In this paper an argon filled coaxial dielectric barrier discharge (DBD) has been studied to understand the detail of power transfer from a unipolar square pulse to plasma during discharge. A dielectric barrier discharge based diffuse pulse discharge and its electrical characteristics are investigated. A quartz coaxial DBD tube filled at different pressures is used in the experiment. A unipolar pulse voltage of different peak voltages and frequencies has been applied to the discharge electrodes for the generation of microdischarges. Two current pulses are used for two consecutive discharges per applied voltage pulse. The second discharge, which occurs at the falling flank of the voltage pulse, is induced by the charges stored on the dielectric barrier during the first discharge. It has been deduced that the power supplied to ignite the first discharge is partly stored to ignite the second discharge when the applied voltage decays. This process ultimately leads to much improved power transfer to the plasma. The knowledge obtained from dynamic processes of the DBDs in the discharge gap explains quantitatively the mechanism of ignition, development and extinction of the DBDs.