Build the Permanent Magnet Model for Analysis

Step 1: Build the Model of No-Load Air-Gap Magnetic Flux Density. The interior structure of a typical PM brushless DC motor is shown in Figure 2. To determine the motor’s no-load air-gap magnetic flux density, firstly it is required to obtain the air-gap magnetic flux density by neglecting  the teeth/slots influence. The Schwarz-Christoffel transformation is used to build the relative magnetic permeability function of the air gap. Without considering the influence of teeth/slots, the analytical equation of the no-load air-gap magnetic flux density can be determined. After that, by taking the teeth/slots effect into consideration, the analytical equation of the no-load air-gap magnetic flux density can also be determined. The magnetic field distribution can then be obtained based on these equations and the boundary conditions that have been configured. Furthermore, the radial component and the tangential component of the air-gap magnetic flux density can be obtained as well. By solving the above-mentioned partial differential equations, the relative magnetic permeability function, the analytical model and the analytical solution of the motor’s no-load air-gap magnetic flux density can also be obtained. At this stage, the air-gap magnetic flux density of teeth/slots is negligible. However, the motor’s load air-gap magnetic flux density should not be neglected since the magnetic field induced by the motor’s coil (i.e., armature winding) is a pulsed rotating magnetic field when the motor is operating under load. The coil is then generating inductance within its armature winding. When the armature reacts in a way of pulsed magnetic field, this means the direction of the electric current within the coil’s armature winding is alternating. The armature’s transformed reaction magnetic field generates the induced electromotive force within the coil winding. This phenomenon affects the current waveform of the coil’s armature winding and the motor’s torque characteristics.

Step 2: Build the Model of the Air-Gap Magnetic Flux Density and the Induced Electromotive Force and Determine the Essentials for Analysis. The next step is to build the PM brushless DC motor’s air-gap magnetic flux density and conduct the finite element analysis of the induced electromotive force. The problem to be solved by the analytical method has been simplified and the requirements of boundary conditions are more stringent. In other words, complex boundary conditions might be difficult to converge. As a result, it is required to assume the slots are infinitely deep when solving for the relative magnetic permeability function. Furthermore it also require to assume the slots as rectangular ones, which this makes it not easy to analyze the influence of different slot shapes and further confines its application to practical product developments. Therefore, the analytical methods often fail to obtain a solution. On the other hand, the FEM is provided with theoretical foundation and has been extensively applied in various fields as an effective numerical
analysis method. Moreover, it is also an effective approach of analysis for different boundary conditions and complicated problems with irregular structures and shapes. It divides an enclosed domain which contains the continuous function in partial differential representation into a limited number of small areas. That is, each of the small areas is represented by a selected approximation function so that the entire domain function can be discretized. Thereby we can obtain a group of similar algebraic equations which lead to simultaneous equations that can be solved for the approximate value. At the moment, the FEM is the most extensively used approach for calculating the electromagnetic field for motors. The advantage of using FEM is fourfold:

 1) The coefficient matrix
is symmetrical, positive definite, and sparse and can help save a considerable amount of calculation
time. 

2) It is very convenient when dealing with the type 2 boundary condition and the interface condition with the internal medium. No further processing is required for the type 2 boundary condition and the medium interface condition with no surface current density. 

3) It allows flexible
mesh configuration, which makes solving geometrically complex shapes such as motors easy.

 4) It is very convenient to handle the characteristics of any non-linear medium such as the core magnetic saturation. No excessive assumption is applied to the physical model of the motor during the calculation process. This approach is provided with higher accuracy and can accurately calculate a motor’s performance and parameters under the effect of teeth/slots.

Step 3: Build the Model of Air-Gap Magnetic Flux Density, No-Load Magnetic Field, and Load Magnetic Field and Determine the Essentials for Analysis. For a motor with no load, only a permanent magnet takes effect and generates air-gap magnetic flux density. Therefore, it is required to build a mathematical model of the permanent magnet for further analysis. Due to the fundamental relationship between the electric current and the magnetic flux density, any magnetic field can be regarded as the product of the current distribution. A permanent magnet can be modeled by two types of current simulation approaches as follows.

 1) Simulation of a permanent magnet by volume current.

Therefore, only the equivalent surface current exists. To simplify the calculation of a motor’s no-load
magnetic field, we made assumptions during the analysis as follows. 1) The material is isotropic and
the hysteresis effect of ferromagnetic materials can be neglected

 2) The PM material has been
uniformly magnetized.

 3) The variation in the motor’s axial magnetic flux density can be neglected.
The above-mentioned approach can be utilized to quickly and accurately calculate the magnetic vector potential of each motor node. Furthermore, it helps obtain the no-load air-gap magnetic flux density of each of the motor nodes so as to obtain the distribution of the air-gap magnetic flux density waveform. Fourier transform was then applied to the air-gap magnetic flux density and the fundamental wave and other orders of harmonic waves of the air-gap magnetic flux density were obtained, along with the single-conductor fundamental-wave electromotive force and the phase winding magnetoresistance fundamental-wave electromotive force. To facilitate modeling and analyzing efforts and to reduce development time by design assessments, the JMAG software was utilized for the magnetic field analysis and control circuit simulation so as to obtain a motor’s no-load and load air gap magnetic field and waveform. The results were cross verified with the calculation results obtained by the FORTRAN program earlier. During the calculation of a motor’s inductance parameters, the winding layout is regular and symmetric if the number of slot winding is an integer since the slot counts for each pole or each phase are integers. On the other hand, if the fractional slot winding approach is used, its winding layout will be different from the integer slot winding designs. The winding layout of a fractional slot winding design is more flexible. For the simulation in this study, the model was built by the fractional slot winding approach so as to generalize the motor design for future applications. When calculating the winding inductance parameters, only the
winding current effect can generate the air-gap magnetic flux density. In other words, a permanent magnet’s boundary equivalent surface current density in the magnetic vector equation equals to zero, i.e., Js=0. The vector magnetic potential of a unit node when only the electric current exists can be obtained. The winding inductance can thus be obtained from vector magnetic potential, stator core length, and the turns-in-series within each phase winding. In addition, the slot average magnetic vector potential can be obtained from the weighted average of the magnetic vectors for all of the units within a slot.