Analytical Calculation of Hybrid Excitation Transverse Flux Permanent Magnet Synchronous Linear Motor Based on Schwarz–Christoffel Mapping

An analytical calculation method of thrust and suspension force based on Schwarz–Christoffel (SC) mapping is presented for hybrid excitation transverse flux permanent magnet synchronous linear motor (HETFPMSLM) applied to a maglev train. The effectiveness of the analytical calculation is verified by three-dimensional finite element analysis. Firstly, the structure and principle of HETFPMSLM are analyzed, and its longitudinal two-dimensional SC mapping model is proposed. The magnetic field of HETFPMSLM is calculated by MATLAB’s SC Toolbox. Then the suspension force and thrust are calculated by the Maxwell stress tensor method, which is compared with the 3D finite element to verify the accuracy of the analytical calculation.


Introduction
Maglev train is a new type of rail without wheel-rail contact.It is suspended, driven, and guided by electromagnetic force.Compared with the traditional wheel-rail train, it has better performance in speed, climbing ability, turning radius, and noise, which means better ride comfort, friendliness to the environment, and less maintenance cost [1].It is expected to make a great contribution to the new transportation system in the 21st century.
Although since 1960, Germany, Japan, China, South Korea, and other countries have been conducting research on maglev train technology with a variety of technical routes, the application is limited due to factors such as technology, cost, and national transportation development policies [2] [3].Therefore, it is of great significance to continuously optimize the current maglev train technology and explore new technologies and new structures in order to enhance the value of the commercial operation.
The transverse flux linear motor has a larger equivalent magnetic pole area and a shorter equivalent magnetic circuit because the plane of the magnetic circuit is perpendicular to the moving direction of the mover, which realizes the decoupling between the magnetic and the electrical field [4] [5].Therefore, it can achieve a larger thrust density.It can use its unilateral magnetic pull as the suspension force of the maglev system, as well as the end effect of the core to develop the self-recovery ability when the lateral offset occurs and complete the guidance function.This means it can realize the integration of drive, suspension, and guidance and has great potential for application on maglev trains.On the other hand, an integrated suspension guidance and drive system of the maglev train based on HETFPMSLM can reduce the weight of the maglev train.It is worth mentioning that for the current main technical route of the maglev train system, the levitation principle is different.It does not integrate suspension guidance and a drive system.Furthermore, the permanent magnet is used to realize the basic suspension of the train under rated conditions.The electromagnetic coil-assisted suspension control method can greatly reduce the volume, weight, and heating of the electromagnet of the traditional electric excitation magnetic levitation device and improve reliability.
The transverse flux linear motor also presents a three-dimensional structure due to its special core structure, which means it is more difficult to analyze the HETFPMSLM based on specific parameters.Using the finite element method to analyze the three-dimensional structure is a common method for this type of motor.Although with high calculation accuracy, it usually takes a lot of time to model and conduct the finite element analysis.Therefore, it is of great significance to study the characteristics of HETFPMSLM with the analytical calculation of the electromagnetic field in the early stage.
Therefore, this paper analyzes the structure and principle of HETFPMSLM and puts forward its longitudinal two-dimensional Schwarz-Christoffel (SC) mapping model.The complicated magnetic field analysis model is converted into rectangles by SC mapping, and the air gap magnetic field is calculated by SC Toolbox in MATLAB.Then the suspension force and thrust are calculated with the Maxwell stress tensor method, which is compared with Ansys Maxwell's three-dimensional finite element simulation results to verify the effectiveness of the analytical algorithm.

Analysis model of HETFPMSLM
Figures 1 and 2 show the structure of HETFPMSLM.The structure of the stator and mover unit adopts a U-shaped iron core and two concentrated windings to form an electromagnet.A pair of permanent magnets with opposite polarity are installed on the surface of the magnetic pole of the mover unit.The stator unit is laid equidistantly under the rail to form a long stator structure.Two mover units constitute a mover pair corresponding to the length of three stator units and are installed on the train suspension frame.Figure 3 shows that the permanent magnet on the same side is staggered.The mover and stator cores are both laminated by silicon steel sheets to reduce eddy effect.The main magnetic circuit is generated by the permanent magnet of the mover unit, on which the train can basically achieve the suspension state.The permanent magnet alone cannot keep the train in suspension stably.Therefore, the controllable coil on the mover is needed to dynamically adjust the circuit to maintain a stable suspension of the train; when the mover moves left and right relative to the stator, due to the transverse end effect of the U-shaped core, the guiding force can be generated to make the train return to the middle of the track; further, three-phase alternating current is introduced into the stator coil.According to the operation principle of the synchronous motor, the magnetic field generated by the stator coils interact with the excitation magnetic field generated by the mover, and a continuous thrust is generated on the mover to drive the train forward.
Due to the complex magnetic field boundary formed by the moving stator slot, it is very difficult to solve the partial differential equation of the air gap magnetic field directly.According to SC mapping, the irregular polygon in the w complex plane can be converted into a rectangle in the Z plane [6].The specific expression is as follows: Through Equation (1), SC can create a conformal mapping from the interior of a complex polygon in plane W to the upper half plane region of another complex plane Z, and vice versa.In this motor model, by finding a suitable transformation function, the irregular polygon boundary in the complex plane W can be transformed into the upper half plane of the plane and then further modified by another mapping function, which can be mapped to the rectangular plane Z.The points in the W plane correspond to the points in the Z plane, respectively.

𝑤 = 𝑓(𝑧) = 𝐴 + 𝐶 (𝜁 − 𝑧 ) 𝑑𝜁
MATLAB integrates the functions () to calculate the mapping with a numerical scheme in the SC toolbox [7], which can transform the complex W plane into a simple plane.
Due to the symmetry of the HETFPMSLM structure, the two-dimensional structure model is given in Figure 3, and the main structural parameters of HETFPMSLM are given in Table 1  The magnetic field excitation source of the HETFPMSLM includes three parts: stator windings, mover permanent magnets, and mover windings.The stator excitation winding is equivalent to a line current I s with N turns and is equidistantly arranged on both sides of the stator magnetic pole.The current direction on both sides is related to the armature current required for the actual train travel.In order to simplify the calculation, the mover rectangular permanent magnet can be equivalent to line current models pm I arranged at both sides of the permanent magnet on the surface of the mover pole line current [8], each of which is: where M is the magnetization of the permanent magnet, pm N is the discrete number, and p h is the length of the magnetization direction of the permanent magnet.Figure 4 gives the mapping model from the W plane to the Z plane.

Solution of magnetic field
A longitudinal structure cross section of HETFPMSLM is shown in Figure 4.When we analyze and calculate the magnetic field of the model, the following assumptions are necessary to simplify the calculation: 1) The motor has no lateral displacement and guiding force; 2) PM has an ideal linear B-H curve; 3) the permeability of the iron core is infinite; and 4) the effect of transverse length is ignored.Due to the conformality of the mapping, the solution of the analytical equation is unchanged before and after the mapping.Therefore, after the line current source in the plane is converted to the corresponding position in the plane, the current value of the line current source remains unchanged.That is, after SC mapping, the solution of the magnet field in the original complex polygon multi-excitation source is transformed into a rectangular field.Among them, the infinite boundaries ABCD on both sides of the W plane and EFGH correspond to the rectangular boundaries AD and EH in the Z plane, so each point on AD and EH has the same vector magnetic potential [9].
Therefore, the magnetic vector potential of point (x, y) in the rectangle generated by the potential line current i I at the point (x 1 , y 1 ) is: where 0 μ is the permeability of air.

Because of A B
∇ × = , the flux density of point ( , ) x y can be given by: Then, the flux density B in the air gap generated by the line current i I in W complex plane can be calculated by: ( ) where ( ) f x ′ can also be calculated with related functions in the toolbox.According to the superposition theorem, the flux density produced by all of line currents can be calculated by: Maxwell stress tensor method usually is used to calculate the electromagnetic force of a region [10] on a plane obtaining a target object in the air gap.As shown in Figure 4, efgh is the line close to the mover.Therefore, thrust force and suspension force under the condition that the motor does not have lateral displacement can be calculated by:  With the current control method of the 0 d-axis current, the analytical and finite element results are shown in Figures 5 and 6 when the three-phase current of the stator is 1500 ampere turns, the current of the mover coil is 0 and the motor running speed is 120 km/h.The error is acceptable, which proves the feasibility of the algorithm.suspension forces are calculated according to the Maxwell stress tensor method.Furthermore, the 3D finite element method of HETFPMSLM is carried out and compared with the SC method.The conclusion is that the magnetic field and thrust waveforms calculated by SC and FEM are in good agreement without transverse offset of the motor, which shows the feasibility of the SC method and lays the foundation for further study of motor characteristics and optimization.

Figure 4 .
Figure 4. Solving HETFPMSLM magnetic field with SC mapping

Figure 5 .Figure 6 .
Figure 5. Suspension force obtained by the SC mapping and FEM