# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
###### Abstract  
There are three principal sources of noise and vibration in electrical machines: electromagnetic sources, mechanical sources, and aerodynamic sources. Nowadays, one of the major advantages of permanent-magnet synchronous machines is their torque density. This density is achieved through a high magnetic flux density in the air gap, which is achieved through hard magnets. Unfortunately, in these machines, electromagnetic forces have been identified as the main source of vibration and noise, and high magnetic flux densities make these vibrations and noises more significant. With the aim of better understanding the relationship between electromagnetic forces and design variables, this article, which is the continuation of previous work, firstly describes a study of the sources of magnetic forces in permanent-magnet synchronous machines. Subsequently, an analytical model for the computation of the radial forces originating from electromagnetic sources in permanent-magnet synchronous machines is stated. This model analyzes the forces on both the rotor surface and the base of the stator tooth. The analytical results were corroborated through simulations using the finite element method (FEM) and also by experimental tests performed over two prototypes. The results achieved by the analytical model show good agreement with both FEM results and experimental measurements. F 2021 11 10865 11.11.110865 11.11.110865 11.11.110865 11.110865  
In electrical machines that do not use the reluctance principle, the electromechanical energy conversion is based on an interaction between two magnetic fields in the air gap [4]. Through the analysis of these magnetic fields, many of the characteristics of the machine, such as the power, torque, and electromotive forces (EMF), can be obtained [6; 7; 8]. In [9; 10], researchers employed FEM to design electrical machines, and in [9], FEM was also used to calculate magnetic forces. The main advantage of FEM simulations is the high accuracy of the results. However, the computational load is high, and this problem is accentuated when the mesh is not properly defined. In order to avoid this problem, analytical models can be used instead of FEM. The low computational load of the analytical models makes them a very interesting tool, especially when a substantial number of iterations need to be performed, as is usual in design processes for electrical machines. In addition, analytical models give insight into the behavior of electrical machines which is more difficult to obtain via FEM simulations. Several methods for developing analytical models can be found in the literature. Probably the most extended methods are the lumped parameter models [11; 12] and models developed by Fourier series [13; 14]. The lumped parameter models can be described as very simplified FEM calculations, since the problem is divided into big elements and several parameters, such as flux sources and reluctances, are associated with each element. Once all elements are defined and the circuit is assembled, the problem is solved by applying numerical methods, as in FEM. Thus, the higher the number of elements the higher the computational load. On the other hand, when using Fourier series, spectra are considered and all harmonics can be described by analytical expressions. These expressions depend on design parameters such as dimensions, number of turns, and remnant fields of magnets, so they give insight into the problem, which is particularly important for designers. It is well known that the performance of electrical machines, for instance magnetic forces or torque ripple, depends strongly on the flux density in the air gap. Thus, in order to obtain accurate results, it is essential to properly define the spatial distribution of this variable. One possible approximation is to suppose a rectangular distribution. Few researchers employ this method because the distribution is not exactly rectangular and this assumption may lead to significant errors [15].