## 1 Introduction
### Overview of Literature on the Design of Synchronous Homopolar Machines  
On non-fully electrified rail lines, undercar generators can be used to power passenger cars. Often, synchronous homopolar generators (SHGs) without permanent magnets with concentrated excitation coils on the stator are used in this application, due to their relatively high performance and highly reliable brushless design that is resistant to shaking and shock loads and to the absence of windings on the rotor [8]. SHMs also find uses in other applications such as traction drives, flywheel storage, wind turbines, on-board generators, and welding generators [9; 10; 11; 12; 13]. In addition, the excitation current of the SHG is adjustable, as in a conventional electrically excited generator. The analysis of the magnetic field of the SHG and, consequently, its optimization, is complicated by the complex three-dimensional configuration of its magnetic core. In some parts of the machine, the flux flows in the axial direction, and in others do so in the transverse plane (in the tangential and radial directions) and changes its direction to an axial one when moving from the laminated parts to the solid stator housing or magnetic sleeve on the rotor shaft. The applicability of the traditional two-dimensional finite element model is limited, due to the three-dimensional nature of the magnetic field of the SHG. Researchers have developed a number of models that take into account the three-dimensional structure of the SHG magnetic field using one-dimensional magnetic circuits [14; 15], models based on the two-dimensional finite element method (FEM) [16; 17], and three-dimensional FEM models [18; 19]. In [17], a technique for modeling the SGM was presented, which considers the three-dimensional structure of the magnetic field within the framework of a two-dimensional model by introducing an additional term into the equations of the vector magnetic potential, and jointly solving them with the one-dimensional equation of the axial magnetic circuit of the excitation flux. An experimental verification of this technique is carried out. In [8; 9; 14; 16; 19; 20; 21; 22], SHM performance analysis for generators and flywheel energy storage applications is carried out. Due to the complexity of the configuration of the magnetic system, many different methods for calculating the performance of SHG have been proposed. In [8], an SHG design using the 2D FEA mode was considered in order to improve the performance properties of a mass-produced undercar generator. Manual optimization of the number of rotor teeth, as well as the parameters of the rotor and stator teeth, was applied. Through experimental verification it is shown that the optimized design has a higher no-load EMF value. In [9], modeling of the transient process in a low-power train-lighting alternator based on SHM is considered, taking into account the eccentricity of the rotor. In [19], the analysis of a low-power high-speed SHM with an external rotor is considered, the design of which is carried out using analytical expressions. The SHM performance at idle and at rated load is calculated using 3D FEA. In [14; 16], an equivalent 2D finite element SHM model with virtual excitation on the rotor is presented, which is verified using 3D FEA and an experimental study. In [20], the design of a high-speed high-power motor/generator based on SHM is considered. The layout of a superconducting excitation winding, and a water-cooled armature winding made of Litz wire is described. For performance analysis, an approximate SHM model is used in which the magnetic field is analyzed using a 2D finite element analysis in a plane containing the axis of rotation of the machine. In [18], analysis using 3D FEM and manual optimization of the no-load electromotive force (EMF) for a three-phase SHG with a speed control range of 500-7000 rpm are presented. Characteristics of the SHM are evaluated with different combinations of stator and rotor teeth: 9/6, 15/10 and 18/12. An experimental verification was provided.