**Preface to "Mathematical Approaches to Modeling, Optimally Designing, and Controlling Electric Machine"**

An electric machine is the main core of electric drives in industrial, transportation, and domestic applications, as well as in traditional and renewable energy generation systems. A pre-experimental evaluation of the electric machine performance for a given application is always based on a mathematical model. The mathematical model accuracy and methodology vary depending on the application requirements. In addition, the methods of optimal design of electric machines significantly facilitate reaching these requirements. Most of the requirements for electrical machine design are in contradiction to each other (reduction in volume or mass, increase in efficiency and power density, etc.). Therefore, finding the optimal design that will achieve all of them can be a massive task due to a large number of varied parameters whose effects on the machine performance and quality of the design are strongly coupled. Therefore, the optimal design methodology of electric machines is always necessary. The reliable and efficient operation of the electrical machine is impossible without precise control. Therefore, the control strategies, state observers, and their mathematical models which help to check the approaches of optimal and efficient control, are important. The main topics of this Special Issue include, but are not limited to:


**Vladimir Prakht, Mohamed N. Ibrahim, Aleksey S. Anuchin** *Editors*

*Article*

## **Optimal Design of a High-Speed Flux Reversal Motor with Bonded Rare-Earth Permanent Magnets**

**Vladimir Prakht, Vladimir Dmitrievskii and Vadim Kazakbaev \***

> Department of Electrical Engineering, Ural Federal University, 620002 Yekaterinburg, Russia; va.prakht@urfu.ru (V.P.); vladimir.dmitrievsky@urfu.ru (V.D.)

**\*** Correspondence: vadim.kazakbaev@urfu.ru; Tel.: +7-343-375-4507

**Abstract:** Single-phase flux reversal motors (FRMs) with sintered rare-earth permanent magnets on the stator for low-cost high-speed applications have a reliable rotor and a good specific power. However, to reduce eddy current loss, the sintered rare-earth magnets on the stator have to be segmented into several pieces and their cost increases with the number of magne<sup>t</sup> segments. An alternative to the sintered magnets can be bonded magnets, in which eddy current loss is almost absent. The remanence of bonded magnets is lower than that of sintered magnets, and they are prone to demagnetization. However, the cost of low-power motors with bonded magnets can be lower because of the simpler manufacturing technology and the lower material cost. This paper discusses various aspects of the optimal design of FRM with bonded magnets, applying the Nelder–Mead method. An objective function for optimizing an FRM with bonded magnets is designed to ensure the required efficiency, reduce torque oscillations, and prevent the bonded magnets from demagnetizing. As a result, it is shown that the FRM with bonded magnets has approximately the same efficiency as the FRM with sintered magnets. In addition, the peak-to-peak torque ripple is minimized and the minimal instantaneous torque is maximized.

**Keywords:** demagnetization; electric machine; flux reversal machine; high-speed electrical machine; high-speed electrical motor; Nelder–Mead method; optimal design
