## 1 Introduction
These make it possible to apply the switchers of a lower power in the frequency converter for the FRM. In addition, the DC-AC converter for an FRM has a lower power loss. Therefore, a smaller heat sink can be used, and the sizes of the converter can be smaller than in the case of the HSRM [9]. In recent years, many examples of three-phase FRMs have been discovered [10; 11; 12; 13]. The three-phase motors, however, demand a more costly power converter and a more complex control. For this reason, the use of a single-phase FRM with the simple power converter and control configurations is preferred in the targeted applications. Besides, all known three-phase FRMs [10; 11; 12; 13] with an even number of the stator and rotor teeth have many poles and cannot be applied in high-speed applications due to high supply frequency, high losses in the steel and in the magnets. Design of a single-phase FRM is not so often discussed in the literature. At present, only several various configurations of a single-phase FRM are known. The first basic configuration of a single-phase FRM was described in [14; 15]. This motor design includes two teeth on the stator and three teeth on the rotor. Permanent magnets forming two poles on each of the stator teeth are installed on their surface. However, the single-phaseFRM configuration presented in [14] demonstrates the following drawbacks: (1) Only two-thirds of the internal surface of the stator are used; (2) the lifetime of the bearings decreases due to the radial force arisen from the asymmetry of the motor structure. These problems of the FRM [14] were resolved by another single-phase FRM described in [16; 17]. In [17], a single-phase high-speed FRM was optimized for fan load in order to reduce losses (increase its efficiency). The torque pulsations in [17] were not included into the target function, therefore, their value is extremely large and equal to 335%. In addition, in the FRM [17], there is an interval at which the motor torque is negative. In [21], a new design of a single-phase FRM with the asymmetric rotor was described to reduce torque pulsations. The new single-phase FRM with the asymmetric rotor was compared with the single-phase FRM with the symmetric rotor. In the rated mode, the calculated torque ripple for an asymmetric rotor FRM was reduced by 1.6 times compared to a symmetrical rotor of the FRM. However, in [21], the FRM was designed without using optimization methods; therefore, the characteristics of the FRM can be improved. The aim of this work is to minimize the torque ripple and increase its minimum instantaneous value, as well as to reduce FRM losses. Few papers are dedicated to the optimization of single-phase high-speed machines. The optimization of the 1.5 kW, 40 krmp single-phase with the response surface method is considered in [30]. With the help of the optimization, the authors [30] tried to increase the efficiency, however, the torque oscillations were not considered. The 500 W, 100 krmp single-phase motor with magnets on the rotor is investigated in [31]. The authors decreased the cogging torque and the EMF (electromotive force) ripple, but the data on the torque ripple and the efficiency are not provided in [31]. In [32], the authors decreased the torque ripple, and achieved the positive torque throughout the electric period, but the efficiency was not investigated in [32]. In this work, to solve the optimization problem, an objective function is proposed, which includes parameters characterizing the pulsations of the torque and the loss of FRM for two load conditions. To optimize the single-phase FRM and minimize the function, the Nelder-Mead method was applied. After optimization, the torque ripple in the FRM in the nominal loading mode decreased by 1.7 times, and in the mode with reduced load by 2.7 times. In addition, after optimization, it was possible for the FRM to exclude sections with a negative value of the torque in the period.