We found that a generation of a uniform discharge in atmospheric-pressure air was possible at a frequency range from 32Hz to 1.1kHz using alumina as a barrier material of the DBD (Dielectric Barrier Discharge) device. We also found that there is a domain in voltages on the generation of the uniform discharge. The domain bocomes wider with the increase of the frequency. If the applied voltage is a bit higher than the upper limit voltage of stable generation of the uniform discharge, the uniform discharge changes to a non-uniform FD (Filamentary Discharge). In order to clarify the mechanism of the generation of the uniform discharge, we investigated how the barrier materials change the discharge mode by both the photographic observation and analysis of the electric circuit phenomena. We also carried out the experiments of a DBD device using a combination barrier of soda-glass and alumina. In this case, the uniform discharge and the FD appeared alternately in every half cycle. As the result, it was found that the uniform discharge was generated when the barrier of alumina was used and it becomes a cathode. From the analysis of the gap voltage and the current, it was also found that the uniform discharge is the APTD (Atmospheric Pressure Townsend Discharge) characterized by the electron avalanche. This paper describes the detail of experimental results and discussions.