**1. Introduction**

Despite the growing popularity of permanent magnet electric machines, induction motors still remain the main components of industrial electric drives. This is due to the simplicity of construction, reliability and the development of static converters that allow the shaping of static characteristics and dynamic parameters of the drive system. Among the various designs, the induction machine with solid rotor deserves special attention, being one of the simplest and oldest AC electric machines, although requiring entirely different design routines that have been in development for nearly a century [1–5]. The mechanical properties of the solid rotor, as well as their resistance to aggressive chemical compounds, are incomparable to any other structure, which makes them a suitable source of drive-in applications requiring high or very high rotational speed such as those shown in [6–13].

The analysis of the properties and design of the discussed machine is still considered a difficult task, mainly because of the very complex electromagnetic phenomena occurring in the solid rotor. The only method that allows for the overall effect of these phenomena to be taken into account is to utilize a non-linear three-dimensional numerical model formulated in the time domain. Unfortunately, as the calculations would take a long time, this method is difficult to apply in practice. For this reason, alternative approaches have been developed which, in principle, can be divided into two groups: methods based on the analytical solution of the field problem in the solid rotor presented in [14–18], and

**Citation:** Garbiec, T.; Jagiela, M. Accounting for Slot Harmonics and Nonsinusoidal Unbalanced Voltage Supply in High-Speed Solid-Rotor Induction Motor Using Complex Multi-Harmonic Finite Element Analysis. *Energies* **2021**, *14*, 5404. https://doi.org/10.3390/ en14175404

Academic Editor: Armando Pires

Received: 19 July 2021 Accepted: 27 August 2021 Published: 30 August 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

methods utilizing the two-dimensional numerical models formulated as time-domain or frequency-domain models used in works [7,9–11,19–21].

The basic problem related to the analysis of high-speed induction machines with solid rotors using two-dimensional numerical models formulated in the time domain is a very long computation time resulting from a significant ratio of the electromagnetic time constant of the machine to the supply voltage period. In the case where the subjects of consideration are the steady states of the machine, the computation time can be shortened by applying special modifications of the model formulated in the time-domain, such as: elimination of the DC flux linkage in [22], elimination of the DC current at each step in [23], transient magnetic solution initialized with proper initial values [24] or initial current extraction in [19].

A very promising alternative to the models formulated in the time domain is the use of modified time-harmonic models, such as, for example, the multi-harmonic field model with a strong coupling, which takes into account the influence of the spatial harmonics due to the slotting or winding distribution on the losses in the solid rotor and the value of the developed electromagnetic torque [25–31]. In these models these effects can be accounted for using the modified slip

$$s'=1-\nu(1-s)\tag{1}$$

where *<sup>ν</sup>* is the ordinal number of spatial harmonic (*<sup>ν</sup>* <sup>∈</sup> <sup>Z</sup>) and *<sup>s</sup>* is the fundamental slip.

As shown in the works [30,31], with the appropriate formulation of the stator–rotor coupling scheme and the method of modelling materials non-linearity, it is possible to obtain high accuracy of the results of steady-state calculations in a very short time of model execution. This is because relatively coarse mesh densities are sufficient for the representation of the most important air gap magnetic field harmonics.

In this moment, the application of this type of model was limited to the case of the symmetrical sinusoidal voltage supply of the stator winding. In the present work, the concept of multi-harmonic effective magnetic permeability proposed in [31] was used to develop an original approach that allows for taking into account the nonsinusoidal voltage supply waveforms (that appear, for example, when supplying the machine through a quasisquare voltage inverter) in a multi-harmonic model of a solid rotor induction machine. It is shown that the steady-state time-harmonic model with only a few dominating timeharmonics of the magnetic field attributed to the voltage supply provides solutions very similar to ones obtained from a time-demanding time-domain solution. This is in terms of the magnitude and phase relationships of the waveforms as well as the computed torques and power losses. The execution time of the former is however only a fraction of that of the latter. Moreover, the model is demonstrated to work correctly in cases where the voltage supply exposes noticeable asymmetry that appears during inverter fault conditions. The above means that in the mathematical sense the model is capable of accounting for the space-harmonics and the time-harmonics that come from the power supply, but not from the motional effects which are more important in the squirrel cage machines. It is thus most suitable to analyze induction motors with solid rotors. To the best of the authors' knowledge such a model has not been proposed until this moment.

The basic operating characteristics of the machine were calculated for four different supply voltage waveforms, and then compared with the results of calculations obtained via the comprehensive model formulated in the time domain which in the steady-state condition considers the whole spectrum of magnetic field harmonics. Original contributions of this work include:


allowing for a more detailed analysis of the impact of stator slotting and power supply harmonics on machine efficiency.
