CN106252873A - 一种共形承载天线远场功率方向图的区间分析方法 - Google Patents

Abstract

本发明涉及一种共形承载天线远场功率方向图的区间分析方法；其特征是：至少包括如下步骤：第一步，确定复合材料天线罩材料厚度的误差区间；第二步，引入变量：X＝cos(Vd),Y＝sin(Vd)，计算变量X和Y的上下边界分别为；第三步，计算传输复数矩阵的区间上下边界；第四步，计算系数的区间上下边界；第六步，计算单个单元的系数F_i(θ,φ)＝T_MiE_i(θ,φ)的区间上下边界；第七步，计算全部单元系数的区间上下边界；第八步，计算功率方向图区间上下边界。本发明将区间分析应用于天线罩远场方向图的分析中，可在给定材料厚度误差区间的基础上，通过一次分析，即可得到对应的远场方向图区间，大大节省了分析时间和计算资源。

Description

一种共形承载天线远场功率方向图的区间分析方法

技术领域

本发明涉及一种共形承载天线，特别是一种共形承载天线远场功率方向图的区间分析方法。

背景技术

共形承载天线(Conformal Load-bearing Antenna Structure,CLAS)是一种兼具天线电磁功能和结构承载功能的新型天线结构。可以融入飞行器外表面蒙皮中，在实现天线电性能的同时，得到光滑平顺的机身表面，完全不影响飞行器的气动性能，并有利于隐身。因此得到了广泛的应用。

常用的CLAS一般是将微带天线集成到可以承载的透波复合材料中。复合材料在实现承载功能的同时，还影响着电性能。在天线与复合材料集成过程中，不可避免的会产生制造误差或制造缺陷，如胶层厚度不均匀或有空洞、粘接的脱胶分层等，在影响结构承载性能的同时，也影响电性能，如副瓣电平、增益损耗、波束宽度、指向精度等，这些电性能均可以由远场功率方向图提取。因此对一体化集成后、含有制造误差或缺陷的CLAS功率方向图的分析就十分重要。

传统的天线电性能分析方法以确定性的方法居多，也有基于统计的概率方法，由于材料厚度误差的分布具有很强的随机性，因此通常需要进行大量的分析和计算，十分耗费时间和计算资源。

本发明的目的是提供一种可大大节省分析时间和计算资源的共形承载天线远场功率方向图的区间分析方法。

发明内容

本发明是这样实现的，一种共形承载天线远场功率方向图的区间分析方法，其特征是：至少包括如下步骤：

1)确定防护结构中复合材料厚度的误差区间；

实际厚度所处的区间为d∈[d^inf；d^sup]，角标inf和sup分别表示区间的上下边界，d为材料实际厚度，d₀为理想设计厚度；

2)确定防护结构中复合材料参数的误差区间；

实际相对介电常数的所在区间为ε′∈[ε^′inf；ε^′sup],磁损耗角的区间为tanδ∈[(tanδ)^inf；(tanδ)^sup],此时参数ε＝ε′(1-jtanδ)的区间为:

${\mathrm{\ϵ}}^{\mathrm{Re}}\⋐\[{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf};{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}\]$

${\mathrm{\ϵ}}^{\mathrm{Im}}\⋐\[-{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}{\left(tan\mathrm{\δ}\right)}^{sup};-{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf}{\left(tan\mathrm{\δ}\right)}^{inf}\]$

因为该参数为复数量，其实部和虚部的区间分别给出，用上角标Re和Im分别表示。

3)确定变量[Vd]区间的上下边界；

当存在厚度误差区间时，其边界为：

(Vd^Re)^inf＝min((V^Re)d^inf,(V^Re)d^sup)

(Vd^Re)^sup＝max((V^Re)d^inf,(V^Re)d^sup)

(Vd^Im)^inf＝min((V^Im)d^inf,(V^Im)d^sup)

(Vd^Im)^sup＝max((V^Im)d^inf,(V^Im)d^sup)

当存在材料参数误差区间时，其边界为：

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{inf}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′inf}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{sup}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′sup}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{inf}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{sup}\right)}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{sup}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{inf}\right)}^{1/2}$

其中，γ为罩体表面入射角，λ为波长，下角标H和V分别表示电磁波的水平和垂直极化分量。

4)引入变量：X＝cos(Vd),Y＝sin(Vd)，计算变量X和Y区间的上下边界分别为；

(X^Re)^inf＝min{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Re)^sup＝max{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Im)^inf＝-max{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(X^Im)^sup＝-min{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Re)^inf＝min{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sin(Vd^Re)^sup·cosh(Vd^Im)^inf,sin(Vd^Re)^sup·cosh(Vd^Im)^sup}

(Y^Re)^sup＝max{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sⁱn(Vd^Re)^sup·cos^h(Vd^Im)^inf,sⁱn(Vd^Re)^sup·cos^h(Vd^Im)^sup}

(Y^Im)^inf＝min{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Im)^sup＝max{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}

5)计算传输复数矩阵区间的上下边界：

其中

A^inf/sup＝C^inf/sup＝X^inf/sup

(B^Re)^inf＝min{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-max{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Re)^sup＝max{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-min{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Im)^inf＝min{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+min{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(B^Im)^sup＝max{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+max{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(C^Re)^inf＝min{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-max{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Re)^sup＝max{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-min{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Im)^inf＝min{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+min{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}

(C^Im)^sup＝max{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+max{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}

6)引入变量T₁＝A+B/Z_∞+CZ_∞+D，并计算其区间的上下边界；：

(T₁ ^Re)^inf＝(A^Re)^inf+(B^Re)^inf/Z_∞+(C^Re)^inf·Z_∞+(D^Re)^inf

(T₁ ^Re)^sup＝(A^Re)^sup+(B^Re)^sup/Z_∞+(C^Re)^sup·Z_∞+(D^Re)^sup

(T₁ ^Im)^inf＝(A^Im)^inf+(B^Im)^inf/Z_∞+(C^Im)^inf·Z_∞+(D^Im)^inf

(T₁ ^Im)^sup＝(A^Im)^sup+(B^Im)^sup/Z_∞+(C^Im)^sup·Z_∞+(D^Im)^sup

其中，

7)计算变量T₁ ²区间的上下边界；

${\left(T_1^{2\mathrm{Re}}\right)}^{inf}=\left\{\begin{array}{cc}min\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\},& 0\∈\[X\]\\ 0& 0\∉\[X\]\end{array}\right.$

${\left(T_1^{2\mathrm{Re}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\}$

${\left(T_1^{2\mathrm{Im}}\right)}^{inf}=\left\{\begin{array}{cc}min\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\},& 0\∈\[X\]\\ 0& 0\∉\[X\]\end{array}\right.$

${\left(T_1^{2\mathrm{Im}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\}$

8)计算透射系数T区间的上下边界；

${\left(T^{\mathrm{Re}}\right)}^{inf}=2min\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\}$

${\left(T^{\mathrm{Re}}\right)}^{sup}=2max\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\}$

${\left(T^{\mathrm{Im}}\right)}^{\inf }=-2\max \left\{\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\right\}$

${\left(T^{\mathrm{Im}}\right)}^{\sup }=-2\min \left\{\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\right\}$

9)计算天线表面单个单元的远场场值x分量F_xi的区间上下边界，

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf }=\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\right\}-\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\right\}-\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\sup }=\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\right\}-\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\right\}-\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf }=\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}+\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}+\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\sup }=\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}+\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}+\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

其中，

$B_x=(A_{ex}^x{E}_t{t}_{bx}+A_{ey}^x{E}_t{t}_{by}+A_{ez}^x{E}_t{t}_{bz}+A_{hx}^x{H}_b{n}_{bx}+A_{hy}^x{H}_b{n}_{by}+A_{hz}^x{H}_b{n}_{bz})e^{{\mathrm{jk}}_0(xsin\mathrm{\θ}cos\mathrm{\φ}+ysin\mathrm{\θ}\sin \mathrm{\φ}+zcos\mathrm{\θ})}$

$C_x=\left(A_{ex}^x{E}_b{n}_{bx}+A_{ey}^x{E}_b{n}_{by}+A_{ez}^x{E}_b{n}_{bz}+A_{hx}^x{H}_t{t}_{bx}+A_{hy}^x{H}_t{t}_{by}+A_{hz}^x{H}_t{t}_{bz}\right)e^{{\mathrm{jk}}_0\left(x\sin \mathrm{\θ}\cos \mathrm{\φ}+y\sin \mathrm{\θ}\sin \mathrm{\φ}+z\cos \mathrm{\θ}\right)}$

$A_{ex}^x=-{\mathrm{cos\θn}}_{rz}-{\mathrm{sin\θsin\φn}}_{ry},A_{ey}^x={\mathrm{sin\θsin\φn}}_{rx},A_{ez}^x={\mathrm{cos\θn}}_{rx}$

$A_{hx}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}\left({\mathrm{sin\θcos\θcos\φn}}_{ry}-{\sin }^2{\mathrm{\θsin\φcos\φn}}_{rz}\right)$

$A_{hy}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}(-{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{rz}-{\cos }^2{\mathrm{\θn}}_{rz}-{\mathrm{sin\θcos\θcos\φn}}_{rx})$

$A_{hz}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}({\sin }^2{\mathrm{\θsin\φcos\φn}}_{rx}+{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{ry}+{\cos }^2{\mathrm{\θn}}_{ry})$

E_b和E_t为天线罩内表面电场和磁场在切平面上的分量，

$E_b=E_x^{in}{n}_{bx}+E_y^{in}{n}_{by}+E_z^{in}{n}_{bz}$

$E_t=E_x^{in}{t}_{bx}+E_y^{in}{t}_{by}+E_z^{in}{t}_{bz}$

$H_b=H_x^{in}{n}_{bx}+H_y^{in}{n}_{by}+H_z^{in}{n}_{bz}$

$H_t=H_x^{in}{t}_{bx}+H_y^{in}{t}_{by}+H_z^{in}{t}_{bz}$

天线罩内表面的电场Eⁱⁿ和磁场Hⁱⁿ为已知量，由天线基本尺寸参数可计算得到，其分量形式：

$E^{in}={\mathrm{iE}}_x^{in}+{\mathrm{jE}}_y^{in}+{\mathrm{kE}}_z^{in}$

$H^{in}={\mathrm{iH}}_x^{in}+{\mathrm{jH}}_y^{in}+{\mathrm{kH}}_z^{in}$

n_b和t_b分别标示天线罩外表面切平面上两个互相垂直的分量，下标i表示天线表面离散后的第i个单元，ρ,φ,θ是球坐标下的半径、方位角和俯仰角，参见图2；

10)计算天线表面全部单元场值x分量F_x的区间上下边界：

${\left(F_x^{\mathrm{Re}}\right)}^{inf/sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf /sup})$

${\left(F_x^{\mathrm{Im}}\right)}^{\inf /sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf /sup})$

n是天线表面离散单元的数量；

11)计算功率方向图区间上下边界

其上边界为：

当时，下边界为

当时，下边界为

当且时，下边界为P_x ^inf(θ,φ)＝0

其余情况下，其下边界为：

本发明的优点是：本发明将区间分析应用于天线罩远场方向图的分析中，可在给定厚度或材料参数误差区间的基础上，通过一次分析，即可得到对应的远场方向图区间，相对于基于概率的蒙特卡罗方法大大节省了分析时间和计算资源。

下面结合实施例附图对本发明用详细说明：

附图说明

图1是共形承载天线的参数示意图；

图2共形承载天线的阵面电场分布图；

图3区间分析方法和蒙特卡罗方法计算的方向图对比；

图4不同介电常数误差区间的功率方向图。

具体实施方式

一种共形承载天线远场功率方向图的区间分析方法，至少包括如下步骤：

1)确定防护结构中复合材料厚度的误差区间；

实际厚度所处的区间为d∈[d^inf；d^sup]，角标inf和sup分别表示区间的上下边界，d为材料实际厚度，d₀为理想设计厚度；

2)确定防护结构中复合材料参数的误差区间；

实际相对介电常数的所在区间为ε′∈[ε^′inf；ε^′sup],磁损耗角的区间为tanδ∈[(tanδ)^inf；(tanδ)^sup],此时参数ε＝ε′(1-jtanδ)的区间为:

${\mathrm{\ϵ}}^{\mathrm{Re}}\⋐\[{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf};{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}\]$

${\mathrm{\ϵ}}^{\mathrm{Im}}\⋐\[-{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}{\left(tan\mathrm{\δ}\right)}^{sup};-{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf}{\left(tan\mathrm{\δ}\right)}^{inf}\]$

因为该参数为复数量，其实部和虚部的区间分别给出，用上角标Re和Im分别表示。

3)确定变量[Vd]区间的上下边界；

当存在厚度误差区间时，其边界为：

(Vd^Re)^inf＝min((V^Re)d^inf,(V^Re)d^sup)

(Vd^Re)^sup＝max((V^Re)d^inf,(V^Re)d^sup)

(Vd^Im)^inf＝min((V^Im)d^inf,(V^Im)d^sup)

(Vd^Im)^sup＝max((V^Im)d^inf,(V^Im)d^sup)

当存在材料参数误差区间时，其边界为：

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{inf}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′inf}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{sup}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′sup}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{inf}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{sup}\right)}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{sup}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{inf}\right)}^{1/2}$

其中，γ为罩体表面入射角，λ为波长，下角标H和V分别表示电磁波的水平和垂直极化分量。

4)引入变量：X＝cos(Vd),Y＝sin(Vd)，计算变量X和Y区间的上下边界分别为；

(X^Re)^inf＝min{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Re)^sup＝max{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Im)^inf＝-max{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(X^Im)^sup＝-min{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Re)^inf＝min{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sin(Vd^Re)^sup·cosh(Vd^Im)^inf,sin(Vd^Re)^sup·cosh(Vd^Im)^sup}

(Y^Re)^sup＝max{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sin(Vd^Re)^sup·cosh(Vd^Im)^inf,sin(Vd^Re)^sup·cosh(Vd^Im)^sup}

(Y^Im)^inf＝min{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Im)^sup＝max{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}

5)计算传输复数矩阵区间的上下边界：

其中

A^inf/sup＝C^inf/sup＝X^inf/sup

(B^Re)^inf＝min{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-max{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Re)^sup＝max{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-min{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Im)^inf＝min{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+min{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(B^Im)^sup＝max{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+max{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(C^Re)^inf＝min{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-max{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Re)^sup＝max{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-min{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Im)^inf＝min{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+min{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}

(C^Im)^sup＝max{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+max{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}

6)引入变量T₁＝A+B/Z_∞+CZ_∞+D，并计算其区间的上下边界；：

${\left(T_1^{\mathrm{Re}}\right)}^{inf}={\left(A^{\mathrm{Re}}\right)}^{inf}+{\left(B^{\mathrm{Re}}\right)}^{inf}/Z_{\mathrm{\∞}}+{\left(C^{\mathrm{Re}}\right)}^{inf}\·Z_{\mathrm{\∞}}+{\left(D^{\mathrm{Re}}\right)}^{inf}$

${\left(T_1^{\mathrm{Re}}\right)}^{sup}={\left(A^{\mathrm{Re}}\right)}^{sup}+{\left(B^{\mathrm{Re}}\right)}^{sup}/Z_{\mathrm{\∞}}+{\left(C^{\mathrm{Re}}\right)}^{sup}\·Z_{\mathrm{\∞}}+{\left(D^{\mathrm{Re}}\right)}^{sup}$

${\left(T_1^{\mathrm{Im}}\right)}^{inf}={\left(A^{\mathrm{Im}}\right)}^{inf}+{\left(B^{\mathrm{Im}}\right)}^{inf}/Z_{\mathrm{\∞}}+{\left(C^{\mathrm{Im}}\right)}^{inf}\·Z_{\mathrm{\∞}}+{\left(D^{\mathrm{Im}}\right)}^{inf}$

${\left(T_1^{\mathrm{Im}}\right)}^{sup}={\left(A^{\mathrm{Im}}\right)}^{sup}+{\left(B^{\mathrm{Im}}\right)}^{sup}/Z_{\mathrm{\∞}}+{\left(C^{\mathrm{Im}}\right)}^{sup}\·Z_{\mathrm{\∞}}+{\left(D^{\mathrm{Im}}\right)}^{sup}$

其中，

7)计算变量T₁ ²区间的上下边界；

${\left(T_1^{2\mathrm{Re}}\right)}^{inf}=\left\{\begin{array}{cc}min\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\},& 0\∈\[X\]\\ 0& 0\∉\[X\]\end{array}\right.$

${\left(T_1^{2\mathrm{Re}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\}$

${\left(T_1^{2\mathrm{Im}}\right)}^{inf}=\left\{\begin{array}{cc}min\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\},& 0\∈\[X\]\\ 0& 0\∉\[X\]\end{array}\right.$

${\left(T_1^{2\mathrm{Im}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\}$

8)计算透射系数T区间的上下边界；

${\left(T^{\mathrm{Re}}\right)}^{inf}=2min\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\}$

${\left(T^{\mathrm{Re}}\right)}^{sup}=2max\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\}$

${\left(T^{\mathrm{Im}}\right)}^{\inf }=-2\max \left\{\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\right\}$

${\left(T^{\mathrm{Im}}\right)}^{\sup }=-2\min \left\{\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2\mathrm{Im}}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2\mathrm{Im}}\right)}^{\inf }}\right\}$

9)计算天线表面单个单元的远场场值x分量F_xi的区间上下边界，

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf }=\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\right\}-\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\right\}-\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\sup }=\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\right\}-\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\right\}-\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf }=\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}+\min \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}+\min \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\sup }=\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}+\max \left\{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\right\}\\ +\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}+\max \left\{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\right\}\end{array}$

其中，

$B_x=(A_{ex}^x{E}_t{t}_{bx}+A_{ey}^x{E}_t{t}_{by}+A_{ez}^x{E}_t{t}_{bz}+A_{hx}^x{H}_b{n}_{bx}+A_{hy}^x{H}_b{n}_{by}+A_{hz}^x{H}_b{n}_{bz})e^{{\mathrm{jk}}_0(xsin\mathrm{\θ}cos\mathrm{\φ}+ysin\mathrm{\θ}\sin \mathrm{\φ}+zcos\mathrm{\θ})}$

$C_x=\left(A_{ex}^x{E}_b{n}_{bx}+A_{ey}^x{E}_b{n}_{by}+A_{ez}^x{E}_b{n}_{bz}+A_{hx}^x{H}_t{t}_{bx}+A_{hy}^x{H}_t{t}_{by}+A_{hz}^x{H}_t{t}_{bz}\right)e^{{\mathrm{jk}}_0\left(x\sin \mathrm{\θ}\cos \mathrm{\φ}+y\sin \mathrm{\θ}\sin \mathrm{\φ}+z\cos \mathrm{\θ}\right)}$

$A_{ex}^x=-{\mathrm{cos\θn}}_{rz}-{\mathrm{sin\θsin\φn}}_{ry},A_{ey}^x={\mathrm{sin\θsin\φn}}_{rx},A_{ez}^x={\mathrm{cos\θn}}_{rx}$

$A_{hx}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}\left({\mathrm{sin\θcos\θcos\φn}}_{ry}-{\sin }^2{\mathrm{\θsin\φcos\φn}}_{rz}\right)$

$A_{hy}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}(-{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{rz}-{\cos }^2{\mathrm{\θn}}_{rz}-{\mathrm{sin\θcos\θcos\φn}}_{rx})$

$A_{hz}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}({\sin }^2{\mathrm{\θsin\φcos\φn}}_{rx}+{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{ry}+{\cos }^2{\mathrm{\θn}}_{ry})$

E_b和E_t为天线罩内表面电场和磁场在切平面上的分量，

$E_b=E_x^{in}{n}_{bx}+E_y^{in}{n}_{by}+E_z^{in}{n}_{bz}$

$E_t=E_x^{in}{t}_{bx}+E_y^{in}{t}_{by}+E_z^{in}{t}_{bz}$

$H_b=H_x^{in}{n}_{bx}+H_y^{in}{n}_{by}+H_z^{in}{n}_{bz}$

$H_t=H_x^{in}{t}_{bx}+H_y^{in}{t}_{by}+H_z^{in}{t}_{bz}$

天线罩内表面的电场Eⁱⁿ和磁场Hⁱⁿ为已知量，由天线基本尺寸参数可计算得到，其分量形式：

$E^{in}={\mathrm{iE}}_x^{in}+{\mathrm{jE}}_y^{in}+{\mathrm{kE}}_z^{in}$

$H^{in}={\mathrm{iH}}_x^{in}+{\mathrm{jH}}_y^{in}+{\mathrm{kH}}_z^{in}$

n_b和t_b分别标示天线罩外表面切平面上两个互相垂直的分量，下标i表示天线表面离散后的第i个单元，ρ,φ,θ是球坐标下的半径、方位角和俯仰角，参见图2；

10)计算天线表面全部单元场值x分量F_x的区间上下边界：

${\left(F_x^{\mathrm{Re}}\right)}^{inf/sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf /sup})$

${\left(F_x^{\mathrm{Im}}\right)}^{\inf /sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf /sup})$

n是天线表面离散单元的数量；

11)计算功率方向图区间上下边界P_x(θ,φ′)＝|F_x(θ,φ′)|²＝|F_x ^Re|²+|F_x ^Im|²，

其上边界为：

当时，下边界为

当时，下边界为

当且时，下边界为P_x ^inf(θ,φ)＝0

其余情况下，其下边界为：

为检验上述方法，特设计一共形承载天线，具体参数见图1。该天线基板宽90mm、长120mm，阵面印制4个相同的微带天线及功分网络，如图中红色部分，工作频率12.5GHz。防护结构的玻璃钢蒙皮厚度0.5mm，介电常数4.2，磁损耗角0.026。天线介质基板厚度0.5mm，介电常数2.2，磁损耗角0.0009。天线表面的电场和磁场由HFSS软件计算得到，如图2所示。

算例1：

设防护结构中玻璃钢蒙皮存在厚度误差，误差区间为[d^inf；d^sup]＝[0^.995；1.005]d_skin，d_skin＝0.5mm为设计的理想厚度。计算结果得到的方向图区间参见图3。同时使用蒙特卡罗方法计算T＝3000次蒙皮存在相同误差区间内的随机误差情况下的方向图，并取平均值与本发明的区间分析方法的平均值(P^inf+P^sup)/2对比。可见两个平均值非常接近，表明本文方法通过一次计算，即可取得蒙特卡罗方法数千次计算的效果。

算例2：

设防护结构中玻璃钢蒙皮存在材料参数误差，并选取不同的误差区间，分别为[0.985；1.015]ε_skin,[0.98；1.02]ε_skin。理想介电常数ε_skin＝4.2。计算结果得到的方向图区间参见图4，相关电性能参数参见表1。

表1是图4中方向图的主要电性能参数

可见，较大的误差区间的方向图包含了较小的误差区间方向图。

本发明将区间分析应用于天线罩远场方向图的分析中，可在给定厚度或材料参数误差区间的基础上，通过一次分析，即可得到对应的远场方向图区间，相对于基于概率的蒙特卡罗方法大大节省了分析时间和计算资源。

Claims (3)

1.一种共形承载天线远场功率方向图的区间分析方法，其特征是：至少包括如下步骤：

1)确定防护结构中复合材料厚度的误差区间；

实际厚度所处的区间为d∈[d^inf；d^sup]，角标inf和sup分别表示区间的上下边界，d为材料实际厚度，d₀为理想设计厚度；

2)确定防护结构中复合材料参数的误差区间：

实际相对介电常数的所在区间为ε′∈[ε′^inf；ε′^sup],磁损耗角的区间为tanδ∈[(tanδ)^inf；(tanδ)^sup],此时参数ε＝ε′(1-jtanδ)的区间为:

${\mathrm{\ϵ}}^{\mathrm{Re}}\⋐\[{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf};{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}\]$

${\mathrm{\ϵ}}^{\mathrm{Im}}\⋐\[-{\left({\mathrm{\ϵ}}^{\′}\right)}^{sup}{\left(tan\mathrm{\δ}\right)}^{sup};-{\left({\mathrm{\ϵ}}^{\′}\right)}^{inf}{\left(tan\mathrm{\δ}\right)}^{inf}\]$

因为该参数为复数量，其实部和虚部的区间分别给出，用上角标Re和Im分别表示；

3)确定变量[Vd]区间的上下边界；

4)引入变量：X＝cos(Vd),Y＝sin(Vd)，计算变量X和Y区间的上下边界分别为：

(X^Re)^inf＝min{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Re)^sup＝max{cos(Vd^Re)^inf·cosh(Vd^Im)^inf,cos(Vd^Re)^inf·cosh(Vd^Im)^sup,cos(Vd^Re)^sup·cosh(Vd^Im)^inf,cos(Vd^Re)^sup·cosh(Vd^Im)^sup}

(X^Im)^inf＝-max{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(X^Im)^sup＝-min{sin(Vd^Re)^inf·sinh(Vd^Im)^inf,sin(Vd^Re)^inf·sinh(Vd^Im)^sup,sin(Vd^Re)^sup·sinh(Vd^Im)^inf,sin(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Re)^inf＝min{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sin(Vd^Re)^sup·cosh(Vd^Im)^inf,sin(Vd^Re)^sup·cosh(Vd^Im)^sup}

(Y^Re)^sup＝max{sin(Vd^Re)^inf·cosh(Vd^Im)^inf,sin(Vd^Re)^inf·cosh(Vd^Im)^sup,sin(Vd^Re)^sup·cosh(Vd^Im)^inf,sin(Vd^Re)^sup·cosh(Vd^Im)^sup}

(Y^Im)^inf＝min{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}

(Y^Im)^sup＝max{cos(Vd^Re)^inf·sinh(Vd^Im)^inf,cos(Vd^Re)^inf·sinh(Vd^Im)^sup,cos(Vd^Re)^sup·sinh(Vd^Im)^inf,cos(Vd^Re)^sup·sinh(Vd^Im)^sup}；

5)计算传输复数矩阵区间的上下边界：其中

Aⁱⁿf^/sup＝C^inf/sup＝X^inf/sup

(B^Re)^inf＝min{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-max{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Re)^sup＝max{(jZ₁)^Re·(Y^Re)^inf,(jZ₁)^Re·(Y^Re)^sup}-min{(jZ₁)^Im·(Y^Im)^inf,(jZ₁)^Im·(Y^Im)^sup}

(B^Im)^inf＝min{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+min{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(B^Im)^sup＝max{(jZ₁)^Re·(Y^Im)^inf,(jZ₁)^Re·(Y^Im)^sup}+max{(jZ₁)^Im·(Y^Re)^inf,(jZ₁)^Im·(Y^Re)^sup}

(C^Re)^inf＝min{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-max{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Re)^sup＝max{(j/Z₁)^Re·(Y^Re)^inf,(j/Z₁)^Re·(Y^Re)^sup}-min{(j/Z₁)^Im·(Y^Im)^inf,(j/Z₁)^Im·(Y^Im)^sup}

(C^Im)^inf＝min{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+min{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}

(C^Im)^sup＝max{(j/Z₁)^Re·(Y^Im)^inf,(j/Z₁)^Re·(Y^Im)^sup}+max{(j/Z₁)^Im·(Y^Re)^inf,(j/Z₁)^Im·(Y^Re)^sup}；

6)引入变量T₁＝A+B/Z_∞+CZ_∞+D，并计算其区间的上下边界，

(T₁ ^Re)^inf＝(A^Re)^inf+(B^Re)^inf/Z_∞+(C^Re)^inf·Z_∞+(D^Re)^inf

(T₁ ^Re)^sup＝(A^Re)^sup+(B^Re)^sup/Z_∞+(C^Re)^sup·Z_∞+(D^Re)^sup

(T₁ ^Im)^inf＝(A^Im)^inf+(B^Im)^inf/Z_∞+(C^Im)^inf·Z_∞+(D^Im)^inf

(T₁ ^Im)^sup＝(A^Im)^sup+(B^Im)^sup/Z_∞+(C^Im)^sup·Z_∞+(D^Im)^sup

其中，

7)计算变量区间的上下边界；

${\left(T_1^{2\mathrm{Re}}\right)}^{inf}=\left\{\begin{array}{c}min\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\}\\ 0\end{array}\right.,\begin{array}{c}0\∈\[X\]\\ 0\∉\[X\]\end{array}$

${\left(T_1^{2\mathrm{Re}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Re}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Re}}\right)}^{sup}\right)}^2\}$

${\left(T_1^{2\mathrm{Im}}\right)}^{inf}=\left\{\begin{array}{c}min\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\}\\ 0\end{array}\right.,\begin{array}{c}0\∈\[X\]\\ 0\∉\[X\]\end{array}$

${\left(T_1^{2\mathrm{Im}}\right)}^{sup}=max\{{\left({\left(T_1^{\mathrm{Im}}\right)}^{inf}\right)}^2,{\left({\left(T_1^{\mathrm{Im}}\right)}^{sup}\right)}^2\}$

8)计算透射系数T区间的上下边界：

${\left(T^{\mathrm{Re}}\right)}^{inf}=2min\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}}\}$

${\left(T^{\mathrm{Re}}\right)}^{\sup }=2\max \{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}}\}$

${\left(T^{\mathrm{Im}}\right)}^{inf}=2min\{\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}},\frac{{\left(T_1^{\mathrm{Re}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}}\}$

${\left(T^{\mathrm{Im}}\right)}^{\sup }=2min\{\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{inf}+{\left(T_1^{2Im}\right)}^{sup}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{inf}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}},\frac{{\left(T_1^{\mathrm{Im}}\right)}^{sup}}{{\left(T_1^{2\mathrm{Re}}\right)}^{\sup }+{\left(T_1^{2Im}\right)}^{inf}}\};$

9)计算天线表面单个单元的远场场值x分量F_xi的区间上下边界，

10)计算天线表面全部单元场值x分量F_x的区间上下边界：

${\left(F_x^{\mathrm{Re}}\right)}^{inf/sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf /sup})$

${\left(F_x^{\mathrm{Im}}\right)}^{inf/sup}=\underset{i=1}{\overset{n}{\Σ}}({\mathrm{\ΔS}}_i\·{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf /sup})$

n是天线表面离散单元的数量；

11)计算功率方向图区间上下边界

其上边界为：

当时，下边界为

当时，下边界为

当且时，下边界为

其余情况下，其下边界为：

2.根据权利要求1所述的一种共形承载天线远场功率方向图的区间分析方法，其特征是：所述的步骤3)确定变量[Vd]区间的上下边界包括：

当存在厚度误差区间时，其边界为：

(Vd^Re)^inf＝min((V^Re)d^inf,(V^Re)d^sup)

(Vd^Re)^sup＝max((V^Re)d^inf,(V^Re)d^sup)

(Vd^Im)^inf＝min((V^Im)d^inf,(V^Im)d^sup)

(Vd^Im)^sup＝max((V^Im)d^inf,(V^Im)d^sup)

当存在材料参数误差区间时，其边界为：

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{inf}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′inf}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{sup}=\frac{2\mathrm{\π}}{\mathrm{\λ}}d{({\mathrm{\ϵ}}^{\′sup}-{\sin }^2\mathrm{\γ})}^{1/2}$

${\left({\mathrm{Vd}}^{Im}\right)}^{inf}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{sup}\right)}^{1/2}$

${\left({\mathrm{Vd}}^{Im}\right)}^{sup}=-\frac{2\mathrm{\π}}{\mathrm{\λ}}d{\left({\left(tan\mathrm{\δ}\right)}^{inf}\right)}^{1/2}$

其中，ε＝ε′(1-jtanδ)，γ为罩体表面入射角，λ为波长，下角标H和V分别表示电磁波的水平和垂直极化分量。

3.根据权利要求1所述的一种共形承载天线远场功率方向图的区间分析，其特征是：所述的步骤9)计算天线表面单个单元的远场场值x分量F_xi的区间上下边界通过如下算法完成，

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf }=\min \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\}-\max \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}\\ +\min \{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\}-\max \{{\left(C_{xi}^{\′}\right)}^{Im}\·{\left(T_V^{Im}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{Im}\·{\left(T_V^{Im}\right)}^{\sup }\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Re}}\right)}^{\inf }=\min \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Re}}\right)}^{\sup }\}-\min \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}\\ +\max \{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Re}}\right)}^{\sup }\}-\min \{{\left(C_{xi}^{\′}\right)}^{Im}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Im}}\·{\left(T_V^{Im}\right)}^{\sup }\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\inf }=\min \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}+\max \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}\\ +\min \{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\}+\max \{{\left(C_{xi}^{\′}\right)}^{Im}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{Im}\right)}^{\sup }\}\end{array}$

$\begin{array}{c}{\left(F_{xi}^{\mathrm{Im}}\right)}^{\sup }=\min \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}+\max \{{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\inf },{\left(B_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_H^{\mathrm{Im}}\right)}^{\sup }\}\\ +\max \{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\sup }\}+\max \{{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{\mathrm{Im}}\right)}^{\inf },{\left(C_{xi}^{\′}\right)}^{\mathrm{Re}}\·{\left(T_V^{Im}\right)}^{\sup }\}\end{array}$

其中，

$B_x=(A_{ex}^x{E}_t{t}_{bx}+A_{ey}^x{E}_t{t}_{by}+A_{ez}^x{E}_t{t}_{bz}+A_{hx}^x{H}_b{n}_{bx}+A_{hy}^x{H}_b{n}_{by}+A_{hz}^x{H}_b{n}_{bz})e^{{\mathrm{jk}}_0(xsin\mathrm{\θ}cos\mathrm{\φ}+ysin\mathrm{\θ}\sin \mathrm{\φ}+zcos\mathrm{\θ})}$

$C_x=(A_{ex}^x{E}_b{n}_{bx}+A_{ey}^x{E}_b{n}_{by}+A_{ez}^x{E}_b{n}_{bz}+A_{hx}^x{H}_t{t}_{bx}+A_{hy}^x{H}_t{t}_{by}+A_{hz}^x{H}_t{t}_{bz})e^{{\mathrm{jk}}_0(xsin\mathrm{\θ}cos\mathrm{\φ}+ysin\mathrm{\θ}\sin \mathrm{\φ}+zcos\mathrm{\θ})}$

$A_{ex}^x=-{\mathrm{cos\θn}}_{rz}-{\mathrm{sin\θsin\φn}}_{ry},A_{ey}^x={\mathrm{sin\θsin\φn}}_{rx},A_{ez}^x={\mathrm{cos\θn}}_{rx},$

$A_{hx}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}({\mathrm{sin\θcos\θcos\φn}}_{ry}-{\sin }^2{\mathrm{\θsin\φcos\φn}}_{rz})$

$A_{hy}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}(-{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{rz}-{\cos }^2{\mathrm{\θn}}_{rz}-{\mathrm{sin\θcos\θcos\φn}}_{rx})$

$A_{hz}^x=\sqrt{{\mathrm{\μ}}_0/{\mathrm{\ϵ}}_0}({\sin }^2{\mathrm{\θsin\φcos\φn}}_{rx}+{\sin }^2{\mathrm{\θsin}}^2{\mathrm{\φn}}_{ry}+{\cos }^2{\mathrm{\θn}}_{ry})$

E_b和E_t为天线罩内表面电场和磁场在切平面上的分量，

$E_b=E_x^{in}{n}_{bx}+F_y^{in}{n}_{by}+E_z^{in}{n}_{bz}$

$E_t=E_x^{in}{t}_{bx}+E_y^{in}{t}_{by}+E_z^{in}{t}_{bz}$

$H_b=H_x^{in}{n}_{bx}+H_y^{in}{n}_{by}+H_z^{in}{n}_{bz}$

$H_t=H_x^{in}{t}_{bx}+H_y^{in}{t}_{by}+H_z^{in}{t}_{bz}$

天线罩内表面的电场Eⁱⁿ和磁场Hⁱⁿ为已知量，由天线基本尺寸参数可计算得到，其分量形式：

$E^{in}={\mathrm{iE}}_x^{in}+{\mathrm{jE}}_y^{in}+{\mathrm{kE}}_z^{in}$

$H^{in}={\mathrm{iH}}_x^{in}+{\mathrm{jH}}_y^{in}+{\mathrm{kH}}_z^{in}$

n_b和t_b分别标示天线罩外表面切平面上两个互相垂直的分量，下标i表示天线表面离散后的第i个单元，ρ,φ,θ是球坐标下的半径、方位角和俯仰角。