﻿ 浅海条件下垂直时谐电偶极子在空气中的磁场
 舰船科学技术  2019, Vol. 41 Issue (3): 127-130 PDF

1. 海军工程大学 兵器工程系，湖北 武汉 430031;
2. 陆军军官学院，安徽 合肥 230031;
3. 海军工程大学 导航工程系，湖北 武汉 430031;
4. 海军海洋测绘研究所，天津 300061

The magnetic field in air produced by a vertically-directed time-harmonic dipole in shallow sea
SUN Yu-hui1,2, LIN Chun-sheng1, WU Hai-bing2, ZHAI Guo-jun3,4
1. Department of Weapon Engineering, Naval University of Engineering, Wuhan 430033, China;
2. Army Officer Academy, Hefei 230031, China;
3. Department of Navigation Engineering, Naval University of Engineering, Wuhan 430033, China;
4. Naval Institute of Hydrographic Surveying and Charting, Tianjin 300061, China
Abstract: In order to analyze the shaft-rate magnetic field of a ship in air, the shaft-rate current is equivalent to a vertically-directed time-harmonic dipole. In three-layer medium, the magnetic field in air generated by a dipole is derived. First, based on Maxwell's equations and boundary conditions of electromagnetic field, the magnetic vector potential of the dipole is modeled. Then, the magnetic field expressions of the dipole are deduced and a numerical computation is conducted by fast Hankel transform while the expressions contain Bessel integration. At last, the practicalities and validities of the calculation results are proved by contrasting with the carbon electrodes experiment in sea pool.
Key words: ship shaft-rate magnetic field     vertically-directed time-harmonic electric dipole     Maxwell's equations     magnetic vector potential
0 引　言

1 船舶轴频磁场产生机理

 图 1 轴转动调制腐蚀相关电流 Fig. 1 Corrosion related current modulated by shaft rotating
2 垂直时谐电偶极子模型

 图 2 3层介质模型下垂直时谐电偶极子示意图 Fig. 2 Diagram of the vertically-directed time-harmonic dipole in three-layer medium

 $\left\{ \begin{array}{l} \nabla \times {{{H}}_i} = {{{J}}_s} + ({\sigma _i} + {{j}}\omega {\varepsilon _i}){{{E}}_i}{\text{，}}\\ \nabla \times {{{E}}_i} = - {{j}}\omega {{{B}}_i} = - {{j}}\omega {\mu _i}{{{H}}_i}{\text{，}}\\ \nabla \cdot {{{B}}_i} = {\mu _i}\nabla \cdot {{{H}}_i} = 0{\text{，}}\\ \nabla \cdot {{{D}}_i} = \rho {\text{。}} \end{array} \right.$ (1)

 $\left\{ \begin{array}{l} {{B}} = \nabla \times {{A}} {\text{，}}\\ {{E}} = - \nabla \phi - {{j}}\omega {{A}} {\text{，}}\\ \nabla \cdot {{A}} - {k^2}\phi /{{j}}\omega = 0 {\text{。}} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} {\nabla ^2}{{{A}}_0} + {k^2}{{{A}}_0} = 0 {\text{，}}\\ {\nabla ^2}{{{A}}_1} + {k^2}{{{A}}_1} = - \mu {{{J}}_{\rm{S}}} {\text{，}}\\ {\nabla ^2}{{{A}}_2} + {k^2}{{{A}}_2} = 0 {\text{。}} \end{array} \right.$ (3)

 ${{{A}}_{1p}} = \frac{{{\mu _1}Il}}{{4{\text{π}} }}\int_0^{ + \infty } {\frac{\lambda }{{{\upsilon _1}}}{e^{ - {\upsilon _0}\left| {z - {z_0}} \right|}}{J_0}(\lambda r)} {\rm d}\lambda \cdot {{{e}}_{{z}}}{\text{，}}$ (4)

${{{A}}_0}$ ${{{A}}_1}$ ${{{A}}_2}$ 的可分别表示为：

 \left\{ \begin{aligned} &{{{A}}_0}(\rho ,\phi ,z) = \displaystyle\sum\limits_{m = - \infty }^\infty {{e^{jm\phi }}\int_0^\infty {{J_m}(\rho \xi )[{b_0}(\xi ,m){e^{z{\upsilon _0}}} + }} \\ & \qquad\qquad\quad {d_0}(\xi ,m){e^{ - z{\upsilon _0}}}]{\rm d}\xi \cdot {{{e}}_z} {\text{，}} \\ & {{{A}}_1}(\rho ,\phi ,z) = {{{A}}_{1p}} + {{{A}}_{1s}} = \\ & \qquad\qquad \dfrac{{{\mu _1}Il}}{{4{\text{π}} }}\int_0^{ + \infty } {\dfrac{\lambda }{{{\upsilon _1}}}{e^{ - {\upsilon _0}\left| {z - {z_0}} \right|}}{J_0}(\lambda r)} {\rm d}\lambda \cdot {{{e}}_{{z}}} + \\ & \qquad\qquad \displaystyle\sum\limits_{m = - \infty }^\infty {{e^{jm\phi }}\int_0^\infty {{J_m}(\rho \xi )[{b_1}(\xi ,m){e^{z{\upsilon _1}}} +} } \\ & \qquad\qquad{d_1}(\xi ,m){e^{ - z{\upsilon _1}}}]{\rm d}\xi \cdot {{{e}}_z}{\text{，}}\\ & {{{A}}_2}(\rho ,\phi ,z) =\\ & \qquad\qquad \displaystyle\sum\limits_{m = - \infty }^\infty {{e^{jm\phi }}\int_0^\infty {{J_m}(\rho \xi )[{b_2}(\xi ,m){e^{z{\upsilon _2}}} + } } \\ & \qquad\qquad {d_2}(\xi ,m){e^{ - z{\upsilon _2}}}]{\rm d}\xi \cdot {{{e}}_z}{\text{。}} \end{aligned} \right. (5)

 $\left\{ \begin{array}{l} {[\dfrac{{\nabla \cdot {{{A}}_1}}}{{k_1^2}} - \dfrac{{\nabla \cdot {{{A}}_0}}}{{k_0^2}}]_{z = 0}} = 0{\text{，}}\\ {[\dfrac{{{{{A}}_{1z}}}}{{{\mu _1}}} - \dfrac{{{{{A}}_{0z}}}}{{{\mu _0}}}]_{z = 0}} = 0{\text{。}} \end{array} \right.$ (6)

 $\left\{ \begin{array}{l} {[\dfrac{{\nabla \cdot {{{A}}_2}}}{{k_2^2}} - \dfrac{{\nabla \cdot {{{A}}_1}}}{{k_1^2}}]_{z = d}} = 0{\text{，}}\\ {[\dfrac{{{{{A}}_{2z}}}}{{{\mu _2}}} - \dfrac{{{{{A}}_{1z}}}}{{{\mu _1}}}]_{z = d}} = 0 {\text{。}} \end{array} \right.$ (7)

$z \to \pm \infty$ ${{{A}}_0}$ ${{{A}}_2}$ 为有限值，因此可得 ${d_0}(\xi ,m) =$ ${b_2}(\xi ,m) = 0$

 $\left\{ \!\!\!\!\! \begin{array}{l} {{{A}}_0}(\rho ,\phi ,z) = \displaystyle\int_0^\infty {{J_0}(\rho \xi )[{b_0}(\xi ,0){e^{z{\upsilon _0}}}]{\rm d}\xi } \cdot {{{e}}_z}{\text{，}}\\ {{{A}}_1}(\rho ,\phi ,z) = \dfrac{{{\mu _1}Il}}{{4{\text{π}} }}\displaystyle\int_0^{ + \infty } {\dfrac{\lambda }{{{\upsilon _1}}}{e^{ - {\upsilon _0}\left| {z - {z_0}} \right|}}{J_0}(\lambda r)} {\rm d}\lambda \cdot {{{e}}_{{z}}} + \\ \begin{array}{*{20}{c}} {}&{} \end{array}\!\!\displaystyle\int_0^\infty \!\!{{J_0}(\rho \xi )[{b_1}(\xi ,0){e^{z{\upsilon _1}}} + {d_1}(\xi ,0){e^{ - z{\upsilon _1}}}]{\rm d}\xi } \cdot {{{e}}_z}{\text{，}}\\ {{{A}}_2}(\rho ,\phi ,z) = \int_0^\infty {{J_0}(\rho \xi )[{d_2}(\xi ,0){e^{ - z{\upsilon _2}}}]{\rm d}\xi } \cdot {{{e}}_z}{\text{。}} \end{array} \right.$ (8)

 ${{\rm{A}}_0}(\rho ,\phi ,z) = \dfrac{{\mu Il}}{{2{\text{π}} }}\int_0^\infty {\xi {G_0}K{e^{z{\upsilon _0}}}{J_0}(\rho \xi ){\rm d}\xi } \cdot {{\rm{e}}_z}{\text{。}}$ (9)

 $\begin{split} & K \!=\!\! \dfrac{{({G_1}{\upsilon _2} \!+\! {\upsilon _1}){e^{{\upsilon _1}(d \!-\! {z_0})}} \!-\! ({G_1}{\upsilon _2} \!-\! {\upsilon _1}){e^{ - {\upsilon _1}(d - {z_0})}}}}{{({G_0}{\upsilon _1} \!+\! {\upsilon _0})({G_1}{\upsilon _2} \!+\! {\upsilon _1}){e^{d{\upsilon _1}}} \!+\! ({G_0}{\upsilon _1} \!-\! {\upsilon _0})({G_1}{\upsilon _2} \!-\! {\upsilon _1}){e^{ - d{\upsilon _1}}}}}{\text{，}}\\ & {G_0} = {\raise0.7ex\hbox{${k_0^2}$} / \!\lower0.7ex\hbox{${k_1^2}$}},{G_1} = {\raise0.7ex\hbox{${k_1^2}$} \!/ \!\lower0.7ex\hbox{${k_2^2}$}}{\text{。}} \end{split}$

 $\left\{ \begin{array}{l} {B_{0x}} = - \dfrac{{\partial \rho }}{{\partial y}}\dfrac{{\mu Il}}{{2{\text{π}} }}\int_0^\infty {{\xi ^2}{G_0}K{e^{z{\upsilon _0}}}{J_1}(\rho \xi ){\rm d}\xi }{\text{，}} \\ {B_{0y}} = \dfrac{{\partial \rho }}{{\partial x}}\dfrac{{\mu Il}}{{2{\text{π}} }}\int_0^\infty {{\xi ^2}{G_0}K{e^{z{\upsilon _0}}}{J_1}(\rho \xi ){\rm d}\xi } {\text{，}}\\ {B_{0z}} = 0 {\text{。}} \end{array} \right.$ (10)
3 仿真计算

 图 3 空气层磁场的x分量 Fig. 3 The x component of magnetic field in air

 图 4 空气层磁场的y分量 Fig. 4 The y component of magnetic field in air

4 水池试验

 图 5 碳棒电极试验示意图 Fig. 5 Diagram of carbon electrodes experiment

 图 6 碳棒电极产生的磁场x分量 Fig. 6 The x component of ELF magnetic field generated by the carbon electrodes

 图 7 碳棒电极产生的磁场y分量 Fig. 7 The y component of ELF magnetic field generated by the carbon electrodes
5 结　语

本文分析船舶轴频磁场产生机理，建立浅海条件下垂直时谐电偶极子在空气层产生的极低频磁场模型，并采用快速汉克尔变换进行数值运算。碳棒电极在海水水池中的试验结果验证了模型有效性。磁传感器技术的发展使得其分辨率已达到pT级，能够检测到远距离目标的磁场信号。从文中的分析可知利用信号的频谱信息能够更好地实现检测，在此基础上，谱线增强技术和测量平台的磁补偿技术需要进一步研究。

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