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This paper studies the propellant and levitation forces of a prototype maglev system where the propellant forces are provided by a linear motor system. For this purpose, the mathematical model and method using finite element method coupled to external circuit model is developed. The details of the propellant and levitation forces for a prototype maglev system under different operating conditions are investigated, and some directions are given for practical engineering applications.

Due to the exclusive salient advantage of non-contact surfaces, an ever-increasing effort has been dedicated to the application of the magnetic levitation technology in engineering disciplines such as maglev trains [

Generally speaking, to address the aforementioned issues of the electromagnetic problem of a maglev system, it is essential to use three dimensional finite element methods [

The transient electromagnetic fields in the propellant and levitation systems are determined using 2D time-stepping finite element method. To determine the field distribution at each time step in a Maglev system, the 2D transverse section spanning one pole pitch of the linear synchronous motor is studied. As shown in

where, n is the reluctivity, s is the conductivity, J is the

source current density, A(x,y,t) is the magnetic vector potential.

For a moving conductor with a velocity v, the induced electric density is

Using the Galerkin approach to discretize (1), one obtains

where,

, ,.

where, a is the number of the parallel branches of the winding, N_{1 }is the number of total serial turns in one coil, S_{b }is the slot area, [N]_{e} is the shape function of the finite element method.

In general, the source current density in (1) is unknown. Thus, the external electric circuit model is coupled to the finite element formulation in this paper to consider the voltage source. Moreover, the end effect of the propellant and levitation systems is also modeled in this circuit formulation. _{e}, and resistance R of the windings, is

where, [u] = [-u_{f} u_{a} u_{b} u_{c} u_{d}]^{T}, [I] = [i_{f} i_{a} i_{b} i_{c} i_{d}]^{T}, [R] = diag [r_{f} r_{a} r_{b} r_{c} r_{d}], [L_{e}] = diag [L_{s}_{f} L_{s}_{a} L_{s}_{b} L_{s}_{c} L_{s}_{d}] is the leakage inductance considering the end effect of the machine and levitation magnet, subscript f denotes the winding of the levitation system/the rotor of the linear motor.

To express e in terms of A(x,y,t), one has

where L_{ef} is the effective length of the core of the propellant and levitation systems.

Integrating (3) and (5) as a whole, one reads

(6)

Applying Crank-Nicolson algorithm to (6) with respect to the time variable, one obtains the following equation set of the time-stepping finite element coupled to external circuit model

Rearranging (7) into a symmetric form, one has

where, , h is the size of the time step.

To consider the relative movement of a maglev system, the moving boundary is used [

boundaries are the boundaries connecting the stator and rotor meshes of the propellant system, and will vary with the movement. The nodes in the stator side and their counterparts in the rotor side satisfy either periodic or semi-periodic boundary conditions. For example, for a specific relative position of the stator and rotor as depicted in

Once the electromagnetic field is determined in each time step, the corresponding propellant and levitation forces of the maglev system in that time instant can be determined from

In the case study, f_{x} is the propellant force, and f_{z} the levitation force.

To predict the transient performances of a maglev system, the computer codes using the proposed models and methods are developed by the authors. The codes are programmed in Fortran language.

To validate the proposed mathematical model and method as well as the computer codes, they are firstly used to compute the no-load propellant and levitation forces of a prototype Maglev train in Shanghai commercial Maglev Line. The mesh of total 4784 nodes and 9197 elements as shown in

After the accuracy and feasibility of the proposed model and method are confirmed, the transient performances of the propellant and levitation forces of a prototype Maglev system under different operating conditions are studied. Firstly, the effect of the torque angle of the linear motor on the propellant and levitation forces is investigated. As shown in

in electrical degrees between the axes of the magnetic fields generated by the stator winding (windings A, Z, B in

the forces as given in these figures are normalized to their rated values. From these numerical results, it is obviously that 1) The torque angle of the linear motor has a significant effect on both the propellant and levitation forces;

2) The averaged propellant and levitation forces will change periodically with the torque angle.

Therefore, the running state of a Maglev system can be controlled by changing the torque angle of the linear motor.

Since the front end of the linear motor is generally a steady electronic device, typical a PWM source for high precise motion control, the high frequency harmonics are unavoidable in engineering applications of a Maglev system. In this regard, the effects of harmonics on the propellant and levitation forces are then studied using the proposed numerical model and method. In the computer simulation, the harmonic voltages are formulated as

where V_{f} and f_{f} are, respectively, the amplitude and frequency of the fundamental voltage of the source.

In the numerical implementation, the torque angle of the linear motor is set to different values. Figures 9 and 10 depict, respectively for torque angle d = 0° and d = 10°, the differences of the transient propellant and levitation forces of the prototype maglev system between the normal operating condition and the aggregation of harmonic voltages and currents. From these numerical results, it is clear that:

1) The averaged values of the differences of the transient propellant and levitation forces for one period will approach zero, this means that the harmonics of the sources have almost no effect on the steady state performances of the Maglev system;

2) In view of the transient propellant and levitation forces, the aggregation of harmonic voltages and currents will result in that the forces oscillate around their rated values, resulting in degradation in the transient performances of the Maglev system;

3) Relatively, the effect of a small harmonics on the levitation forces can be neglected compared to that on the

propellant forces.

A model and method for computing the transient propellant and levitation forces of a Maglev system, with relative error being smaller than 5%, is proposed based on time stepping finite element method coupling to external circuit models. In addition to having the advantages to consider complications such as the relative movement of different components of the linear electrical machines, the saturation of iron materials, the proposed model and method can also take into account the interaction of an external voltage source which is very common with the increasing use of power electronics devices in the front end of general electromagnetic devices. Based on the computer simulation of this case study, for a maglev system, it is concluded that:

1) The torque angle of the linear motor has a significant effect on both the propellant and levitation forces;

2) The averaged propellant and levitation forces will change periodically with the torque angle;

3) The addition of small harmonics in the sources has almost no effect on the steady performances;

4) However, the aggregation of small harmonic voltages and currents in the sources will result in degradation in the transient performances.

This work is supported by the Science and Technology Department of Zhejiang Province under Grant No. 2008C31021.