Winding Tensor Approach for the Analytical Computation of the Inductance Matrix in Eccentric Induction Machines

Abstract

Induction machines (IMs) are critical components of many industrial processes, what justifies the use of condition-based maintenance (CBM) systems for detecting their faults at an early stage, in order to avoid costly breakdowns of production lines. The development of CBM systems for IMs relies on the use of fast models that can accurately simulate the machine in faulty conditions. In particular, IM models must be able to reproduce the characteristic harmonics that the IM faults impress in the spatial waves of the air gap magneto-motive force (MMF), due to the complex interactions between spatial and time harmonics. A common type of fault is the eccentricity of the rotor core, which provokes an unbalanced magnetic pull, and can lead to destructive rotor-stator rub. Models developed using the finite element method (FEM) can achieve the required accuracy, but their high computational costs hinder their use in online CBM systems. Analytical models are much faster, but they need an inductance matrix that takes into account the asymmetries generated by the eccentricity fault. Building the inductance matrix for eccentric IMs using traditional techniques, such as the winding function approach (WFA), is a highly complex task, because these functions depend on the combined effect of the winding layout and of the air gap asymmetry. In this paper, a novel method for the fast and simple computation of the inductance matrix for eccentric IMs is presented, which decouples the influence of the air gap asymmetry and of the winding configuration using two independent tensors. It is based on the construction of a primitive inductance tensor, which formulates the eccentricity fault using single conductors as the simplest reference frame; and a winding tensor that converts it into the inductance matrix of a particular machine, taking into account the configuration of the windings. The proposed approach applies routine procedures from tensor algebra for performing such transformation in a simple way. It is theoretically explained and experimentally validated with a commercial induction motor with a mixed eccentricity fault.

Analytical Model of Induction Machines with Multiple Cage Faults Using the Winding Tensor Approach

“…Only the radial component of the main flux that crosses the smooth air gap is considered in this work, and the iron permeability has been considered as infinite. The analytical computation of mutual inductances considering also the tangential component of the flux can be found in [ 25 ], and with non-uniform air-gap in [ 29 ]. A higher precision can be achieved using numerical methods, such as those based in FEM [ 20 , 47 ], but at the cost of an increased computing complexity.…”

“…In [ 26 ], the calculation of self and mutual winding inductances with spatial harmonics was made by first establishing the mutual inductance between two elementary coils, in different relative positions, and then transforming it into winding inductances using a transformation matrix, as in (5). In [ 29 ], a similar procedure was presented, but using a single conductor instead of a coil as the most basic unit, which avoids the need to establish a different winding function for each relative position between two coils.…”

“…In [ 25 ], two layers of conductors have been considered instead, one placed on the inner stator surface and the other one placed on the outer rotor surface. The maximum spatial harmonic of the winding that can be represented in this reference frame is , and, therefore, a high value of N is selected ( in [ 29 ]).…”

“…Nevertheless, this approach requires complex winding functions that depend on the relative position of the coils, on the coils MMF functions, on the permeance function of the air gap, and on the rotor position, leading to triple integrals for each pair of coils [ 15 ]. On the contrary, the winding tensor function approach [ 29 ] uses the conductor as the most basic unit, instead of the coil, and gives the self and mutual inductances of the IM windings applying routine tensor algebra functions, following Kron’s method [ 26 , 30 ].…”

“…The methodology proposed in this work greatly simplifies the process for calculating the parameters of the SCIM model under different rotor asymmetry conditions, using a novel approach: instead of obtaining directly the parameters of the SCIM under faulty conditions, which is a difficult computation that must be done for each type of fault or combination of faults, only the parameters of the healthy SCIM are obtained, using the winding tensor approach [ 29 ]. The parameters of the SCIM under any type of rotor asymmetry (bar breakages, end-ring breakages) are then obtained using simple connection tensors, whose elements are only zeros, ones or minus ones, and applying routine tensor algebra operations.…”