## 1 Introduction  
Flux Reversal Machines (FRMs) are doubly salient machines with permanent magnets (PMs) glued on the stator. The stator windings are of a concentrated type, and the flux linkage of the PMs reverses its polarity in the stator windings when the rotor rotates [1]. A first basic configuration of single-phase FRMs was described in [1] (Figure 1a). This FRM has a single phase, two stator teeth with two PM poles on each of them, and three rotor teeth. The FRM of this kind was designed for a high-speed vacuum cleaner [2]. Thanks to their geometry design, single-phase FRMs have several merits compared to the conventional permanent magnet synchronous machines (PMSMs), as follows: They are characterized with a simple and robust rotor structure due to the absence of PMs and windings in the rotor, which makes them easier for manufacturing and reduces their cost. Therefore, they can work properly and reliably in high-speed applications (over 10 k\(\mathrm{r}\mathrm{p}\mathrm{m}\)) [1; 2; 3; 4; 5]. However, the single-phase FRM design (Figure 1a), presented in [1; 2], has the following disadvantages:1. Only two thirds of the stator internal surface are used, which decreases the power density and efficiency of the motor;
2. There is no geometrical symmetry of rotating FRM as a whole by 180\({}^{\circ}\), which causes radial force acting in the rotor. This radial force decreases the lifetime of the bearings and increases the acoustic noise and losses in the motor. In this paper, in order to overcome the above disadvantages of the single-phase FRM shown in Figure 1a, a single-phase FRM shown in Figure 1b proposed in [3] is used to develop a three-phase FRM. A two-dimensional (2D) finite element model (FEM) is constructed to simulate the performance of the proposed motor. Furthermore, an optimization strategy based on the Nelder-Mead method is coupled with FEM to improve the performance of the proposed geometry. The single-phase FRM prototype for the proposed FRM has four stator teeth, four rotor teeth, and two PM poles on each stator tooth as described in [3] (see Figure 1b). In contrast to the FRM described in [1], the nearest magnets mounted on the adjacent teeth are magnetized in the same direction. This FRM operates at a very high speed (about 28,000 rpm) as a drive of an angular grinder (power tool). The FRM prototype is described in [4]. In order to avoid the induced eddy currents in the magnets and the heating thereof, each magnet is divided into several electrically insulated segments. Such single-phase FRMs have several [3] advantages compared to those of [1], as follows: (1) Increasing the power density and the efficiency because of better use of the stator surface; (2) the radial forces are absent, and the bearing lifetime is increased because of an even number of rotor teeth, resulting in motor symmetry. Single-phase motors have a small starting torque and large torque ripple. It causes noise and vibrations, not only in the machine, but also in the mechanisms coupled therewith, and shortens their lifetime. Such machines have a limited range of applications, such as, for example, in circular saws, vacuum cleaners, fans, blowers, etc., in which a large starting torque is not required. In contrast, three-phase motors have a much lower torque ripple and can be used in a wide range of applications. A three-phase FRM (Figure 2) is described in [6]. The FRM as a motor [6] operates from 180 rpm up to 18,000 rpm and has eight teeth on the rotor. The operating frequency is equal to 2.4 kHz at 18,000 rpm. Further increase of speed is complicated because of increasing the operating frequency. Besides, the three-phase FRM [7] is not used in industry, because of the following drawbacks: (1) The torque ripple is rather high (\(\approx\)25%) [7] when a conventional three-phase frequency converter is used; (2) only two thirds of the stator internal surface are used, which decreases the power density and efficiency of the motor.