## Performance Analysis of a Transverse Flux Wheel Motor by a Non-Linear Mathematical Model

Wheel electric motors have gained an ever-growing interest in road electric vehicle applications, due to the absence of gearboxes. The configurations with an outer rotor directly coupled to the rim are particularly favorable, thanks to their compactness and to the lightness of the rotary components, resulting in an increase of the transportation capacity and the operating reliability. The main design constraints are related to the wheel weight and volume, to the large air-gap and to the performance optimization at low speed operation with high starting torque. The permanent magnet synchronous machine is able to fulfill such requirements and in particular the transverse flux configuration with surface permanent magnets (TFPM motor) can be a suitable solution for this kind of applications (Fig. 1, 2). In addition to its rotor lightness and cooling efficiency, such arrangement allows to compensate the torque ripple by displacing the magnets on the two sides of the rotor and to increase the torque by using two rotor modules, the radial size of the motor being maintained. The steady state running and the transient behavior of this motor can be analyzed by a non-linear mathematical model, in which the motor electromagnetic parameters are obtained by means of suitable magnetostatic FEM analysis. This technique can be profitable also in the preliminary design for the accurate definition of the drive rated quantities (motor and supply converter), taking into account the high variability of the operating conditions of urban vehicles.

Mathematical model: To solve the phase voltage equations, the winding flux linkages are expressed by analytical functions of the phase currents and of the rotor position. The h-phase flux linkage generally depends on three variables (radial position(Î¸) and two phase currents, since a wye connection is assumed). Nevertheless, to simplify the analytical function formulation, an approach which allows consideration ofÂ  only two state variables is used: the flux is evaluated assuming a single-phase supply with an equivalent current reproducing the effect of the mutual fluxes due the other phases. After sampling the flux data by sequences of 3D FEM solutions, an interpolation procedure expresses the dependence on current by polynomials and the dependence on Î¸ by truncated Fourier series. The time variation of the torque is also obtained by a suitable elaboration of the results of the same FEM analysis. After the calculation of the torque acting on the rotor when only the h-phase is supplied, the total torque is obtained by summing up the effects of the three phases and detracting the doubled value of the no-load cogging torque (otherwise its contribution would be considered three times).

Example of application: The performance of a 24-pole TFPM motor with rare earth permanent magnets is analyzed by means of the proposed methodology (geometrical sizes are given in Fig.2). The phases consist of six branches in parallel, each composed of two coils wound around the same core and series-connected. In order to calculate the model parameters, 16 values of the rotor positions Î¸ in the interval 0Â°, 7.5Â° and 13 values of the coil current in the range â€“60 , 60 A are considered; the simulations are executed by means of FLUX3D Magnetostatic module using a model which consists of two polar pairs and half stator core. Itâ€™s worth to point out that the data interpolation has been remarkably simplified by the regular profile of the results provided by FLUX3D, also with high saturation condition.

First, a current source supply is analyzed (I r.m.s. value, Æ’ frequency, Î³ phase angle) by considering a quadrature condition between magnet and armature fields (Î³=0Â°) and a condition of partial demagnetisation (Î³=45Â°). The profiles of the flux linkage related to one coil pair of phase 2 (Fig.3) show good correspondence between the proposed method and the FEM simulation with three-phase winding supply. Also the torque profile (Fig.4) is properly reproduced; the slight difference in some values can be mainly attributed to a poor reproduction of the mutual interaction between the phases and to a difference between the no-load and load cogging torque.

Afterwards an analysis with voltage source supply is carried out (U r.m.s. value, Î± phase angle). The steady-state waveforms of the current given in Fig.5 are in good agreement with the correspondent FEM transient solution obtained by of FLUX3D Transient magnetic module. Itâ€™s worth to point out that the FEM transient code requires no less than 35 hours on a ordinary PC to perform the above calculation, while the proposed method takes about 30 minutes for the solution on the same PC. Taking into account that the total elaboration time required to determine the mathematical model is about 60 hours due to the very fine mesh (Fig.6), the adoption of the proposed procedure is convenient in the design phase when several operating conditions have to be analyzed.

Finally, the motor performance as a function of speed is also determined by applying the mathematical model assuming a variable frequency current source supply (Fig.7). For speed values lower than the base speed (nb1=510 rpm), the current r.m.s. value I is kept constant with Î³=0Â° to obtain the mean torque Tem0 needed for the starting condition. Once the voltage reaches the rated value (U=Ur= 238 V) for n=nb1, it is kept constant for further speed increase. Then a flux-weakening operation is considered by varying the load angle Î³ to obtain a quasi-constant electromagnetic power in a restricted speed interval (nb1â‰¤nâ‰¤nb2 with nb2=680 rpm). In order to arrest the torque decrease for nâ‰¥nb2, all the coils are parallel-connected (nb3=755 rpm) when the current Iq is equal to Ir/2: after the connection change and by imposing Î³=0Â° the motor produces half the rated torque.

Conclusion: The methoddescribed above makes the analysis of a TFPM motor possible. Then this can be conveniently used as a wheel-hub motor. Though the methodology requires 3D FEM simulations to define a general mathematical model, the developed procedure considerably reduces the overall calculation time for the performance analysis with respect to FEM transient solutions, thanks to the flexibility and the accuracy provided by FLUX3D Magnetostatic module. Therefore the proposed methodology represents an efficient tool to define the ratings of the supply system and to investigate the control strategies more suitable to the various operating conditions of electric buses.