Dielectric barrier discharges (DBDs) are self-extinguishing discharges due to charge accumulation on dielectric surfaces. In order to take advantage of these surface charges also at a low repetition frequency, high-voltage unipolar square pulses (amplitude up to 15 kV, rise and fall time less than 20 ns) are applied to drive DBDs. For electrical diagnostics of this novel excitation method, a temporally dynamic model for diffuse DBDs is introduced, from which equations were derived which allow the calculation of internal electrical quantities in the discharge gap from measured external electrical quantities. It was found, following a primary discharge at the rising front or at the top of the voltage pulse, that a secondary discharge is induced at the end of the falling voltage flank without simultaneously consuming energy from the external circuit. The energy needed is provided by the accumulated surface and space charges left by the primary discharge, which are totally or partially lost under normal low-frequency sine or square wave excitation. Secondary discharges are observed in a wide range of electrode configurations, gases and gas pressures for both homogeneous and filamentary discharges. In the case of filamentary modes, secondary discharges are found to develop along the remaining channels of the preceding primary discharges. Experiments for ozone synthesis show an improved energy efficiency of 8-9 eV per ozone molecule, which is about 30% better than that achieved with sine wave excitation.