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The long stator track for high speed maglev trains has a tooth-slot structure. The sensor obtains precise relative position information for the traction system by detecting the long stator tooth-slot structure based on nondestructive detection technology. The magnetic field modeling of the sensor is a typical three-dimensional (3-D) electromagnetic problem with complex boundary conditions, and is studied semi-analytically in this paper. A second-order vector potential (SOVP) is introduced to simplify the vector field problem to a scalar field one, the solution of which can be expressed in terms of series expansions according to Multipole Theory (MT) and the New Equivalent Source (NES) method. The coefficients of the expansions are determined by the least squares method based on the boundary conditions. Then, the solution is compared to the simulation result through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution agrees approximately with the numerical solution. Finally, based on electromagnetic modeling, a difference coil structure is designed to improve the sensitivity and accuracy of the sensor.

High speed maglev trains are driven by a linear synchronous motor [

The operating principle of the relative position detection sensor is based on nondestructive examination technology [

This paper studies the electromagnetic modeling of the sensor analytically, which lays an essential foundation for the sensor design work. Nowadays, with the development of PC technology, complex electromagnetic modeling problems can be realized through numerical methods precisely and conveniently, whereas, analytical methods are complicated, time consuming, and may even have no solution. Usually, in order to get a solution, model simplification is needed, which decreases the precision of the solution. However, analytical methods can get the expression of the field distribution function, represent less of a calculation load for computers, and are beneficial for the development of theories to some extent. They can be used to do some preliminary analysis and get a rough estimate for a problem to guide the principle design work. Numerical methods can be introduced later to guide the detailed design and parameter optimization.

Usually, it's difficult to solve 3-D electromagnetic problems with complex boundaries directly through separation of variables or mirror methods,

The electromagnetic modeling of the relative position detection sensor is studied analytically in this paper. Firstly, the electromagnetic field in air due to an inductive excitation by a rectangular coil is calculated and analyzed. Based on this, a simplified model of the sensor is established. Then the magnetic induction density distribution of the simplified model is calculated semi-analytically according to the MT and NES methods, and a corresponding numerical solution is obtained through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution approximately agrees with the numerical solution. Finally, based on the modeling, a difference coil structure is designed to improve the sensitivity and precision of the sensor.

The excitation frequency of the detecting coil is chosen to be about 2 MHz. The corresponding wavelength is

According to the electromagnetic field superposition principle, the magnetic induction intensity in free space due to the rectangular coil is obtained as follows:

The coil-stator system model is shown in

The length and width of the coil and the width of a slot or a tooth denoted by _{w}_{s}_{s}

The model is too complex for an analytical solution of the field, so some reasonable simplification is needed. Supposing

According to _{w}

Furthermore, considering the periodicity, the coil inductance is figured out only when 0< _{d}_{w}

In a 3-D static field comprising several domains separated by medium boundaries, there exists a certain scalar

The domain in which

The outside poles are set according to the boundaries. If all the inner angles between adjacent boundaries of the valid domain are less than 180°, each boundary should be assigned to an outside pole. For a spherical boundary surface, the pole should be set at the centre of the spherical surface to make the series expansion converge fast. For a plane boundary, it can be treated as a part of a spherical surface with an infinite radius. In this case, the series expansion corresponding to the plane boundary is reduced to a constant. Furthermore, if there exist inner angles greater than 180°,

Consider the situation shown in _{1}. The outside pole is set at _{2}, the centre of spherical surface Γ_{2}. The solution to the Laplace _{i}_{p}_{p}_{op}_{l}_{i}_{l}^{m}_{i}_{i}_{i}_{1}. _{op}_{op}_{2}).

Experientially, to get a precise enough solution, the order of a pole corresponding to a spherical surface boundary can be set to 6, and the order of an inside pole corresponding to a cubic domain can be set to 10.

Suppose the domain to be solved is the space outside the cube shown in

A circular equivalent source is shown in _{k}_{k}_{0})^{2}+_{0}^{2}+^{2})/2_{0}_{k}_{-1/2} is half-integer degree Legendre function [_{l}

Thus, similar to the spherical equivalent sources, the invalid domain can be filled with auxiliary circular sources to replace the actual boundaries.

Usually, it is hard to get the solution of a 3-D electromagnetic field problem by solving the Maxwell's equations directly, so Smythe [_{1} and _{2} are scalar components. The vector potential

In 3-D passive static electromagnetic field, the vector potential ^{2} = −j

According to _{1} and _{2} satisfy the Laplace equations:

The magnetic induction intensity

Formula (14) results in the following expressions in spherical coordinate system:

According to _{1} which satisfies the Laplace

For the simplified problem model shown in _{I1} and _{II1} are calculated as follows, respectively:

Domain I

In domain I, there exists an excitation coil, so the vector potential
_{I}

The induction intensity due to the excitation coil in the free space can serve as a particular solution as

Considering domain I as the valid domain and according to the analysis above, it does not satisfy the expansion condition, so equivalent sources are introduced to replace the cubic boundary. The sectional view of the arrangement of equivalent sources in the cube is shown in

The cube is filled with six equivalent sources. Source no. 1 is a spherical equivalent source and sources no. 2, 3, 4, 5 and 6 are circular equivalent sources. To calculate the inductance of the coil above the cube, the magnetic induction intensity near the upper surface of the cube should be precise enough, so equivalent sources are mainly set at the upper part of the cube to make the upper surface of equivalent sources more approximate to the upper plane of the cube. Because domain I is an open domain, there is no inside pole.

Let _{I1}_{i}

As shown in _{I1} can be expressed as follows:
_{i}_{1}) are undetermined coefficients, and _{1} is the term number. The meanings of other symbols are the same as in

Domain II

There is no excitation source in domain II, so the vector potential
_{II1} can be expressed according to _{i}_{2}) are undetermined coefficients, _{2} is the term number. By substituting

Coefficients Determination

The undetermined coefficients of _{0}) denotes the permeability of silicon steel, _{I}_{n}, B_{II}_{n}_{I}_{t}_{1}, _{II}_{t}_{1}, _{I}_{t}_{2}, _{II}_{t}_{2} denote the normal components and tangential components of the magnetic induction intensity, respectively. The LS method is adopted to minimize the error between the solutions of the two domains on the boundary. The target function is given as follows:
_{LS}_{i}_{1}) and _{i}_{2}). In order to minimize _{LS}_{LS}

There are (_{1} + _{2}) equations in _{1} + _{2}) undetermined coefficients can be determined. After substituting the coefficients _{i}_{1}) in the expression of

The Finite Element Analysis (FEA) software Ansoft Maxwell is used to simulate the electromagnetic field distribution of the relative position detection sensor. The simulation model is shown in

The calculation results of the coil inductance at different positions along the x-axis are shown in

According the electromagnetic modeling results, the relationship between the coil inductance and the tooth-slot structure is shown in

The curve shown in

Thus, the precision of the sensor can be improved through the table-lookup method based on the relationship between the inductance difference and the relevant displacement. In addition, the difference operation can also counteract common mode disturbances such as temperature drift and enhance the performance of the sensor.

The relative position detection sensor obtains the relative displacement information between the train and the long stator by detecting the tooth-slot structure of the long stator based on nondestructive technology. This paper researches the electromagnetic field modeling of the sensor to guide the coil design and improvement. The electromagnetic field model of the coil-stator system is a complex three-dimensional (3-D) electromagnetic problem, so at first, the problem is simplified based on the analysis of the electromagnetic field distribution of a rectangular coil in free space. And then, MT and the NES method are adopted to get the series expansion of the magnetic induction intensity of the simplified model. The relationship between the coil inductance and the relative position is figured out afterwards. The inductance is also simulated through FEA. The comparison results show that the semi-analytical solution approximately agrees with the numerical solution. Finally, based on the modeling work, a difference coil structure is designed to improve the sensitivity and precision of the relative position detection sensor.

This work was performed at the Engineering Research Center of Maglev Technology at National University of Defense Technology with funding from National Natural Science Foundation of China under grant no. 60874015.

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Diagram of the sensor coil and the long stator.

A rectangular coil in a Cartesian coordinate system.

The coil-stator system.

The Z-direction component of the magnetic induction intensity due to the rectangular coil excitation at (

Simplified model of the coil-stator system.

The final simplified model of the sensor.

A field with two spherical boundaries.

Spherical equivalent source.

Circular equivalent source.

Configuration of the equivalent sources.

Simulation model of the numerical method.

Relationship between coil inductance and relative position.

The relationship between coil inductance and the tooth-slot structure.

Difference coil structure.

(