Due to the high cost and the predicted shortage of rare earth elements in the near future, the task of developing energy-efficient electric machines without rare earth magnets is of great importance. This article presents a comparative analysis of optimized designs of a ferrite-assisted synchronous reluctance machine (FaSynRM) and a ferrite-assisted synchronous homopolar machine (FaSHM) in a 370-kW subway train drive. The objectives of optimizing these traction machines are to reduce their losses, maximum armature current, and torque ripple. The optimization considers the characteristics of the machines in the subway train moving cycle. The problem of the risk of irreversible demagnetization of ferrites in the FaSynRM and FaSHM is also considered. To reduce the computational burden, the Nelder-Mead method is used for the optimization. It is shown that the FaSHM demonstrates better field weakening capability, which can reduce the maximum current, power, and cost of the inverter power modules. At the same time, the FaSynRM requires less permanent magnet mass for the same torque density and is more resistant to irreversible demagnetization, which can reduce costs and improve the reliability of the electric machine.