**3. Calculation Results**

The above approach was utilised to calculate the basic operating characteristics (phase current RMS and electromagnetic torque) of the machine described in Section 2.1 when supplied by three different waveforms presented in Figure 3 accompanied by their total harmonic distortion (THD) coefficients. The root mean square (RMS) value of the stator phase current *I* and the electromagnetic torque *Te* are determined on the basis of the following formula:

$$I = \sqrt{\sum\_{h1}^{hN} I\_{hn'}^2} \tag{6}$$

$$T\_{\mathfrak{E}} = \sum\_{h1}^{hN} T\_{elm} \tag{7}$$

where *Ihn* and *Tehn* are, respectively, the RMS value and the electromagnetic torque, obtained as a result of solving the model related to the *hn* harmonic of the supply voltage.

**Figure 3.** Three different types of supply waveforms considered in computations: square (**a**), six-step (**b**) and trapezoidal (**c**).

The calculated characteristics were compared with the results obtained with the timedomain model and by adopting only one fundamental voltage harmonic with the RMS value equal to the RMS value of the original supply waveform (note that all the waveforms shown have the same 50 V RMS value). All the considered models were implemented by the authors in the Matlab scripting language (Mathworks, Natick, MA, USA) [38]. To discretise the computational area an open-source generator (GMSH) was used [39]. In each case, the five most significant harmonics of the supply waveforms and two main

slot harmonics were used (−11, +13). Due to the star-connected winding without neutral wire, harmonics that are multiplicities of the third harmonic were not taken into account. As a result, in addition to the fundamental harmonics, harmonics 5, 7, 11 and 13 were also included. The voltage harmonics 1, 7 and 13 formed positive sequence of voltages, whereas the 5 and 11 formed a negative sequence of voltages. The obtained results of the calculations are shown in Figure 4.

**Figure 4.** Results of computations: electromagnetic torque (**a**) and RMS stator current (**b**) for square supply waveforms (see Figure 3a); electromagnetic torque (**c**) and RMS stator current (**d**) for sixstep supply waveforms (see Figure 3b); electromagnetic torque (**e**) and RMS stator current (**f**) for trapezoidal supply waveforms (see Figure 3c). Results obtained using the single-harmonic model and the proposed model are practically the same.

When analysing the results of the calculations one can notice a very high consistency between the characteristics calculated using the proposed approach in relation to the results of the calculations using the standard model formulated in the time domain. At the same time, one can see that by providing equality of the real and modelled waveforms the use of the RMS value is sufficient when the supply waveform distortion is small (THD < 10%). The developed methodology is also characterized by a relatively short calculation time. When the models related to individual harmonics are solved sequentially using a standard PC (Intel Core i7-5820K CPU @ 3.30 GHz, 16 GB RAM), the solution time for the speed 24 krpm with the square waveform supply is 2.5 min. For the sake of comparison, the timedomain model solution time is 4 h 16 min.

In addition to determining the static characteristics of the analysed machine, the developed method also allows for the recovering of steady-state current waveforms and making an assessment, for example, of electromagnetic torque resulting from the interaction of individual harmonics, both related to the supply waveforms and the magnetic circuit grooves. The steady-state current waveform can be determined by solving individual component models related to the individual harmonics of the supply waveforms (see Figure 5) as

**Figure 5.** Comparison of the phase current waveforms calculated with the use of the time-domain model (red line) with the waveforms calculated with the use of the developed method (blue line) for a rotational speed of 24,000 rpm: square supply waveform (**a**), six-step supply waveform (**b**), trapezoidal supply waveform (**c**).

Since the electromagnetic torque is calculated as the sum of the components coming from individual slot harmonics (within the model related to the considered harmonic of the supply wave), it is possible to perform a detailed analysis of the influence of these harmonics on its value and sign. An example of such an analysis for synchronous speed with a square wave is presented in Table 2. Because the torque components correspond with different harmonic slips, these are not directly proportional to power dissipation in the rotor. The results of the rotor harmonic power dissipation computation are presented in Table 3. These clearly show that the disadvantageous effect of distorted voltage wave (first row in Table 3) is comparable to the disadvantageous effect of slotting (first column in Table 3). These results best demonstrate the machine design areas where the proposed modelling framework can be especially useful.

**Table 2.** The results of the electromagnetic torque computations for the synchronous speed (30,000 rpm) in mNm produced by the interaction of the harmonics of the magnetic field in the gap (*n*—time harmonic number of voltage supply, *m*—spatial harmonic ordinal number). Value of total electromagnetic torque obtained from the time-domain analysis is −4.256 mNm.


(S): rotor in sync, (B): braking mode operation, (G): generating mode operation, (M): motoring mode operation, (+) means positive phase sequence, (−) negative phase sequence relative to the stator.

**Table 3.** The results of the rotor power dissipation computations for the synchronous speed (30,000 rpm) in Watt (*n*—time harmonic number of voltage supply, *m*—spatial harmonic ordinal number). Value of total rotor power dissipation obtained from the time-domain analysis is 19.97 W.


(S): rotor in sync, (B): braking mode operation, (G): generating mode operation, green: motoring mode operation, (+) means positive phase sequence, (−) negative phase sequence relative to the stator.

The above analysis was carried out at synchronous speed because only in such a case can the computed value of loss torque be compared with predictions obtained from a comprehensive time-domain model. It should be, however, noticed that the proposed model can be used at any speed allowing for a more detailed investigation on power dissipation in the rotor.
