WO2020015339A1 - 一种求全空间有限长谐变线电流源场的闭合形式精确解的方法 - Google Patents

一种求全空间有限长谐变线电流源场的闭合形式精确解的方法 Download PDF

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WO2020015339A1
WO2020015339A1 PCT/CN2019/070322 CN2019070322W WO2020015339A1 WO 2020015339 A1 WO2020015339 A1 WO 2020015339A1 CN 2019070322 W CN2019070322 W CN 2019070322W WO 2020015339 A1 WO2020015339 A1 WO 2020015339A1
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陈文卿
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薛国强
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江苏大学
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    • G01V3/00—Electric or magnetic prospecting or detecting; Measuring magnetic field characteristics of the earth, e.g. declination, deviation
    • G01V3/08—Electric or magnetic prospecting or detecting; Measuring magnetic field characteristics of the earth, e.g. declination, deviation operating with magnetic or electric fields produced or modified by objects or geological structures or by detecting devices
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  • the invention belongs to the field of electromagnetic exploration in the frequency domain of an artificial source, and particularly relates to a method for obtaining a closed form accurate solution of a current source field of a finite-space harmonic line in full space.
  • the harmonic line current source In the field of geophysical electromagnetic exploration, an electric dipole approximation is adopted for the harmonic line current source: the source current is regarded as a uniform distribution, and the line-of-sight current source is a point source to obtain a closed form solution. Formula suitable for far field applications.
  • the present invention proposes a method for finding a closed form accurate solution of a current source field of a finite-length harmonic line in full space.
  • the key is to express the uniform current distribution as a cosine.
  • the integral form of the field of the source point vector can be elementary quadrature, and the closed-form accurate solution of the full-space finite-length harmonic line current source field is obtained. It is not only applicable to the whole field area, but also can express the nature of the distribution of harmonic currents along the line. Attributes.
  • a method for obtaining a closed form accurate solution of a full-space finite-length harmonic line current source field including:
  • Step 1 List the vector magnetic potential integral formula containing the source point position vector of the finite-length harmonic line current source
  • Step 2 Cosine the uniform current
  • Step 3 The current in the vector magnetic potential integral formula containing the position vector of the source point is expressed by a cosine function
  • Step 4 Calculate the integral containing the position vector of the source point to obtain the closed form accurate solution of the current source field of the finite space harmonic line;
  • a z ( ⁇ , z) is the z component of the magnetic vector bit A (r). Due to the symmetry A z is only three coordinate variables of the cylindrical coordinate system.
  • ⁇ is the angular frequency of the source current (unit rad / s)
  • f is the frequency of the source current (unit Hz)
  • ⁇ is the dielectric constant (unit F / m)
  • ⁇ is the conductivity (unit S / m).
  • the cosine processing of the uniform current is: set the current at the midpoint of the line current source to I 0 ,
  • I 0 is the peak value of the harmonic current
  • Step 4.1 Expand the relationship between the vector magnetic potential A (r) and the magnetic field strength H (r) containing the position vector of the source point in a cylindrical coordinate system, and obtain the magnetic field strength H (r) as follows:
  • Step 4.2 Substituting the integral formula (3) of the vector magnetic potential A (r) with the position vector of the source point into the magnetic field strength H (r) formula (4) to obtain:
  • Step 4.3 Use the Euler formula and the homogeneous Maxwell equation to obtain the exact solution:
  • the cosine representation of the line current source of the present invention is more uniform and approximate, and can better reflect the fundamental properties of the harmonic current. Regardless of the ratio of the wavelength ⁇ to the line length l (electric dipoles that can be regarded as point sources in the far-field field, or finite-line current sources), the currents along the lines are truly reflected in the environment of geophysical electromagnetic exploration ( Weak or significant) fluctuations; the cosine current representation of the present invention has a wider applicability to changes in the operating frequency and the conductivity of the space medium than the original uniform current representation.
  • the closed-form accurate solution of the full-space finite-length harmonic line current source field is obtained, which is applicable to the full-field region.
  • the closed-form solution composed of elementary functions can intuitively reveal the changing law of the electromagnetic field, visually show the relationship between the field quantity and the parameters, and play a central role in the theoretical research of antenna and radio wave propagation.
  • the cosine representation of the harmonic line current source of the present invention can well embody the assumption that the electric dipole current is uniformly distributed in free space; the closed-loop field of the obtained full-space finite-length harmonic line current source
  • the exact form solution has the same beneficial effect on underwater communication.
  • Figure 1 Line current source and coordinate system
  • Figure 2 Current distribution of a finite-length harmonic line current source in free space
  • Figure 3 Current distribution of a finite-length harmonic line current source in a wet soil environment
  • Figure 4 Current distribution generated by a finite-length harmonic line current source in a seawater environment
  • FIG. 5 is a flowchart of a method for obtaining an accurate solution of a closed form current source field of a finite-space harmonic line in full space.
  • a method for obtaining a closed form accurate solution of a current source field of a finite-space harmonic line in full space includes:
  • Step 1 List the vector magnetic potential integral formula containing the source point position vector of the finite-length harmonic line current source
  • Step 2 Cosine the uniform current
  • Step 3 The current in the vector magnetic potential integral formula containing the position vector of the source point is expressed by a cosine function
  • Step 4 Calculate the integral containing the position vector of the source point to obtain the closed form accurate solution of the current source field of the finite space harmonic line;
  • a z ( ⁇ , z) is the z component of the magnetic vector bit A (r). Due to the symmetry A z is only three coordinate variables of the cylindrical coordinate system.
  • ⁇ is the angular frequency of the source current (unit rad / s)
  • f is the frequency of the source current (unit Hz)
  • ⁇ is the dielectric constant (unit F / m)
  • ⁇ is the conductivity (unit S / m).
  • the cosine processing of the uniform current is: set the current at the midpoint of the line current source to I 0 ,
  • I 0 is the peak value of the harmonic current
  • step 4.1 the relationship between the vector magnetic potential A (r) and the magnetic field strength H (r) containing the position vector of the source point is developed in a cylindrical coordinate system, and the magnetic field strength H (r) is obtained as follows:
  • Step 4.2 Substituting the integral formula (3) of the vector magnetic potential A (r) with the position vector of the source point into the magnetic field strength H (r) formula (4) to obtain:
  • Step 4.3 Use the Euler formula and the homogeneous Maxwell equation to obtain the exact solution:
  • the present invention provides a method for obtaining a closed-form accurate solution of a full-space finite-length harmonic line current source field, and obtains an accurate solution of the full-space finite-length harmonic line current source field.
  • the solution is not only applicable in the far, middle and near regions, but also better reflects the distribution of the harmonic current along the line source.
  • the cosine current representation of the present invention is better than the original uniform current representation for the operating frequency. It has strong adaptability to changes in the conductivity of space media.

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Abstract

本发明公开了一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，列出有限长谐变线电流源含源点位置矢量的矢量磁位公式，对均匀电流做余弦化处理，将谐变线电流源磁矢量位公式中的电流用余弦函数表示；由此使含源点位置矢量的矢量磁位公式可以初等函数求积，得到全空间有限长谐变线电流源场的闭合形式精确解，线电流源的余弦化表示，更能体现电偶极子和导电全空间中线电流源的谐变电流的根本属性，所获得的全空间有限长谐变线电流源场的闭合形式精确解，是全场区适用的。

Description

一种求全空间有限长谐变线电流源场的闭合形式精确解的方法 技术领域
本发明属于人工源频率域电磁法勘探领域，尤其涉及一种求全空间有限长谐变线电流源场的闭合形式精确解的方法。
背景技术
在地球物理电磁勘探领域，对谐变线电流源采取了电偶极子近似的处理方式：将源电流视为均匀分布，视线电流源为点源求得闭合形式的解，这种偶极近似公式，适合在远区场应用。
随着勘探方法的发展，在观察点从远区经中区向近区推进的过程中，需要获得有限长谐变线电流源场的精确解，作为发展全场区勘探的理论基础。不过，有限长谐变线电流源精确解的积分表达式，无法闭合形式求积。虽然设端点电流为0的电流正弦分布精确解的积分可闭合求值，但正弦表示与地球物理用谐变线电流源的实际电流分布不符。
发明内容
本发明根据现有技术的不足与缺陷，提出了一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其中的关键在于将均匀电流分布做余弦化表示，目的是使含源点位置矢量的场的积分式可以初等求积，所获全空间有限长谐变线电流源场的闭合形式精确解，不仅是全场区适用的，且能够表达谐变电流沿线分布的本质属性。
本发明所采用的技术方案为：
一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，包括：
步骤1，列出有限长谐变线电流源的含源点位置矢量的矢量磁位积分式；
步骤2，对均匀电流的余弦化处理；
步骤3，将含源点位置矢量的矢量磁位积分式中的电流，用余弦函数表示；
步骤4，求含源点位置矢量的积分，得到全空间有限长谐变线电流源场的闭合形式精确解；
进一步，取圆柱坐标系，线电流源的中点与坐标原点O重合，沿z轴放置，如图1所示，在均匀、线性、各向同性和时不变的无界媒质中，列出含源点位置矢量的谐变线电流源的磁矢量位公式A(r)：
Figure PCTCN2019070322-appb-000001
其中，A z(ρ,z)为磁矢量位A(r)的z分量，由于对称性A z仅为圆柱坐标系3个坐标变量
Figure PCTCN2019070322-appb-000002
中(ρ,z)的函数，
Figure PCTCN2019070322-appb-000003
为沿z轴方向的单位矢量，r为场点位置矢量，r′为源点位置矢量，z′为源点位置坐标，μ为磁导率，l为线电流源的长度，I(z′)为电流分布函数，j为虚数单位，R为源点到场点的距离，k为波数，ρ为圆柱坐标系中径向距离。
Figure PCTCN2019070322-appb-000004
其中，ω为源电流角频率(单位rad/s)，f为源电流频率(单位Hz)，ε为介电常数(单位F/m)，σ为电导率(单位S/m)。
进一步，均匀电流的余弦化处理为：设线电流源中点的电流为I 0，
Figure PCTCN2019070322-appb-000005
其中，I 0为谐变电流的峰值，
进一步，将式(2)代入式(1a)，得到含源点位置矢量的积分表示为：
Figure PCTCN2019070322-appb-000006
进一步，获得精确解的过程为：
步骤4.1，将含源点位置矢量的矢量磁位A(r)与磁场强度H(r)的关系式在圆柱坐标系中展开，得到磁场强度H(r)如下：
Figure PCTCN2019070322-appb-000007
式中，
Figure PCTCN2019070322-appb-000008
是沿
Figure PCTCN2019070322-appb-000009
方向的单位矢量。
步骤4.2，将含源点位置矢量的矢量磁位A(r)的积分式(3)代入磁场强度H(r)公式(4)得到：
Figure PCTCN2019070322-appb-000010
步骤4.3，利用Euler公式和齐次Maxwell方程，得到精确解：
由Euler公式：
Figure PCTCN2019070322-appb-000011
将(5b)代入公式(5a)展开，得到
Figure PCTCN2019070322-appb-000012
Figure PCTCN2019070322-appb-000013
由齐次Maxwell方程：
Figure PCTCN2019070322-appb-000014
得电场表达式：
Figure PCTCN2019070322-appb-000015
Figure PCTCN2019070322-appb-000016
其中，波阻抗
Figure PCTCN2019070322-appb-000017
还有
Figure PCTCN2019070322-appb-000018
Figure PCTCN2019070322-appb-000019
本发明的有益效果：
本发明的线电流源余弦化表示较均匀近似表示，更能体现谐变电流的根本属性。无论波长λ与线长l比是多少(在远区场成立可视为点源的电偶极子，或有限长线电流源)，在地球物理电磁勘探的环境中，真实地体现了沿线电流(微弱或显著)的波动性；本发明的 余弦电流表示，较原有的均匀电流表示，对工作频率和空间媒质电导率变化，有更为广泛的适用性。
通过对有限长线电流源的余弦化表示，获得的全空间有限长谐变线电流源场的闭合形式精确解，是全场区适用的。以初等函数构成的闭式解，能够直观揭示电磁场的变化规律，直观展示场量与参数之间的关系，在天线与电波传播的理论研究中，起到核心的作用。
此外，本发明的谐变线电流源的余弦表示，对自由空间中电偶极子电流均匀分布的假设，也能很好地体现；所获得的全空间有限长谐变线电流源场的闭合形式精确解，对水中通信具有相同的有益效果。
附图说明
图1线电流源与坐标系；
图2自由空间中有限长谐变线电流源的电流分布；
图3湿土环境中有限长谐变线电流源的电流分布；
图4海水环境中有限长谐变线电流源产生的电流分布；
图5一种求全空间有限长谐变线电流源场的闭合形式精确解的方法流程图。
具体实施方式
为了使本发明的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本发明进行进一步详细说明。应当理解，此处所描述的具体实施例仅用于解释本发明，并不用于限定本发明。
如图5，一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，包括：
步骤1，列出有限长谐变线电流源的含源点位置矢量的矢量磁位积分式；
步骤2，对均匀电流的余弦化处理；
步骤3，将含源点位置矢量的矢量磁位积分式中的电流，用余弦函数表示；
步骤4，求含源点位置矢量的积分，得到全空间有限长谐变线电流源场的闭合形式精确解；
进一步，取圆柱坐标系，线电流源的中点与坐标原点O重合，沿z轴放置，如图1所示，在均匀、线性、各向同性和时不变的无界媒质中，列出含源点位置矢量的谐变线电流源的磁矢量位公式A(r)：
Figure PCTCN2019070322-appb-000020
Figure PCTCN2019070322-appb-000021
其中，A z(ρ,z)为磁矢量位A(r)的z分量，由于对称性A z仅为圆柱坐标系3个坐标变量
Figure PCTCN2019070322-appb-000022
中(ρ,z)的函数，
Figure PCTCN2019070322-appb-000023
为沿z轴方向的单位矢量，r为场点位置矢量，r′为源点位置矢量，z′为源点位置坐标，μ为磁导率，l为线电流源的长度，I(z′)为电流分布函数，j为虚数单位，R为源点到场点的距离，k为波数，ω为圆柱坐标系中径向距离。
Figure PCTCN2019070322-appb-000024
其中，ω为源电流角频率(单位rad/s)，f为源电流频率(单位Hz)，ε为介电常数(单位F/m)，σ为电导率(单位S/m)。
进一步，均匀电流的余弦化处理为：设线电流源中点的电流为I 0，
Figure PCTCN2019070322-appb-000025
其中，I 0为谐变电流的峰值，
进一步，将式(2)代入式(1a)，得到含源点位置矢量的积分表示为：
Figure PCTCN2019070322-appb-000026
进一步，获得精确解的过程为：
步骤4.1，将含源点位置矢量的矢量磁位A(r)与磁场强度H(r)的关系式在圆柱坐标系中展开，得到磁场强度H(r)如下：
Figure PCTCN2019070322-appb-000027
式中，
Figure PCTCN2019070322-appb-000028
是沿
Figure PCTCN2019070322-appb-000029
方向的单位矢量。
步骤4.2，将含源点位置矢量的矢量磁位A(r)的积分式(3)代入磁场强度H(r)公式(4)得到：
Figure PCTCN2019070322-appb-000030
步骤4.3，利用Euler公式和齐次Maxwell方程，得到精确解：
由Euler公式：
Figure PCTCN2019070322-appb-000031
将(5b)代入公式(5a)展开，得到
Figure PCTCN2019070322-appb-000032
Figure PCTCN2019070322-appb-000033
由齐次Maxwell方程：
Figure PCTCN2019070322-appb-000034
得电场表达式：
Figure PCTCN2019070322-appb-000035
其中，波阻抗
Figure PCTCN2019070322-appb-000036
还有
Figure PCTCN2019070322-appb-000037
Figure PCTCN2019070322-appb-000038
实施例一，自由空间
将I 0＝1A，l＝2m，f＝10 6Hz，μ＝μ 0＝4π×10 -7H/m，
Figure PCTCN2019070322-appb-000039
和σ＝0，代入公式(2)，得到自由空间中有限长谐变线电流源的电流分布，如图2所示，
根据精确解公式(6)、公式(8)和公式(9)，有自由空间中f＝10 6Hz时，l＝2m长的谐变线电流源产生的场
Figure PCTCN2019070322-appb-000040
Figure PCTCN2019070322-appb-000041
Figure PCTCN2019070322-appb-000042
由公式(11)，可知上面公式(12)、公式(13)和公式(14)中的
Figure PCTCN2019070322-appb-000043
Figure PCTCN2019070322-appb-000044
由公式(1b)和公式(10)，可知上面公式(12)、公式(13)和公式(14)中自由空间的波数k和波阻抗Z为
Figure PCTCN2019070322-appb-000045
Figure PCTCN2019070322-appb-000046
将场点坐标
Figure PCTCN2019070322-appb-000047
代入公式(12)、公式(13)和公式(14)，便得到自由空间中f＝10 6Hz、l＝2m有限长谐变线电流源产生的场的空间分布。
实施例二，湿土环境
将I 0＝1A，l＝2000m，f＝100Hz，μ＝μ 0＝4π×10 -7H/m，
Figure PCTCN2019070322-appb-000048
和σ＝0.01S/m代入公式(2)，得到湿土环境中有限长谐变线电流源的电流分布，如图3所示，
根据精确解公式(6)、公式(8)和公式(9)，有湿土环境中f＝100Hz时，l＝2000m长的谐变线电流源产生的场
Figure PCTCN2019070322-appb-000049
Figure PCTCN2019070322-appb-000050
Figure PCTCN2019070322-appb-000051
由公式(11)，可知上面公式(15)、公式(16)和公式(17)中的
Figure PCTCN2019070322-appb-000052
Figure PCTCN2019070322-appb-000053
由公式(1b)和公式(10)，可知上面公式(15)、公式(16)和公式(17)中湿土的波数k和波阻抗Z为
Figure PCTCN2019070322-appb-000054
Figure PCTCN2019070322-appb-000055
将场点坐标
Figure PCTCN2019070322-appb-000056
代入公式(15)、公式(16)和公式(17)，便得到湿土环境中f＝100Hz、l＝2000m有限长谐变线电流源产生的场的空间分布。
实施例三，海水环境
将I 0＝1A，l＝500m，f＝100Hz，μ＝μ 0＝4π×10 -7H/m，
Figure PCTCN2019070322-appb-000057
和σ＝4S/m代入公式(2)，有海水环境中有限长谐变线电流源的电流分布，如图4所示。
根据公式(6)、公式(8)和公式(9)，有海水环境中f＝100Hz时，l＝500m长的谐变线电流源产生的场：
Figure PCTCN2019070322-appb-000058
Figure PCTCN2019070322-appb-000059
Figure PCTCN2019070322-appb-000060
Figure PCTCN2019070322-appb-000061
由公式(11)，可知上面公式(18)、公式(19)和公式(20)中的
Figure PCTCN2019070322-appb-000062
Figure PCTCN2019070322-appb-000063
由公式(1b)和公式(10)，可知上面公式(18)、公式(19)和公式(20)中海水的波数k和波阻抗Z为
Figure PCTCN2019070322-appb-000064
将场点坐标
Figure PCTCN2019070322-appb-000065
代入公式(18)、公式(19)和公式(20)，便得到海水环境中f＝100Hz、l＝500m有限长谐变线电流源产生的场的空间分布。
综上所述，本发明所提供的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，获得全空间有限长谐变线电流源场的精确解，该闭合形式的精确解，不仅在远区、中区和近区均是适用的，而且能够更好地反映谐变电流沿线源分布的状况，本发明的余弦电流表示，较原有的均匀电流表示，对工作频率和空间媒质电导率变化，有较强的适应性。
以上实施例仅用于说明本发明的设计思想和特点，其目的在于使本领域内的技术人员能够了解本发明的内容并据以实施，本发明的保护范围不限于上述实施例。所以，凡依据本发明所揭示的原理、设计思路所作的等同变化或修饰，均在本发明的保护范围之内。

Claims (6)

  1. 一种求全空间有限长谐变线电流源场的闭合形式精确解的方法,其特征在于，包括：
    步骤1，列出有限长谐变线电流源的含源点位置矢量的矢量磁位积分式；
    步骤2，对均匀电流的余弦化处理；
    步骤3，将含源点位置矢量的矢量磁位积分式中的电流，用余弦函数表示；
    步骤4，求含源点位置矢量的积分，得到全空间有限长谐变线电流源场的闭合形式精确解。
  2. 根据权利要求1所述的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其特征在于，取圆柱坐标系，线电流源的中点与坐标原点O重合，沿z轴放置，在均匀、线性、各向同性和时不变的无界媒质中，列出含源点位置矢量的谐变线电流源的矢量磁位公式A(r)：
    Figure PCTCN2019070322-appb-100001
    其中，A z(ρ,z)为磁矢量位A(r)的z分量，由于对称性A z仅为圆柱坐标系3个坐标变量
    Figure PCTCN2019070322-appb-100002
    中(ρ,z)的函数，
    Figure PCTCN2019070322-appb-100003
    为沿z轴方向的单位矢量，r为场点位置矢量，r′为源点位置矢量，z′为源点位置坐标，μ为磁导率，l为线电流源的长度，I(z′)为电流分布函数，j为虚数单位，R为源点到场点的距离，k为波数，ρ为圆柱坐标系中径向距离。
  3. 根据权利要求2所述的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其特征在于，所述波数为：
    Figure PCTCN2019070322-appb-100004
    其中，ω为源电流角频率，f为源电流频率，ε为介电常数，σ为电导率。
  4. 根据权利要求1所述的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其特征在于，设线电流源中点的电流为I 0，将均匀电流余弦化处理得到：
    Figure PCTCN2019070322-appb-100005
    其中，I(z′)为电流分布函数，I 0为谐变电流的峰值，k为波数，z′为源点位置坐标。
  5. 根据权利要求1或2所述的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其特征在于，将含源点位置矢量的谐变线电流源的磁矢量位积分中的电流用余弦函数表示为：
    Figure PCTCN2019070322-appb-100006
  6. 根据权利要求1或2述的一种求全空间有限长谐变线电流源场的闭合形式精确解的方法，其特征在于，求含源点位置矢量的积分获得精确解的过程为：
    步骤4.1，将含源点位置矢量的矢量磁位A(r)与磁场强度H(r)的关系式在圆柱坐标系中展开，得到磁场强度H(r)如下：
    Figure PCTCN2019070322-appb-100007
    式中，
    Figure PCTCN2019070322-appb-100008
    是沿
    Figure PCTCN2019070322-appb-100009
    方向的单位矢量。
    步骤4.2，将含源点位置矢量的矢量磁位A(r)代入磁场强度H(r)得到：
    Figure PCTCN2019070322-appb-100010
    步骤4.3，利用Euler公式和齐次Maxwell方程，得到精确解：
    Figure PCTCN2019070322-appb-100011
    Figure PCTCN2019070322-appb-100012
    Figure PCTCN2019070322-appb-100013
    Figure PCTCN2019070322-appb-100014
    其中，
    Figure PCTCN2019070322-appb-100015
    为波阻抗，
    Figure PCTCN2019070322-appb-100016
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