The implementation of log-pol transformation has recently introduced a new boost in electron optics with charged matter vortices, allowing to map conformally between linear and orbital angular momentum (OAM) states and to measure them. That coordinate change belongs to the general framework of Hossack's transformations and it has been recently realized efficiently by means of electrostatic elements. In this letter we show that it is a general property of those conformal transformations to be produced by harmonic phase elements and therefore to admit an electrostatic implementation in the electron optics scenario. We consider a new kind of conformal mapping, the circular-sector transformation, which has been recently introduced for OAM multiplication and division in optics, and discuss how it represents a general solution of Laplace's equation, showing the analogy of the generating phase elements with projected multipole fields and linear charge distributions. Moreover, we demonstrate its capability to perform the sorting of multipole wavefronts, discovering a novel and effective method to measure the strength and orientation of a dipole field in a fast, compact and direct way.

A general framework for conformal transformations in electron optics

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.

Electromagnetic Field Analysis and Modeling of a Relative Position Detection Sensor for High Speed Maglev Trains

The long stator track for high speed maglev train has a special tooth-slot structure. The relative position detection sensor obtains precise relative position information for the operation control system by detecting the changing coil inductance reflecting the long stator tooth-slot structure. The aluminum windings are inlaid in the stator slots, and the magnetic field due to the sensor coils induces eddy current in them, which would change the magnetic field distribution and have a significant influence on the sensor. In this paper, the multipole theory (MT) is adopted to build the analytical electromagnetic model of the sensor. Based on the electromagnetic model, the influence of the eddy current induced in the windings on the coil inductance is studied. The DC offset formula of differential voltage owing to the slight inductance difference of the coils is derived. The influence of the eddy current on the DC offset is analyzed, based on which, a method of no-load voltage compensation and an optimal sensor excitation frequency is proposed to improve the accuracy of the sensor.

Modeling and optimal design of relative position detection sensor for high speed maglev train

A new approach, the multipole theory(MT) method, is presented for computation of the characteristics impedance of the rectangular shielded planar striplines. The rectangular structure is transformed as the cylindrical structure by conformal mapping, the inhomogeneous field domain is treated as a combination of several simple homogeneous regions, and the solution of the boundary-value problem is represented by the generalized MT formula. Representative computational examples, including asymmetric TEM cell, rectangular shielded single microstrip line and coupled microstrip line on multilayered substrates, are given to validate the theory, and to demonstrate the accuracy and flexibility of the MT method.

Multipole theory analysis of rectangular shielded planar striplines

A new approach, the multipole theory (MT) method based on the magnetic scalar potential, is presented for the computation of 3D magnetostatic field problems. The essential concept is to represent the solution of the governing partial differential equation by the generalized MT formula of the 3D Laplace equation. The least squares method reduces the problem to the solution of a set of linear equations. Ferromagnetic sphere and iron cube in uniform magnetic fields, and iron cylinder in the field of a cylindrical coil are considered as examples. The results obtained by the MT method are in excellent agreement with the accurate data reported in the literatures.

Multipole Theory Analysis of 3D Magnetostatic Fields

A new approach, the multipole theory (MT) method, is presented for calculating two-dimensional (2-D) Laplace equation boundary-value problem. By the mathematical deduction, the generalized MT series formula and its applied laws are derived. The numerical analysis procedure and application of the MT method in electromagnetic engineering have been discussed. In order to verify the accuracy of this method, the MT method is applied to calculate the characteristic impedances of transmission lines of various cross-section configurations. The results obtained by the MT method are compared with exact data reported in the literature. It has been proven that the MT method is an effective approach for calculating 2-D Laplace equation boundary-value problems.

D. Hou

Papers

Solution of two-dimensional laplace equation by multipole theory method