Analytical solution for transient, diffusing fields inside a ring conductor with rectangular cross section

Electromagnetic launch involves diffusing fields and currents and finite-element analyses are usually employed to compute the transients in launch systems. The simulation times of interest in launch components such as sliding electrical contact systems and pulsed power systems vary from a few microseconds to a few milliseconds. Numerical solutions in such short time scales require very fine meshes to avoid instabilities. Benchmarking and validation of codes with finite-element analyses of coupled electromagnetic equations require checks on errors and instabilities. For this purpose, analytical series solutions have been derived here to describe the diffusing field transients in a few milliseconds in a ring conductor with rectangular cross section. A linearly ramped or step voltage is imposed on one axial end of the ring conductor, whereas the other axial end is grounded and maintained at zero potential. The coupled transient, one-dimensional diffusion equations have been solved using classical methods in applied mathematics. The distributions of currents and fields inside the conductor and the stored magnetic energy with time have been computed. The method presented here can be applied to any general excitation encountered in electromagnetic launch.