# Synchronous Homopolar Generator without Permanent Magnets for Railway Passenger Cars
## 1 Introduction  
In many countries, less than half of the railways are electrified [1, 2]. Nonelectrified railway lines use diesel or diesel-electric locomotives [3]. One of the options for the electrification of passenger cars in this case is undercar generators [4, 5]. Brushless synchronous homopolar machines (SHMs) with axial excitation flux and stator-fixed concentric coils of the excitation winding are known for their high reliability, due to which they find use in flywheel energy storage and traction motors, as well as in welding automotive and aircraft generators [6, 7, 8, 9]. They can be used at high temperatures and in hazardous environments, such as drives [10] and high-power wind generators [11]. The main benefit of SHMs, in comparison with conventional generators with electrically excited rotor, is high reliability, due to the simple rotor without windings, no sliding contact, and reliable concentric excitation winding coils located on the stator. At the same time, the SHM retains the ability to control the excitation current, as in a conventional generator with an excitation winding on the rotor. It is also a problem to ensure the cooling of a conventional generator with an excitation winding on the rotor, which has significant electrical losses. Generators with a brushless exciter are used in a number of applications, such as aircraft generators and high-power generators [12, 13]. The disadvantages of such generators with a brushless exciter, compared with brushed generators, include an increase in the cost, dimensions, and weight of the machine due to the presence of a brushless exciter. Additionally, in this case, there are problems of the reliability and repair of the brushless exciter assembly. The design of the excitation winding is much simpler and more reliable and requires much less copper that that of a conventional synchronous generator. In addition, becausethe rotor does not have windings and a significant copper loss associated with them, no special measures are required to cool the rotor [14]. Another important advantage of the SHG is the high strength of the rotor to the centrifugal force and large shock loads [15]. SHMs have a complex configuration of the magnetic system, which is inconvenient for calculation and optimization, because the magnetic flux propagates in all three dimensions. The magnetic flux flows in the axial direction in laminated parts of the machine and in the transverse plane in nonlaminated parts of the machine's magnetic core. The complex three-dimensional design of the magnetic circuit causes difficulties when using conventional two-dimensional models based on the finite element analysis (FEA) to compute the characteristics of the SHM. Because analytical models of electrical machines are usually not accurate enough due to the difficulty of taking into account the saturation of the magnetic circuit, it is necessary to use numerical models [16]. For this reason, a number of numerical calculation methods have been proposed for SHMs, taking into account the saturation of the magnetic circuit, including three-dimensional FEA [17; 18; 19], two-dimensional FEA [9; 20], one-dimensional magnetic circuits, and their various combinations [21; 22]. In Reference [9], a simplified SHM modeling technique was proposed, in which the features of the SHM magnetic field are taken into account in a two-dimensional model by introducing the excitation field term into the equations of the magnetic vector potential and solving the 2D magnetostatic problems complemented with an equivalent circuit equation for the excitation flux. Articles [9; 23; 24; 25; 26; 27] present SHM as a traction motor. Article [9] discusses a simplified methodology for estimating the characteristics of the SHM, suitable for use in multi-iteration automatic optimization, and its experimental verification. Articles [23; 24; 25] present techniques for optimizing SHMs for various traction applications. Articles [14; 26] present the results of comparing the characteristics of the SHM with other types of traction motors.