Comparison of Main Magnetic Force Computation Methods for Noise and Vibration Assessment in Electrical Machines

Abstract

This paper presents a comparison of several methods to compute the magnetic forces experienced by the stator teeth of electrical machines. In particular, the comparison focuses on the Virtual Work Principle (VWP) based nodal forces and the Maxwell Tensor (MT) applied on different surfaces. The VWP is set as the reference. The magnetic field is computed either with Finite Element Analysis (FEA) or with the semi-analytical Subdomain Method (SDM). Firstly, the magnetic saturation in iron cores is neglected (linear B-H curve). Then, the saturation effect is discussed in a second part. Homogeneous media are considered and all simulations are performed in 2D. The link between slot's magnetic flux and tangential force harmonics is also highlighted. The comparison is performed on the stator of a Surface-Mounted Permanent Magnet Synchronous Machine (SPMSM). While the different methods disagree on the local distribution of the magnetic forces at the stator surface, they give similar results concerning the integrated forces per tooth, referred as Lumped Forces. This conclusion is mitigated for saturated cases: the time harmonics are correctly computed with any of the presented Lumped Force methods but the amplitude of each harmonic is different between methods. Nonetheless, the use of semi-analytical Subdomain Method remains accurate with Maxwell Tensor in the air-gap even with saturation for design and diagnostic of electromagnetic noise in electrical machines. However, for more accurate studies based on local magnetic pressure, the Virtual Work Principle is strongly recommended.

Analytical study of air-gap surface force – application to electrical machines

“…2ˆ ( ) F R θ ag can be used accurately for vibroacoustic assessment [14,32]. Indeed, the air-gap cylindrical band thickness g is generally very small compared to the stator bore radius R s such that the following approximations can be made:…”

“…where t is the unit tangential field associated with n. Expression (8) is usually applied on a cylindrical surface Γ in the middle of the air gap [5,[12][13][14][15][16][17]32]. An example is provided in Figure 2 with the blue dashed surface Γ.…”

“…As seen in ref. [32], magnetic forces are mainly concentrated on the tip of the stator teeth as surface forces in the case of low saturation (or high relative permeability). To reduce differences with the interface surface forces on the tooth tip (15), the AGSF (10) should be applied at the stator bore radius R s .…”

“…Several models have recently been proposed to compute from the AGSF the equivalent forces which applies to the neutral fiber [30,31]. In some cases, the AGSF is integrated as lumped tooth forces [14,32].…”

Experimental Benchmark for Magnetic Noise and Vibrations Analysis in Electrical Machines

“…In case of under-sampling, force harmonics with high spatial orders may excite lower structural modes due to aliasing effect. This phenomenon exists in topologies where the number of teeth is close to the number of poles [17]; 3) the computation of magnetic forces applied to a slotted structure, regarding the method formulation which depends on the electromagnetic model (mainly Virtual Work Principle for FEM [18] versus Maxwell Stress Tensor for (semi-) analytical models [1], [10]) and also on the modeling assumption (either local (or nodal) forces applied everywhere on the surface or lumped forces (resultant) applied on each tooth) whose equivalence is still under discussion [19]. 4) the noise and vibrations reduction strategies including passive reduction (such as rotor and/or stator skewing [20], [21], rotor and/or stator notching [22], [23], magnetic wedges [7], flux barriers [24], etc.)…”