**6. Use of finite element analysis in PM servomotors**

The finite element method (FEM) is a numerical method for solving the complex electromagnetic field problems and circuit parameters. It is specifically convenient for problems with non-linear material characteristics where mathematical modeling of the system would be difficult. This method involves dividing the servomotor cross section or volume into smaller areas or volumes. It could be 2D objects in the case of 2D FEM analysis or 3D objects in the case of 3D analysis. The variation of the magnetic potential throughout the motor is expressed by non-linear differential equations in finite element analysis. These differential equations are derived from Maxwell equations and written in terms of vector potential where the important field quantities such as flux, flux direction and flux density can be determined.

The FEM can accurately analyze the magnetic systems which involve permanent magnets of any shape and material. There is no need to calculate the inductances, reluctances and torque values using circuit type analytical methods because these values can simply be extracted from the finite element analysis. Another important advantage of using FEM over analytical approach is the ability to calculate the torque variations or torque components such as cogging torque, ripple torque, pulsating torque and average torque accurately without too much effort.

There are various FEA packages used for motor analysis. FEA packages have 3 main mechanisms which are pre-processor, field solver and post-processor. Model creation, material assignments and boundary condition set-up are all completed in the pre-processor part of the software. Field solver part has 4 main steps to solve the numerical problems. After the pre-process, the software generates the mesh, which is the most important part of getting accurate results. User's experience in generating the mesh has also an important effect on the accuracy of the results. Then, the FEA package computes the magnetic field, performs some analysis such as flux, torque, force and inductance, and checks if the error criteria have been met. If not, it refines the mesh and follows the same steps based on the user's inputs until it reaches the specified error limit. This procedure is shown in Fig. 12. In the post-processor, magnetic field quantities are displayed and some quantities such as force, torque, flux, inductance etc. are all calculated.
