一种共形承载天线远场功率方向图的区间分析方法

摘要

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Fi(θ,φ)＝TMiEi(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。

说明书

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:

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

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

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：

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{inf}=\frac{2\pi }{\lambda }d{({ϵ}^{\prime inf}-{\sin }^2\gamma )}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{sup}=\frac{2\pi }{\lambda }d{({ϵ}^{\prime sup}-{\sin }^2\gamma )}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{inf}=-\frac{2\pi }{\lambda }d{\left({\left(tan\delta \right)}^{sup}\right)}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{sup}=-\frac{2\pi }{\lambda }d{\left({\left(tan\delta \right)}^{inf}\right)}^{1/2}$

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T₁²区间的上下边界；

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

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

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

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

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，

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

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

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

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

其中，

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

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

$A_{ex}^x=-{\mathrm{cos\theta n}}_{rz}-{\mathrm{sin\theta sin\phi n}}_{ry},A_{ey}^x={\mathrm{sin\theta sin\phi n}}_{rx},A_{ez}^x={\mathrm{cos\theta n}}_{rx}$

$A_{hx}^x=\sqrt{{\mu }_0/{ϵ}_0}\left({\mathrm{sin\theta cos\theta cos\phi n}}_{ry}-{\sin }^2{\mathrm{\theta sin\phi cos\phi n}}_{rz}\right)$

$A_{hy}^x=\sqrt{{\mu }_0/{ϵ}_0}(-{\sin }^2{\mathrm{\theta sin}}^2{\mathrm{\phi n}}_{rz}-{\cos }^2{\mathrm{\theta n}}_{rz}-{\mathrm{sin\theta cos\theta cos\phi n}}_{rx})$

$A_{hz}^x=\sqrt{{\mu }_0/{ϵ}_0}({\sin }^2{\mathrm{\theta sin\phi cos\phi n}}_{rx}+{\sin }^2{\mathrm{\theta sin}}^2{\mathrm{\phi n}}_{ry}+{\cos }^2{\mathrm{\theta n}}_{ry})$

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：

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

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

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:

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

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

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：

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{inf}=\frac{2\pi }{\lambda }d{({ϵ}^{\prime inf}-{\sin }^2\gamma )}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Re}}\right)}^{sup}=\frac{2\pi }{\lambda }d{({ϵ}^{\prime sup}-{\sin }^2\gamma )}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{inf}=-\frac{2\pi }{\lambda }d{\left({\left(tan\delta \right)}^{sup}\right)}^{1/2}$

${\left({\mathrm{Vd}}^{\mathrm{Im}}\right)}^{sup}=-\frac{2\pi }{\lambda }d{\left({\left(tan\delta \right)}^{inf}\right)}^{1/2}$

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；：

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

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

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

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

其中，

7)计算变量T₁²区间的上下边界；

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

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

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

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

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，

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

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

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

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

其中，

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

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

$A_{ex}^x=-{\mathrm{cos\theta n}}_{rz}-{\mathrm{sin\theta sin\phi n}}_{ry},A_{ey}^x={\mathrm{sin\theta sin\phi n}}_{rx},A_{ez}^x={\mathrm{cos\theta n}}_{rx}$

$A_{hx}^x=\sqrt{{\mu }_0/{ϵ}_0}\left({\mathrm{sin\theta cos\theta cos\phi n}}_{ry}-{\sin }^2{\mathrm{\theta sin\phi cos\phi n}}_{rz}\right)$

$A_{hy}^x=\sqrt{{\mu }_0/{ϵ}_0}(-{\sin }^2{\mathrm{\theta sin}}^2{\mathrm{\phi n}}_{rz}-{\cos }^2{\mathrm{\theta n}}_{rz}-{\mathrm{sin\theta cos\theta cos\phi n}}_{rx})$

$A_{hz}^x=\sqrt{{\mu }_0/{ϵ}_0}({\sin }^2{\mathrm{\theta sin\phi cos\phi n}}_{rx}+{\sin }^2{\mathrm{\theta sin}}^2{\mathrm{\phi n}}_{ry}+{\cos }^2{\mathrm{\theta n}}_{ry})$

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：

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

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

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。