**1. Introduction**

Mechanical coupling adopts a rigid structure, which is simple, reliable and low cost. However, it requires a high alignment accuracy between the power input end and load end during installation. Meanwhile, it cannot isolate harmful vibrations and has no overload protection function [1]. That means the use of a solid medium to transmit power cannot solve the inherent problems of harmful vibration isolation and overload protection at the input and output ends. Figure 1 shows a damaged rotating shaft and bearing due to shafting vibration.

**Citation:** Guo, S.; Yi, Z.; Liu, P.; Wang, G.; Lai, H.; Yu, K.; Xie, X. Analysis and Performance Evaluation of a Novel Adjustable Speed Drive with a Homopolar-Type Rotor. *Mathematics* **2022**, *10*, 3712. https://doi.org/ 10.3390/math10193712

Academic Editors: Camelia Petrescu and Valeriu David

Received: 15 September 2022 Accepted: 9 October 2022 Published: 10 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** A damaged rotating shaft and bearing due to shafting vibration. (**a**) The damaged rotating shaft. (**b**) The damaged bearing.

Compared with mechanical coupling, magnetic adjustable speed drives have attracted a great deal of attention. As the electromagnetic field is used as the power transmission medium, the transmission link has no direct mechanical contact, leading to complete isolation between the power drive end and the load end, which perfectly solves many problems such as the isolation of harmful vibrations, overload protection, motor safety starting with loads, etc. [2–4].

At present, most of the existing magnetic adjustable speed drives contain permanent magnets, and speed adjustment is mainly achieved through the non-rotary mechanical speed regulation structure to adjust the air gap length [5,6]. For permanent magnet adjustable speed drives, because they contain permanent magnets, this type of machine has continuous free-wheeling losses in the form of hysteresis and induced eddy currents, which may lead to high-temperature loss of excitation and vibration loss of excitation. Additionally, because they contain a mechanical speed regulation structure, this also increases the complexity and reduces the reliability of the system.

The homopolar-type rotor is very simple and robust, and has been widely used in high-speed ac machines [7]. In some conditions, it can have a similar performance to the typically used PM machines [8]. For this reason, researchers have paid much attention to this type of machine. The authors of [9] use a bearingless ac homopolar alternator for a flywheel energy storage system, which avoids significant mechanical friction, increases the motor efficiency and increases the operational lifespan. The authors of [10] use a novel permanent magnet homopolar inductor machine with a mechanical flux modulator for a flywheel energy storage system. The authors of [11] use this type of rotor for a bearingless motor, which can be driven up to an ultrahigh speed of 100,000 r/min. The authors of [12] use a homopolar motor in a mining truck with a carrying capacity of 90 tons.

In addition to the above application, this study proposes a new application of the homopolar-type rotor that could be used to regulate the speed of the load, named the homopolar-type rotor adjustable speed drive (HTR-ASD). It usually contains a homopolartype rotor, a squirrel-cage rotor and a non-rotary shell with an excitation winding. This machine has no brush, no permanent magnet, and no mechanical flux regulation device. At the same time, the rotating parts of the device are composed of a solid homopolar-type rotor and a simple squirrel cage structure, making it fit for high-speed or high-temperature occasions [13,14]. Its magnetic excitation is produced by a dc current in a stationary field with the winding fixed to the shell. In the regulation process, the speed and torque can be adjusted only by adjusting the amplitude of excitation current. This current can be completely turned off during idling times to eliminate magnetic losses.

The performance of the adjustable speed drive may be studied by either numerical or analytical approaches [2]. The former, such as the finite element method (FEM), albeit precise, are time consuming. To simplify the analysis of magnetic adjustable speed drives, some researchers use an analytical method. The analytical methods mainly include the field analysis analytical model and the magnetic equivalent circuit (MEC) model. The field analysis analytical model is based on Maxwell's equations and boundary conditions. The MEC is a simple analytical calculation method [15,16]. In recent years, it has been applied in the design of eddy current couplings [17] and retarders [18,19]. The authors of [20] use an analytical method to calculate the torque characteristics of a novel hybrid superconducting magnetic coupling with axial flux. The authors of [21,22] attempt to simplify the actual 3D geometric model as a 2D model for research, aiming to increase the efficient of the preliminary design. However, most of the analyses are for the PM adjustable speed drive and are not suitable for the adjustable speed drive proposed in this study.

The homopolar-type rotor has a three-dimensional magnetic circuit structure with the same pole and the 3D-FEM is often used for its analysis [23]. The use of the 3D-FEM enables an accurate result to be obtained; however, this method is time consuming. Therefore, the 3D-FEM is usually used for the final performance check. To conveniently obtain an effective no-load air gap flux density, a simplified 2D equivalent analysis model of the homopolar machine is proposed by [24]. Through this method, the no-load effective air gap flux density can be calculated. However, whether this method is applicable to the load condition of this machine is not stated. To calculate the load condition of this type of machine, the authors of [25] adopted a 2D simplified analysis method for a PM homopolar inductor machine. However, this model is for PM homopolar machines and is not suitable for electric excitation machines. Aiming to simplify the calculation of a 3D magnetic circuit for this type of machine, some researchers have used MEC [26,27]. As for the parameter calculations, the authors of [27] use the rotor shape function to speed up the calculation and analysis process. However, the authors of [27] ignore the difference between the rotor shape and the air gap permeance, so the corresponding conclusion is not accurate enough.

In this study, an analytical examination of the homopolar-type rotor is performed, which is found to be effective in the preliminary design stages and analysis of electric machines. To analyze the speed regulation characteristics, the equivalent circuit of the HTR-ASD is obtained and the torque of this machine is calculated by using it. The air gap permeance function is analyzed, which is used not only for analyzing the air gap magnetic field parameters, but also for calculating the winging parameters. The air gap permeance function in this study is directly related to the rotor shape, which means that it is much more accurate than in [27]. By using the analytical method proposed in this study, researchers can quickly obtain the primary scheme of the machine and evaluate its performance [28–30].

The remainder of this paper is organized as follows. In Section 2, the operation principle and the flux-modulated mechanism of the HTR-ASD are analyzed in detail. The equivalent circuit of the HTR-ASD is studied and the torque of the HTR-ASD is calculated in Section 3. Then, the analytical method is proposed and key parameters of the HTR-ASD are calculated. In Section 4, an HTR-ASD prototype is designed and the performances of the HTR-ASD are comparatively studied by the analytical method and the finite element method. The comparison of the results shows the accuracy of the analytical method, indicating that the proposed adjustable speed drive can be applied successfully.
