电波科学学报  2019, Vol. 34 Issue (3): 330-335  DOI: 10.13443/j.cjors.2018051503. PDF

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Transmission and coupling characteristics of leaky coaxial cable using the equivalent circuit
School of Optical Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: We studied the electrical characteristics of the buried leaky coaxial cables(LCX) as a distributed sensor, applying to detect the intrusion signal and the landslides early warning signal in tunnel and railway, etc. Taking the vertical slot structure as an example, we set up a LCX model using HFSS, and then analyzed the theory and extracted the equivalent circuit model. Based on the equivalent model, we analyzed the transmission, reflection and coupling characteristics of the leaky coaxial cable; we also calculated the equivalent parameters and the coupling parameters using the ABCD matrix. The theoretical calculation is very consistent with the full-wave simulation, which proves the equivalent circuit is right. Results show that compared with the HFSS simulation, the equivalent circuit occupies less computing resources and greatly decreases the simulation time, which will solve the problem that the HFSS simulation cable length is limited and time-consuming; we also theoretically calculate the equivalent parameters and the coupling parameters between the LCXs, which show the coupling capability between the LCX directly.
Keywords: buried leaky coaxial cable    the full wave simulation    equivalent circuit    ABCD matrix    coupling parameter value

1 理论介绍 1.1 漏缆的结构

 图 1 漏缆的结构[7] Fig. 1 Structure of leaky coaxial cable[7]
1.2 传输线特性参数 1.2.1 特性阻抗Z0

 $Z_{0}=\frac{60}{\sqrt{\varepsilon_{\mathrm{r}}}} \ln \frac{b}{a}.$ (1)

1.2.2 传播常数γ

 $\gamma=\alpha+\mathrm{j} \beta.$ (2)

 $\gamma=j \omega \sqrt{L_{1} C_{1}}.$ (3)

2 等效电路模型搭建 2.1 等效分析

 图 2 单个缝隙单元结构 Fig. 2 Structure of single slot unit
2.2 等效模型 2.2.1 二端口网络

 图 3 单个缝隙单元等效电路模型 Fig. 3 Equivalent circuit model of single slot unit

 $\begin{array}{l} {{\boldsymbol{Y}}_{\rm{S}}} = \\ \left[ {\begin{array}{*{20}{c}} {0.001\;3786 - 9.087\;3{\rm{i}}}&{ - 0.001\;378\;4 - 9.086\;3{\rm{i}}}\\ { - 0.001\;378\;9 - 9.084\;3{\rm{i}}}&{0.001378\;6 - 9.089\;3{\rm{i}}} \end{array}} \right]. \end{array}$ (4)

 图 4 单根漏缆缝隙的等效电路图 Fig. 4 Equivalent circuit diagram of single leaky coaxial cable slot

 $\begin{array}{l} {{\boldsymbol{Y}}_{\rm{T}}} = \left[ {\begin{array}{*{20}{l}} {{Y_{11}}}&{{Y_{12}}}\\ {{Y_{21}}}&{{Y_{22}}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{{\rm{j}}\omega L + R}}}&{\frac{{ - 1}}{{{\rm{j}}\omega L + R}}}\\ {\frac{{ - 1}}{{{\rm{j}}\omega L + R}}}&{\frac{{{\rm{j}}\omega {C_{\rm{L}}}({\rm{j}}\omega L + R) + 1}}{{{\rm{j}}\omega L + R}}} \end{array}} \right]. \end{array}$ (5)

 $A B C D_{\text { slot }}=\left[\begin{array}{cc}{\mathrm{j} \omega C_{\mathrm{L}}(\mathrm{j} \omega L+R)+1} & {\mathrm{j} \omega L+R} \\ {\frac{\mathrm{j} \omega C_{\mathrm{L}}(\mathrm{j} \omega L+R)}{\mathrm{j} \omega L+R}} & {1}\end{array}\right].$ (6)

 $A B C D_{L}=\left[\begin{array}{cc}{\operatorname{ch}(\gamma d)} & {Z_{0} \operatorname{sh}(\gamma d)} \\ {\frac{\operatorname{sh}(\gamma d)}{Z_{0}}} & {\operatorname{ch}(\gamma d)}\end{array}\right].$ (7)

 $ABC{D_{{\rm{periodic - unit}}}} = ABC{D_{\rm{L}}}\cdot ABC{D_{{\rm{slot}}}}\cdot ABC{D_{\rm{L}}}.$ (8)

 $ABC{D_n} = \prod\limits_1^n A BC{D_{{\rm{ periodic - unit }}}}.$ (9)

 $\left[\begin{array}{cc}{A} & {B} \\ {C} & {D}\end{array}\right]=\left[\begin{array}{cc}{\frac{\left(1+S_{11}\right)\left(1-S_{22}\right)+S_{12} S_{21}}{2 S_{21}}} & {Z_{0} \frac{\left(1+S_{11}\right)\left(1+S_{22}\right)-S_{12} S_{21}}{2 S_{21}}} \\ {\frac{1}{Z_{0}} \frac{\left(1-S_{11}\right)\left(1-S_{22}\right)-S_{12} S_{21}}{2 S_{21}}} & {\frac{\left(1-S_{11}\right)\left(1+S_{22}\right)-S_{12} S_{21}}{2 S_{21}}}\end{array}\right].$ (10)

2.2.2 四端口网络

 图 5 双根漏缆缝隙的等效电路图 Fig. 5 Equivalent circuit diagram of double leaky coaxial cable slot

 ${{\boldsymbol{Z}}_{\rm{T}}} =\\ \left[ {\begin{array}{*{20}{c}} {({\rm{j}}\omega L + R) + \frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{{\rm{j}}\omega L{K_1} + \frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}\\ {\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}\\ {{\rm{j}}\omega L{K_1} + \frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{({\rm{j}}\omega L + R) + \frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}\\ {\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{{K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}}&{\frac{{1 + {K_{\rm{c}}}}}{{{\rm{j}}\omega {C_{\rm{L}}}\left( {1 + 2{K_{\rm{c}}}} \right)}}} \end{array}} \right].$ (11)

 $\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{V_2}}\\ {{V_3}}\\ {{V_4}} \end{array}} \right] = {{\boldsymbol{Z}}_{\rm{T}}}\left[ {\begin{array}{*{20}{c}} {{I_1}}\\ {{I_2}}\\ {{I_3}}\\ {{I_4}} \end{array}} \right].$ (12)

 $\left\{ \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {V_1^\prime }\\ {I_1^\prime } \end{array}} \right] = ABC{D_{\rm{L}}}\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{I_1}} \end{array}} \right]\;\;{\kern 1pt} \left[ {\begin{array}{*{20}{c}} {V_2^\prime }\\ {I_2^\prime } \end{array}} \right] = ABC{D_{\rm{L}}}\left[ {\begin{array}{*{20}{c}} {{V_2}}\\ {{I_2}} \end{array}} \right]\\ \left[ {\begin{array}{*{20}{c}} {V_3^\prime }\\ {I_3^\prime } \end{array}} \right] = ABC{D_{\rm{L}}}\left[ {\begin{array}{*{20}{c}} {{V_3}}\\ {{I_3}} \end{array}} \right]\;\;{\kern 1pt} \left[ {\begin{array}{*{20}{c}} {V_4^\prime }\\ {I_4^\prime } \end{array}} \right] = ABC{D_{\rm{L}}}\left[ {\begin{array}{*{20}{c}} {{V_4}}\\ {{I_4}} \end{array}} \right] \end{array} \right..$ (13)

 $\left[ {\begin{array}{*{20}{c}} {V_1^\prime }\\ {V_2^\prime }\\ {V_3^\prime }\\ {V_4^\prime } \end{array}} \right] = {\boldsymbol{Z}}_{\rm{T}}^\prime \left[ {\begin{array}{*{20}{c}} {I_1^\prime }\\ {I_2^\prime }\\ {I_3^\prime }\\ {I_4^\prime } \end{array}} \right].$ (14)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{l}} {V_1^\prime }\\ {V_2^\prime }\\ {V_3^\prime }\\ {V_4^\prime } \end{array}} \right] = {\bf{Z}}_{\rm{T}}^\prime \left[ {\begin{array}{*{20}{c}} {I_1^\prime }\\ {I_2^\prime }\\ {I_3^\prime }\\ {I_4^\prime } \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{l}} {{Z_{11}}}&{{Z_{12}}}&{{Z_{13}}}&{{Z_{14}}}\\ {{Z_{21}}}&{{Z_{22}}}&{{Z_{23}}}&{{Z_{24}}}\\ {{Z_{31}}}&{{Z_{32}}}&{{Z_{33}}}&{{Z_{34}}}\\ {{Z_{41}}}&{{Z_{42}}}&{{Z_{43}}}&{{Z_{44}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {I_1^\prime }\\ {I_2^\prime }\\ {I_3^\prime }\\ {I_4^\prime } \end{array}} \right]; \end{array}$ (15)
 $\left[\begin{array}{c}{V_{1}^{\prime}} \\ {I_{1}^{\prime}} \\ {V_{3}^{\prime}} \\ {I_{3}^{\prime}}\end{array}\right]=\left[\begin{array}{llll}{A_{11}} & {A_{12}} & {A_{13}} & {A_{14}} \\ {A_{21}} & {A_{22}} & {A_{23}} & {A_{24}} \\ {A_{31}} & {A_{32}} & {A_{33}} & {A_{34}} \\ {A_{41}} & {A_{42}} & {A_{43}} & {A_{44}}\end{array}\right]\left[\begin{array}{c}{V_{2}^{\prime}} \\ {-I_{2}^{\prime}} \\ {V_{4}^{\prime}} \\ {-I_{4}^{\prime}}\end{array}\right].$ (16)

3 等效参数求解

 $\left\{ {\begin{array}{*{20}{l}} {R = 0.000\;013\;349\;\Omega }\\ {L = 0.579\;596\;{\rm{nH}}}\\ {{C_{\rm{L}}} = 0.237\;{\rm{pF}}} \end{array}} \right..$ (17)

 $\left\{ {\begin{array}{*{20}{l}} {{K_1} = 0.000\;48}\\ {{K_{\rm{c}}} \approx 8.438\;8 \times {{10}^{ - 6}}} \end{array}} \right..$ (18)

4 仿真结果分析 4.1 传输和耦合特性

 图 6 单根漏缆HFSS和等效电路计算S参数对比图 Fig. 6 Comparison of single LCX S-parameter calculated by HFSS and equivalent circuit

 图 7 两根耦合漏缆的HFSS和等效电路 Fig. 7 Comparison of double coupling LCX S-parameter calculated by HFSS and equivalent circuit

4.2 HFSS仿真与理论计算耗时分析

5 结论

1) 以垂直开槽结构为例, 模型实验仿真与理论分析计算结合, 验证了等效电路思想用于解决HFSS仿真漏缆模型长度受限和耗时太长问题的正确性, 等效思想将大大提高研究应用于隧道、铁路沿线等防止入侵破坏和预警山体塌方信号探测的漏缆的效率, 奠定了进一步研究漏缆电气特性的基础;

2) 研究中通过Mathematical编程完成了对各种仿真数据的处理和分析, 并最终在理论分析的基础之上解出了本文开槽结构的等效电路模型参数及同轴漏缆间耦合电容值和耦合电感值, 使我们更直观准确地分析和了解本文提出的漏缆模型的耦合特性.

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