Analyses of Initial Condition Problem and Linear Loop Modes in the Solution of the Derivative Form of Time-Domain Electric Field Integral Equation

The marching-on-in-time solution of the derivative form of time-domain electric field integral equation has long been suffering from three typical problems: internal resonant modes, initial condition problem, and linear loop modes. This letter focuses on the last two problems. The possible appearance of initial condition problem is attributed to the inappropriate choice of temporal basis function; to avoid it, a constraint condition for temporal basis functions is presented in this letter. Additionally, linear loop modes defined as the constant and linearly increasing components of current density in time are studied in theory. Theoretical analysis indicates that limitation on machine precision makes linear loop modes approximately belong to the null space of discretized derivative form of time-domain electric field integral equation. It is also shown that errors due to space-time discretization (except time step size) and double surface integration have no visible influence on linear loop modes. As numerical experiments will demonstrate, the magnitudes of linear loop modes are approximately proportional to the incident field at the initial time and the reciprocal of time step size.