We highlight some physical phenomena occurring in pulsed eddy current testing, through the modeling of the response of a voltage driven pulsed eddy current sensor for the characterization of the thickness and conductivity of a metallic conductive plate. We use the finite element time domain method coupled to the electric circuit equation in the eddy current sensor which consists of an air cored coil. The characterization parameters are obtained from the current variation in the coil. The effects of the thickness and conductivity are clearly distinguished by considering three metallic plates of different conductivities. Time and frequency approaches are considered. We show that the reconstitution of the coil response signal from its Fourier series expansion leads to a loss of important features of the latter, limiting thus the characterization parameters, since this reconstitution does not take account of the subtransient phenomena, i.e., the time of diffusion of the electromagnetic field in the conductive plate. The numerical results are supported by measurements.