化方式下不同入射角时透射场的表达式. 通过对几个案例的计算以及和全波仿真的比较验证了公式的有效性, 并得到了以下结论.

      1)透射场强幅值与波频率、孔的磁化系数成正比, 与一个周期的面积成反比.

      2)TE极化下, 由于仅磁偶极阵列起作用, 透射场幅值与入射角的关系十分简单, 即与入射角的余弦成正比. 此时, 屏蔽效能随入射角的增加而变大.

      3)TM极化下, 磁偶极阵列和电偶极阵列同时起作用. 透射场幅值与入射角的关系相对复杂, 不仅与入射角的余弦成反比, 还需要乘以一个包括了电极化系数以及入射角正弦值的附加项. 比较而言, 磁偶极比电偶极对透射场的贡献要大. 总体上, 此时屏蔽效能随入射角的增加而变小.

    附录A
    • 对于TE极化, 只存在x方向的磁偶极子, 一个位于x = na, y = pb, z = 0处磁偶极子${{m}} = {m_{{\rm{TE}}}}{{{e}}_x}$在场点(xp, yp, zp)处产生的电场为(只考虑远场)

      其中${\eta _0} = \sqrt {{{{\mu _0}} / {{\varepsilon _0}}}} $, $\lambda {\rm{ = }}{{2{\text{π}}} / k}$, $k = {\omega / c}$, $c = {1/ {\sqrt {{\mu _0}{\varepsilon _0}} }}$, ${{R}} = \left( {{x_p} - x} \right){{{e}}_x} +\! \left( {{y_p} - y} \right){{{e}}_y} +\! \left( {{z_p} - 0} \right){{{e}}_z},R = \left| {{R}} \right|$. 代入(1)式有

      相应的磁场为

      对于TM极化, 存在y方向磁偶极子${{m}} = {m_{{\rm{TM}}}}{{{e}}_y}$z方向电偶极子${{p}} = {p_{{\rm{TM}}}}{{{e}}_z}$.

      磁偶极子产生的电场为

      磁偶极子产生的磁场为

      电偶极子产生的磁场为

      电偶极子产生的电场为

      根据(2)式—(7)式, 不难得出mTMey产生的ExmTEex产生的Ey的关系为:

      pTMez产生的ExmTEex产生的Hz的关系

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