﻿ 多层介质平板结构的传输线等效面模型及相位修正算法
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 雷达学报  2015, Vol. 4 Issue (3): 317-325  DOI: 10.12000/JR15038 0

### 引用本文 [复制中英文]

[复制中文]
Zhang Lei, Dong Chun-zhu, Hou Zhao-guo, et al.. Transmission line equivalent plane model and phase correction algorithm for multilayered dielectric slab structure[J]. Journal of Radars, 2015, 4(3): 317-325. DOI: 10.12000/JR15038.
[复制英文]

### 文章历史

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Transmission Line Equivalent Plane Model and Phase Correction Algorithm for Multilayered Dielectric Slab Structure
, , , ,
Science and Technology on Electromagnetic Scattering Laboratory, Beijing 100854, China
The Second Academy of China Aerospace Science and Industry Corporation, Beijing 100854, China
Information Engineering School, Communication University of China, Beijing 100024, China
Abstract: The Equivalent Plane Model (EPM) and Phase Correction Algorithm (PCA) that are based on Transmission Line Theory (TLT) are proposed to satisfy the resource and efficiency requirements of ElectroMagnetic (EM) scattering analysis of large and complex multilayered dielectric targets. The proposed method accurately predicts the EM scattering characteristics of reference targets. To simplify the analysis, the multilayered dielectric slab structure is considered planar. On the basis of the TLT, the reflection and transmission coefficients of the plane are determined by using network analysis methods typically adopted in circuit analysis. Moreover, the reflection and transmission phases are corrected by considering the thickness of the multilayered dielectric slab and the direction of incidence and observation. Simulation results verify the applicability of the proposed method.
Key words: ElectroMagnetic (EM) scattering    Multilayered dielectric    Ray tracing    Transmission Line Theory (TLT)    Phase correction
1 引言

2 传输线等效模型 2.1 传输线理论

N层均匀介质平板及其传输线等效模型如图 1 所示。图中第m层介质板厚度为tm，相对介电常数和相对磁导率分别为emmm，对应等效传输线长度为tm，等效传播常数为 $\beta _m^{{\rm{TL}}}$ ，等效波阻抗为 $Z_m^{{\rm{TL}}}$ 。

 图 1 N 层介质平板及其传输线等效模型 Fig.1 N-layered slab and its transmission line equivalent model

m层均匀介质平板对应传输线的传输矩阵可表示为[21, 22, 23]

 $\left[\!\!{\begin{array}{*{20}{c}} {{A_m}\ {B_m}}\\ {{C_m}\ {D_m}} \end{array}} \!\! \right] \\ \quad = \left [\!\! {\begin{array}{*{20}{c}} \ \ \ { \cosh ({\rm{j}}{\beta _m^{{\rm{TL}}}}{t_m}) \qquad \quad \; Z_m^{{\rm{TL}}}\sinh ({\rm{j}}{\beta _m^{{\rm{TL}}}}{t_m})} \\ \!\! { \sinh ({\rm{j}}{\beta _m^{{\rm{TL}}}}{t_m})/Z_m^{{\rm{TL}}} \qquad \quad \cosh ({\rm{j}}{\beta _m^{{\rm{TL}}}}{t_m}) } \end{array}} \!\! \right]$ (1)
 $\beta _m^{{\rm{TL}}} = {k_m}\cos {\theta _m} = {k_0}\sqrt {{\mu _m}{\varepsilon _m}} \cos {\theta _m} \hspace{12pt} \hspace{40pt}$ (2)
 $Z_m^{{\rm{TL}}} = \ \!\! \left\{ {\begin{array}{*{20}{l}} \!\! {{\eta _0}\sqrt {{\mu _m}/{\varepsilon _m}} \cos {\theta _m},\quad\ /\!/ }\\ {{\rm{ }} \!\! {\eta _0}\sqrt {{\mu _m}/{\varepsilon _m}} / \cos {\theta _m},\quad \bot } \end{array}} \!\! \right. \hspace{65pt}$ (3)

N层均匀介质平板的总传输矩阵可表示为：

 $\left[\!\! {\begin{array}{*{20}{c}} {{A} \quad {B}}\\ {{C} \quad {D}} \end{array}} \!\! \right] = \left[\!\! {\begin{array}{*{20}{c}} {{A_1} \quad {B_1}}\\ {{C_1} \quad {D_1}} \end{array}} \!\! \right]\left[\!\! {\begin{array}{*{20}{c}} {{A_2} \quad {B_2}}\\ {{C_2} \quad {D_2}} \end{array}} \!\! \right] \cdots \left[\!\! {\begin{array}{*{20}{c}} {{A_N} \quad {B_N}}\\ {{C_N} \quad {D_N}} \end{array}} \!\! \right]$ (4)

 $R = \frac{{\left( {A + B/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right) - Z_0^{{\rm{TL \; }}}\left( { C + D/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right)}}{{\left( {A + B/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right) + Z_0^{{\rm{TL \; }}}\left( {C + D/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right)}}$ (5)
 $T = \frac{2}{{\left( {A + B/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right) + Z_0^{{\rm{TL \; }}}\left( {C + D/Z_{N + 1 \;\; }^{{\rm{TL \; }}}} \right)}}$ (6)

2.2 散射场计算

 ${E\left( r \right) = \oint_S {\left[{ - {\rm{j}}\omega \mu \left( {{e_{\rm{n}}} \times {H_{\rm{s}}}} \right) + \left( {{e_{\rm{n}}} \times {E_{\rm{s}}}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \; {\left. { + \left( {{e_{\rm{n}}} \cdot {E_{\rm{s}}}} \right)\nabla '} \right]{G_0}\left( {r,r'} \right){\rm{d}}S'}$ (7)
 ${H\left( r \right) = \oint_S {\left[{\rm j\omega \varepsilon \left( {{e_{\rm{n}}} \times {E_{\rm{s}}}} \right)+ \left( {{e_{\rm{n}}} \times {H_{\rm{s}}}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \; {\left. { + \left( {{e_{\rm{n}}} \cdot {H_{\rm{s}}}} \right)\nabla '} \right]{G_0}\left( {r,r'} \right){\rm{d}}S' \hspace{40pt}}$ (8)

 ${G_0}\left( {r,r'} \right) = \frac{{{{\rm{e}}^{ - {\rm{j}}k\left| {r - r'} \right|}}}}{{4π \left| {r - r'} \right|}}$ (9)

3 等效面模型及其相位修正算法 3.1 等效面模型

 图 2 多层介质结构等效面模型 Fig.2 Equivalent plane model of multi-layered dielectric structure

 图 3 四室建筑物模型 Fig.3 The building model with 4 rooms

 图 4 射线追踪示意图 Fig.4 The schematic diagram of ray tracing

3.2 相位修正算法

 图 5 多层介质平板传输线等效模型及其等效面模型 Fig.5 Multi-layered dielectric slab’s transmission line equivalent model and equivalent plane model

 ${R^m} = R{e^{{\rm{j}}{k_0}{P_1}Q \cdot \left( {\hat i - \hat s} \right)}}$ (10)
 ${T \;^m} = T\,{e^{{\rm{j}}{k_0}\left( {{ P_1} Q \cdot \hat i - { P_2} Q \cdot \hat s} \right)}} \hspace{33pt}$ (11)
 ${P_1}Q = 0.5{t_{{\rm{total}}}}{\rm{sec}}{\theta _0}\hat i \hspace{41pt}$ (12)
 ${P_2}Q = 0.5{t_{{\rm{total}}}}\sec {\theta _0}\hat i + {t_{{\rm{total}}}}\hat n$ (13)

 ${T \;^m} = T \;{{\rm{e}}^{{\rm{j}}{k_0}\left( {{P_1}Q \cdot \hat i - {P_2}Q \cdot \hat s} \right)}}{{\rm{e}}^{ - {\rm{j}}{k_0}\sin {\theta _0}t'}} \hspace{35pt}$ (14)
 $t' = {t_1}\tan {\theta _1} + {t_2}\tan {\theta _2} + \cdot \cdot \cdot + {t_N}\tan {\theta _N}$ (15)
4 仿真示例与分析

4.1 双层介质平板与金属柜组合结构

 图 6 双层介质平板与金属柜组合结构 Fig.6 Two-layered dielectric slab with metal cabinet

 图 7 两组模型1维距离像对比 Fig.7 Comparison of range profile between initial model and equivalent plane model

4.2 建筑物模型

 图 8 建筑物模型SAR图像对比 Fig.8 Comparison of building model’s SAR images

5 总结

 $\nabla '{G_0}\left( {{{r}},r'} \right) = - {\rm{j}}k\hat s{G_0}\left( {r,r'} \right)$ (A-1)

 $\begin{array}{*{20}{l}} {{E_1}\left( r \right) = \oint_{{S_{{P_1}}}} {\left[{ - {\rm{j}}\omega \mu \left( {{e_ \rm n} \times {H \!_{{P_1}}}} \right) + \left( {{e_ \rm n} \times {E_{{P_1}}}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \ {\left. { + \left( {{e_ \rm n} \cdot {E_{{P_1}}}} \right)\nabla '} \right]{G_0}\left( {r,{{}r^\prime \! \! _{{P_1}}} } \right){\rm{d}} {S^\prime \! \!_{{P_1}}} } \end{array}$ (A-2)

 $\begin{array}{*{20}{l}} {{E_2}\left( r \right) = \oint_{{S_Q}} {\left[{ - {\rm{j}}\omega \mu \left( {{e_ \rm n} \times {H\! _Q}} \right) + \left( {{e_ \rm n} \times {E_Q}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \ {\left. { + \left( {{e_ \rm n} \cdot {E_Q}} \right)\nabla '} \right]{G_0}\left( {r,{r^\prime \! \! _Q}} \right){\rm{d}}{{S}^\prime} \! \! _Q} \end{array}$ (A-3)

 ${H \! _Q} = {H \! _{{P_1}}}{\rm e}^{ - {\rm{j}}k{P_1}Q{ \hat i}} \hspace{55pt}$ (A-4)
 ${E \! _Q} = {E \! _{{P_1}}}{{\rm e}^{ - {\rm{j}}k{P_1}Q \hat i}} \hspace{60pt}$ (A-5)
 ${G_0}\left( {r,{r^\prime \! \! _Q}} \right) = {G_0}\left( {r,{r^\prime \! \! _{{P_1}}}} \right){{\rm e}^{{\rm{j}}k{P_1}Q \hat s}}$ (A-6)

 ${E_2}\left( r \right) = {E_1}\left( r \right){{\rm e}^{ - {\rm{j}}k{P_1}Q\hat i}}{{\rm e}^{{\rm{j}}k{P_1}Q\hat s}}$ (A-7)

 ${R^m} = R{{\rm e}^{{\rm{j}}k{P_1}Q\hat i}}{{\rm e}^{ - {\rm{j}}k{P_1}Q\hat s}} = R{{\rm e}^{{\rm{j}}k{P_1}Q\left( {\hat i - \hat s} \right)}}$ (A-8)

 $\begin{array}{*{20}{l}} {{E_1}\left( r \right) = \oint_{{S_{{P_2}}}} {\left[{ - {\rm{j}}\omega \mu \left( {{e_{\rm{n}}} \times {H_{{P_2}}}} \right) + \left( {{e_{\rm{n}}} \times {E_{{P_2}}}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \ {\left. { + \left( {{e_{\rm{n}}} \cdot {E_{{P_2}}}} \right)\nabla '} \right]{G_0}\left( {r,{r^\prime \! \! _{{P_2}}} } \right){\rm{d}}{S^\prime \! \! _{{P_2}}} } \end{array}$ (A-9)

 $\begin{array}{*{20}{l}} {{E_2}\left( r \right) = \oint_{{S_Q}} {\left[{ - {\rm{j}}\omega \mu \left( {{e_{\rm{n}}} \times {H\! _Q}} \right) + \left( {{e_{\rm{n}}} \times {E\! _Q}} \right) \times \nabla '} \right.} }\\ \quad \qquad \ \ {\left. { + \left( {{e_{\rm{n}}} \cdot {E\! _Q}} \right)\nabla '} \right]{G_0}\left( {r,{r^\prime} \! \! _Q} \right){\rm{d}}{S^\prime \! \! _Q} } \end{array}$ (A-10)

 ${H_Q} = {H_{{P_2}}}{{\rm {e}}^{ - {\rm{j}}k{P_1}Q\hat i}} \hspace{55pt}$ (A-11)
 ${E_Q} = {E_{{P_2}}}{{\rm {e}}^{ - {\rm{j}}k{P_1}Q\hat i}} \hspace{60pt}$ (A-12)
 ${G_0}\left( {r,{r^\prime \! \! _Q}} \right) = {G_0}\left( {r,{r^\prime \! \! _{{{P}_2}}}} \right){e^{{\rm{j}}k{P_2}Q\hat s}}$ (A-13)

 ${E_2}\left( r \right) = {E_1}\left( r \right){{\rm{e}}^{ - {\rm{j}}k{P_1}Q\hat i}}{{\rm{e}}^{{\rm{j}}k{P_2}Q\hat s}}$ (A-14)

 ${T^m} = T \,{{\rm{e}}^{{\rm{j}}k{P_1}Q\hat i}}{{\rm{e}}^{ - {\rm{j}}k{P_2}Q\hat s}} = T \,{{\rm{e}}^{{\rm{j}}k\left( {{P_1}Q\hat i - {P_2}Q\hat s} \right)}}$ (A-15)