## 2 Selection of Optimization Criteria for a Single-Phase Flux Reversal Motor  
The goal of this optimization is to reduce the torque oscillations (TO) of a single-phase FRM. By TO, we mean the deviations of the instantaneous values of the torque from its average value. A traditional measure of this deviation is peak-to-peak value of torque ripple (PPTR). This criterion has proved itself well for multiphase machines, for which TO are poorly expressed and, as a result, negative values of the torque are absent on the entire period of the machine. Single-phase motors [9; 17; 22; 23; 25; 26] have significantly higher TO than three-phase motors [24; 33; 34], and as a result, instantaneous torque values can be negative. In order to increase the minimum value of the torque and, if possible, make it positive, in this paper, we introduce another criterion of TO--average to minimum torque difference (AMinTD), i.e., average torque value minus the minimum torque value in the given mode. Obviously, both the PPTR and AMinTD criteria are positive. In this paper, we consider the optimization of a single-phase FRM operating in the fan load mode where the torque is proportional to the square of the speed. The considered speed range is from 80% to 100% of the rated speed of 18,000 rpm. The motor rated power is 754 W. Table 1 shows the characteristics of 2 considered modes of this range. In general, the average efficiency over operation modes can be calculated as:  
\[\eta=\frac{\sum\limits_{i}p_{i}P_{mech,i}}{\sum\limits_{i}p_{i}P_{A,i}}, \tag{1}\]  
where \(p_{i}\) is the fraction of the time occupied with \(i\)-th mode; \(P_{mech,i}\) and \(P_{A,i}\). It is assumed that the motor most of the time operates with underload, and the fraction of the time occupied by the mode is approximately inversely proportional to real (active) power. On substituting, \(p_{i}-1/P_{A,i}\), (1) turns into arithmetical average. To optimize FRM, the objective function was minimized:  
\[F=A\cdot B=<PPTR+2\cdot AMintD>(1-<Eff>), \tag{2}\]  
where \(A=<PPTR+2\cdot AMintD>\), \(B=1-<Eff>\) are optimization criteria, \(<x>\) = (\(x_{1}+x_{2}\))/2, means the arithmetic average of two modes, \(Eff\) is the motor efficiency. Criterion \(A\) provides a reduction in the torque oscillations, criterion \(B\) provides a reduction in losses and an increase in motor efficiency. Both criteria are positive. The meaning of the multiplication is that a 1% decrease in \(A\) is considered as valuable as a 1% decrease in \(B\). To optimize the single-phase FRM and minimize the target function (2), the Matlab implementation of the Nelder-Mead method (function \"fminsearch\") was used [35], since it does not require a gradient calculation and is applicable to noisy objective functions. The Nelder-Mead method is an unconstrained algorithm, and the region of the allowed parameters are not required. Only the objective function and the initial design must be provided. The noisiness of the objective functions (2) is due to rounding errors in the calculation based on the finite element method, as well as differences in the mesh for different calls to the objective function. Figure 1 shows parameters describing the design of the motor. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode Number** & **Rotational Speed, rpm** & **Torque, N-m** & **Mechanical Power, W** \\ \hline
1 & 14,400 (80\%) & 0.256 & 386 \\ \hline
2 & 18,000 (100\%) & 0.4 & 754 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Motor operating points. Figure 1: Motor design parameters: (**a**) stator; (**b**) rotor. FRM is powered by a pulse-wise voltage of 320 V, 0 V, \(-\)320 V, 0 V. Moreover, the durations of the positive and negative pulses are equal. The periods of zero voltages are also equal to each other. In the simplest case, the middle of the voltage pulses falls on the rotor position in the middle between two positions: when the middle of the rotor arc facing the gap is above the middle of the slot opening, and when it is above the middle of the tooth (Figure 2a).