DIDACTIC MATHEMATICAL DERIVATION OF THE REAL AND IDEAL TRANSFORMER MODELS

This paper presents a potential solution to a pedagogic deficiency in current textbooks on Electric Circuits Theory. Most textbooks on this discipline, extensively address the topic of magnetically coupled coils using phasors. However, when covering the concept of an ideal transformer, these textbooks define it in terms of voltage and current conversions, without establishing a clear logical connection with magnetically coupled circuit theory. This lack of clear connection deprives students from obtaining a deep understanding of transformers. In this paper, we first establish the general equations for two magnetically coupled coils, the primary and the secondary. Then, we consider six possible solutions to these equations, and adopt the solution that best meets our need to describe power transmission from primary to secondary. A topological interpretation of this solution under Kirchhoff's laws provides a useful model for the real transformer. By considering ideal conditions, we show how the model of the real transformer becomes the model for the ideal transformer. This way, we derive- rather than simply define- the conversion relationships that exist between transformer primary and secondary voltages and currents. Such an approach can better support students in developing strong understandings of this subject matter. Finally, this paper assesses student learning to show how students' understanding benefits from a logical derivation that connects magnetically coupled circuit theory to the real and ideal transformer models.