When designing transformers, the system to which transformers are to be used, the electrical power to be transferred, and the continuity and safety of the system must be taken into consideration. Paired physical and mathematical models help to develop a system that is both accurate and easy to implement. In this study, a model was developed to determine the electromagnetic flux distribution, losses, and electromagnetic forces of the transformer. A thermal model was also developed to measure the heat generated and to determine the temperature distribution of the transformer. The performances of the proposed models were compared with the performance of the transformer, which was determined analytically, and the performance of the transformer was also determined experimentally and the results were confirmed. When the results were compared, it was seen that the proposed models gave more optimum results. In this way, electromagnetic flux, electromagnetic force, and thermal distribution at each point of the transformer can be easily determined. On the other hand, in this study, the windings and the core were optimized, the electromagnetic forces formed in the windings were calculated, the heat distribution in the winding and core was determined, the electrostatic field distributions were examined, and the problematic areas were determined. The thermal effects of the total losses of the transformer were investigated and optimum design parameters were determined. It was concluded that the methods proposed in this study were more practical than the methods reported in the literature. (c) 2019 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.