The sine-cosine method for the finite-difference time-domain-based dispersion analysis of periodic structures is extended to incorporate the presence of nonperiodic wideband sources. A new formulation of this method is presented to clearly demonstrate that it can be employed for the characterization of periodic structures over a broad bandwidth. Moreover, its coupling with the array-scanning technique enables the incorporation of nonperiodic sources, thus enabling the fast characterization of driven periodic structures in the time domain via a small number of low-cost simulations. The convergence, accuracy, and efficiency of the proposed method is demonstrated with its application to the analysis of a negative-refractive-index transmission-line ldquoperfect lensrdquo and the successful comparison of simulated with experimental results. Finally, a modified version of this method is proposed for the accelerated simulation of microwave circuit geometries printed on periodic substrates.