Analytical Evaluation of Transient Magnetic Fields Due to RWG Current Bases

A new analytical approach for obtaining the time samples of the magnetic field intensity due to an impulsively excited Rao–Wilton–Glisson (RWG) basis function is presented. The approach is formulated directly in the time domain. It is shown that the magnetic field is related to the arc segments formed by the intersection of the triangular patch of the RWG basis with the sphere that is centered at the observation point and that has a radius of $R = ct$, where $c$ is the speed of light. In particular, the magnetic field can be expressed as the variations of two quantities with respect to $R$. The first quantity is the arc segment length, and the second quantity is the bisecting vector of the arc segment. Analytical representations of these quantities are presented. Contrary to previous studies, these representations do not require the calculation of the intersection points of the sphere with the boundaries of the bases. The validity of the obtained time domain formulae is demonstrated through comparison of the results with those obtained in the frequency domain by using numerical quadrature. Finally, it is demonstrated that the derived formulae yield closed-form expressions when convolved with piecewise polynomial temporal basis functions.