**4. Including the Unbalance of Nonsinusoidal Voltage Waveforms**

The developed approach can be generalized to cases where asymmetry is observed in voltage supply waveforms, including the nonsinusoidal ones. The idea of the presented method is based on an independent solution of coupled sub-models by the application of single effective magnetic permeability distribution. As shown above, despite the nonlinearity of the problem, the use of the superposition principle allows one to obtain very accurate results. Therefore, in the case of supply unbalance, the method of symmetrical components can be used, allowing the presentation of the three-phase asymmetric distribution of a given harmonic of the supply waveform as a superposition of three symmetrical systems, namely the zero, positive and negative phase sequence system. To include the unbalance of the voltage supply, the model creation algorithm presented in the SubSection 2 should be modified as follows:

(I) Perform FFT analysis for the adopted non-linear asymmetrical supply waveforms {*eA*(*t*) , *eB*(*t*), *eC*(*t*)}. Extract the amplitudes {*EAhn*, *EBhn*, *EChn*} and phase angles {*ψAhn*, *ψBhn*, *ψChn*} for *N* of the most significant harmonics of the supply voltage with orders {*h*1, *h*2, . . . , *hN*}.

(II) Determine the amplitudes of three-phase symmetrical systems of zero, positive and negative sequences, respectively {*E*0*hn*, *E*1*hn*, *E*2*hn*} for each most significant harmonics of the supply voltage {*EAhn*, *EBhn*, *EChn*}:

$$
\begin{bmatrix}
\underline{E}\_{\text{Ohm}} \\
\underline{E}\_{\text{L}lm} \\
\underline{E}\_{\text{2}lm}
\end{bmatrix} = \frac{1}{3} \begin{bmatrix}
1 & 1 & 1 \\
1 & a & a^2 \\
1 & a^2 & a
\end{bmatrix} \begin{bmatrix}
\underline{E}\_{\text{Alu}} \\
\underline{E}\_{\text{Blu}} \\
\underline{E}\_{\text{Chu}}
\end{bmatrix} \tag{9}
$$

where *a* = *e<sup>j</sup>* <sup>2</sup> <sup>3</sup> *<sup>π</sup>*.


*Tehn* = *T*0*ehn* + *T*1*ehn* + *T*2*ehn*. For a zero sequence of the supplying voltage, adopt pulsation close to zero.


The above procedure was used to calculate the operational characteristics of the tested machine when supplied by the harmonic-rich square waveforms with the amplitude of one phase reduced by 25%. As shown in Figures 6 and 7, the presented approach allows one to obtain predictions very close to ones obtained from a comprehensive time-domain model, even in the case of a lack of supply symmetry. The reason why the assumptions hold despite the asymmetry is that the circumferential distribution of winding magnetomotive force is always a periodic function as it depends on the circumferential distribution of the winding that has nothing to do with voltage asymmetry. Moreover, the degree of voltage asymmetry does not break the validity of the assumptions.

In the considered case, the solution time using the proposed model increases to 8 min, while the execution time of the complementary time-domain model solution time is similar to the previously presented case and equal to 4 h 30 min. Connected with a very good agreement of results, this result clearly exposes the benefits of the developed approach.

Looking from the practical point of view, it should be noticed that the voltage unbalance problem is effectively reduced in modern fault-tolerant power converters; however, the circumferential unbalance of magnetomotive force due to the faults of the motor winding is a much more common case of the faulty operation of induction motor drives. The corresponding computational problem can be effectively solved by the algorithm in Section 3 with both winding distribution-related and slot-related spatial harmonics taken into account.

**Figure 6.** Results of computations for the square nonsymmetrical supplying waveforms: electromagnetic torque (**a**) RMS stator currents (**b**).

**Figure 7.** Comparison of the phase currents waveforms calculated with the use of the model formulated in the time domain (red line) with the waveforms calculated with the use of the developed method (blue line), as a superposition of harmonic waveforms, for the rotational speed of 24,000 rpm when supplied with square waveforms and 25% amplitude asymmetry.

### **5. Conclusions**

The development of the strongly coupled multi-harmonic field model concept effectively accounted for the nonlinearity and asymmetry of the voltage supply in the calculation of the operating characteristics of a high-speed induction machine with a solid rotor in a steady-state complex-valued finite element modelling framework. According to

the authors' opinions, in combination with the previously developed numerical method of determining the rotor end-effect coefficient [21], the multi-harmonic field-circuit model may become an effective tool in the process of designing the above-mentioned motor type. In particular, this may be an effective tool in the investigations on the reduction of losses due to higher harmonics of the magnetic field of various origins.

Of course, the present work does not cover all aspects of the issue. Further research will be undertaken to consider losses in the stator package, hysteresis losses in the solid rotor and even the influence of the power supply using pulse-width modulation (PWM) converters.

**Author Contributions:** Conceptualization, T.G. and M.J.; methodology, T.G. and M.J.; software, T.G.; investigation, T.G.; resources, T.G.; data curation, T.G.; writing—original draft preparation, T.G.; writing—review and editing, M.J.; visualization, T.G.; supervision, M.J.; project administration, T.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

