This paper presents a simple and efficient magnetic equivalent circuit (MEC) model for surface axial flux permanent magnet synchronous machines. The MEC model is used to solve all the electromagnetic properties of the machine including the no load, full load voltages, cogging torque, torque ripple and stator iron core losses. Moreover, this approach can be extended for all surface permanent magnet synchronous machines. The main novelty of this approach is the development of a static system, which accounts for the rotation. The model takes into account the rotor rotation via time dependent permanent magnet magnetization sources. The static system matrix facilitates a very fast solving. In addition, to take into account the 3D effect, a multi-slicing of the machine in the radial direction is done. This boosts the simulation time to only 60 seconds for 6 slices and 50 time steps including the non-linear behaviour of the stator elements with a great accuracy. Additionally, the number of elements in the MEC can be adjusted to reduce the computational time. This model is verified by means of 3D and 2D multi slice finite element (FE) models. In addition, experimental validations are also provided at the end. Index Terms-Analytical modeling, Axial flux permanent magnet synchronous machines (AFPMSM), Cogging torque, Magnetic equivalent circuit (MEC), Surface permanent magnet synchronous machines (SPMSM), Torque ripple. Peter Sergeant received the M.Sc. degree in electromechanical engineering and the Ph.D. degree in engineering sciences from
This paper presents an analytical solution of the eddy currents in the permanent magnets (PMs) in the Axial Flux Permanent Magnet Synchronous Machine (AFPMSM) using a coupled solution of Maxwell's equations and electric circuit network. This is based on calculating the axial field in an accurate way on the surface of the PM. This method is able to take into account the effect of armature field and slots. The eddy currents are obtained by imposing this solution on the PM which is modeled by a simple electric network. This network is composed of simple resistances and inductances. The inductances are used to model the reaction field effect of the eddy currents flowing through the PM and also the skin effect. To show this effect, the machine is excited with different sources at different speeds. Also, a variant of the model is made that neglects the inductances, in order to show in which conditions this "low-frequency approximation" is acceptable. It is concluded from this paper that inclusion of the reaction field is necessary when the machine is excited by a pulse width modulated (PWM) current, while for a sinusoidal excitation, the reaction field effect has minor contributions to the total eddy losses. In addition, the reaction field has major influence at higher speeds rather than lower speeds for PWM injection. In conclusions, for a preliminary design, the resistance model without the reaction field computation would be enough for the calculation of the PM losses. The circuit model is also capable of obtaining the solution with PM segmentation. In this paper different PM segments were studied from the circuit model and FE model point of view. Compared to the Finite Element (FE) Model, the circuit model has the advantage of flexibility in geometrical machine parameters, less CPU time, and accurate results for the PM losses up to 10%.
I. Introduction Due to their high torque density and an excellent efficiency, axial flux PM machines are favorable for vehicle propulsion and wind energy conversion. Although many effort was already done to simulate the electromagnetic properties of the machine using full 3D or multilayer-2D simulations, the work towards modelling of the thermal behavior is still very limited. As the disc-shaped rotors will operate as a fan during rotation, convective heat transfer in the air gap will have a major influence on the thermal design of the machine. Rather than performing full 3D combined fluid and thermal analysis, empiric analytical expressions based on CFD are used in this work to define the convective heat coefficient in different parts of the machine. The final thermal simulations are carried out for the stator and rotor individually, where only a segment needs to be modelled due to thermal periodicity. As a result, the use of this coupled modelling technique results in a significant decrease of the overall simulation time. The coupled electromagnetic and thermal modelling techniques are illustrated on a 4kW axial flux PM machine having the yokeless and segmented armature (YASA) topology. Finally, both the electromagnetic and thermal simulation results were validated on a preliminary prototype. II. Coupled Electromagnetic and Thermal Modelling In the first step of the analysis, the electromagnetic behavior of the machine is modelled. The results from the electromagnetic field computations are used to calculate the corresponding losses in post-processing. In a second step, these losses are used as the heat sources for the thermal simulations. As the axial flux PM machine has an inherent 3D structure, multilayer-2D simulations [1] are used to obtain the electromagnetic properties such as flux-linkage, back-emf and (cogging) torque. In postprocessing, the magnetic flux density pattern of the different layers in the core is used to calculated the core losses. The solutions for the air gap magnetic flux density pattern of each layer are combined and used for a time harmonic calculation of the eddy current losses in the permanent magnets [1]. Together with the Joule losses in the machine winding, the iron losses in the stator cores and eddy current losses in the PM's are used as the source terms in the thermal simulations. As the thermal process is particularly slow with respect to the electromagnetic one, the time average values of the losses are chosen as a source in the thermal model. Instead of using full 3D combined fluid and thermal analysis, empirical equations derived in [2] are used to model the convective heat transfer from each of the surfaces near the air gap region. Next to an empirical formula for the convective heat coefficients of each surface, an expression for the reference temperatures is also derived using dimensionless numbers. The reference temperature is the bulk fluid temperature of the domain near each surface which can be expressed as a function of the average stator, rotor and ambient temper...
To find the temperature rise for high power density yokeless and segmented armature (YASA) axial flux permanent magnet synchronous (AFPMSM) machines quickly and accurately, a 3D lumped parameter thermal model is developed and validated experimentally and by finite element (FE) simulations on a 4 kW YASA machine. Additionally, to get insight in the thermal transient response of the machine, the model accounts for the thermal capacitance of different machine components. The model considers the stator, bearing, and windage losses, as well as eddy current losses in the magnets on the rotors. The new contribution of this work is that the thermal model takes cooling via air channels between the magnets on the rotor discs into account. The model is parametrized with respect to the permanent magnet (PM) angle ratio, the PM thickness ratio, the air gap length, and the rotor speed. The effect of the channels is incorporated via convection equations based on many computational fluid dynamics (CFD) computations. The model accuracy is validated at different values of parameters by FE simulations in both transient and steady state. The model takes less than 1 s to solve for the temperature distribution.
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