## 1 Introduction  
Single-phase motors are used in low-power, cost-effective, variable-speed applications. Various types of single-phase motors with magnets are known, such as a synchronous motor with magnets on the rotor surface [1; 2; 3; 4; 5] and motors with magnets on the stator [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. The main advantage of single-phase motors compared to three-phase motors is the cheaper single-phase frequency converter. The main disadvantage of the single-phase machines is a low starting torque. Since the single-phase motors are cheap, they are used in low-cost variable speed drivers, such as fans [1; 2], compressors [3], pumps [4], vacuum cleaners [5], and other household appliances and fluid machinery in which a high starting capability is not necessary. The torque ripple of single-phase synchronous motors is much higher than that of three-phase motors. For instance, the torque ripple of the single-phase synchronous motor is 90% in [22] and 168% in [23], while that of the three-phase motor in [24] is 6.3%. The torque ripple of the brushed universal motor (AC commutator motor) is higher and can reach 300%, while the sign of the torque is the same during the entire electric period [25; 26]. So, single-phase synchronous motors with lower torque ripple than of universal motor and with the torque of constant sign is an acceptable replacement for universal motors. In high-speed applications, such as blowers, vacuum cleaners [5], and turbochargers [27; 28], the rotors of these machines are designed with a retaining ring, which is mandatory in machines with permanent magnets on the rotor to withstand centrifugal stress and to ensure rotor durability [29]. However, the use of the retaining ring on the rotor increases the complexity and the cost of the rotor manufacturing. Furthermore, the use of the retaining ring leads to an increase in the equivalent air gap (the distance between the stator teeth and the permanent magnets), and to a decrease in the magnetic flux density in the air gap, the efficiency and the specific torque of the machine. As a replacement to the single-phase synchronous machines with magnets on the rotor, motors with a rugged and reliable toothed rotor are considered for the high-speed applications [6; 7; 8]. One of these motors is a single-phase hybrid switched reluctance motor (HSRM) with magnets fixed on the stator, developed for a blender and described in [8]. In contrast to synchronous motors with magnets on the rotor, the HSRM has a simple toothed rotor without any magnets. The torque ripple of the single-phase HSRM [8] is 168%, and its torque is of constant sign throughout the entire electric period. Regardless of the above-mentioned benefits of the HSRM over traditional single-phase motors, its drawback is that it is supplied by unipolar voltage pulses, while traditional synchronous motors are supplied by the bipolar AC voltage. As a result, to reach the same peak-to-peak value of the HSRM stator current, its root mean square (RMS) and maximum values need to be significantly higher than those of a similar single-phase traditional synchronous motor. Because of this, the cost and the size of the converter for the HSRM increase. The power losses in the converter and in the stator winding also increase. Single-phase flux reversal machines (FRMs) have certain benefits over the HSRM and traditional synchronous motors. In contrast to traditional synchronous motors, an FRM has a simple and reliable toothed rotor similar to that of the HSRM. The use of permanent magnets mounted on the stator reduces its size and mass compared to the HSRM. Moreover, like traditional synchronous motors with magnets on the rotor, the FRM is supplied by bipolar voltage pulses. Current pulses of different signs in the FRM winding can occupy a larger part of the electrical period than in the HSRM. For this reason, the amplitude and RMS value of the FRM stator current decrease compared to the HSRM [9].