Multipolar development of vector potential for parallel wires. Application to the study of eddy currents effects in transformer windings

An analytical method for solving two-dimensional electromagnetic problems is presented. First, the classical multipolar development is adapted to translation unvarying systems. Second, the link between imaginary parts of the complex Poynting vector and complex power is clarified. Then, using both extensions, a simple expression for complex power of any two-dimensional device is derived. To validate these methods, two devices subjected to an external uniform magnetic field are studied: a cylindrical wire alone and a layer of such wires. Analytical results are compared with those given by electromagnetic simulation software. The dipolar approximation is proven to be accurate enough to investigate the wire layer. It leads to simple formulas that can be used for transformer windings.