## 1 Introduction  
The high energy intensity of the modern industry determines the high urgency of increasing the energy efficiency of production processes. Electric motors consume about 70% of the electricity in industrial applications. Therefore, in many countries around the world, legislative restrictions on the use of motors with low energy efficiency classes are gradually being introduced. The widespread use of energy-efficient motors will significantly reduce the energy intensity of the gross national product and greenhouse gas emissions. This is consistent with the targets announced in the energy and environmental strategies of the European Union (European Green Deal [1]), USA (State Energy Program [2]), Switzerland (supporting Paris Climate Agreement [3]), China (supporting Paris Climate Agreement [4]), Japan (Net Zero Energy Building [5]), South Korea (supporting Paris Climate Agreement [6]), and other countries. By now in the European Union, most recently commissioned drives are equipped with induction motors (induction motors, IM, Figure 1a) of energy efficiency class IE3 [7,8]. Moreover, in line with the sustainable development strategy of the European Union (EU), the requirements for the energy efficiency of motors will increase in the coming years. Therefore, already from 1 July 2023, motors with a power of more than 75 kW used in the EU must have an energy efficiency class of at least IE4 [9]. In the future, it is planned to achieve compliance with IE4 class for motors of lower power, as well as achieve compliance with IE5 class for higher power motors [10]. It was already demonstrated that the use of motors with an energy efficiency class higher than IE3 can already be feasible today since they provide significant energy savings and a decrease in greenhouse gas emissions [1]. However, the increase in the demand for motors of higher energy efficiency classes faces significant difficulties in the case of IMs: The sum of the electrical losses in the stator and rotor of IMs is usually greater than the losses in the stator winding of synchronous motors, especially for a power range below 30 kW [8,11]. Increasing the efficiency of induction motors is possible, either by increasing the weight and size of the motor or by using a copper squirrel cage in the rotor [8]. Both of these solutions lead to a significant increase in the cost of the motor. The use of a cast copper rotor cage also significantly complicates the manufacturing process. Therefore, induction motors with a cast copper rotor cage and with direct mains power supply are still not widely used. In addition to induction motors, some manufacturers [12,13] also offer permanent-magnet synchronous motors with direct mains supply (DOL PMSM, Figure 1b). Typically, an aluminum starting cage is installed on the DOL PMSM rotor for the possibility of starting from the mains. The motor starts as an induction motor and then continues to operate as a synchronous motor, thereby eliminating the main electrical losses in the rotor winding. With reduced rotor losses, these motors can be rated as class IE4. DOL PMSM characteristics are described in a number of publications [14, 15, 16, 17, 18, 19]. However, such motors also do not find widespread use since their cost turns out to be significantly higher than that of IE3 IM due to the use of expensive rare-earth permanent magnets. In addition, the use of rare-earth permanent magnets leads manufacturers to depend on supplies from a specific region since almost all raw materials for the manufacture of rare-earth magnets are imported from mainland China [20]. This factor can also lead to volatility and a sharp rise in prices for rare-earth raw materials, as was the case in the early 2010s [20,21]. Another factor that does not contribute to the widespread use of DOL PMSM is the significant environmental damage caused by the mining of rare-earth metals.