A Mathematical Expression to Determine Copper Losses in Switching-Mode Power Supplies Transformers Including Geometry and Frequency Effects

High-frequency copper losses, or copper losses produced during switching in switching-mode power supplies, are very much related to variations of the magnetic field distribution within the core window. When operating at high frequency, the current flowing through the windings of a magnetic component experiences redistribution across the section of the conductor due to skin and proximity effects. Current redistribution depends not only on actual frequency, but also on conductor size and layer distribution. While previous works aim to optimize windings size by considering just the current flowing through that winding, this paper shows that, in most of the cases, current redistribution is strongly affected by the currents at the other transformer windings, which should also be taken into account. This paper derives a mathematical expression to calculate the evolution, in the time domain, of the magnetic field during the switching, which will be later used to obtain the current density distribution and copper losses in the component. An easy-to-calculate expression will be derived that allows magnetic windings to be analyzed and/or optimized, because losses are expressed as a function of the winding geometry and position. An application example is also included. All the equations derived are verified by comparing them with the results obtained from a differential equations solver, and with previous works when applicable. Experimental results are also provided.