A parametric study of an irreversible regenerative Brayton heat engine with isothermal heat addition has been performed with external as well as internal irreversibilities. The external irreversibility is due to the finite temperature differences between the heat engine and the external reservoirs, while the internal irreversiblities are due to other processes, viz. nonisentropic compression and expansion processes in the compressor and turbine, respectively, and the regenerative heat loss. The power output is maximized with respect to the working fluid temperatures, and the effects of different parameters on the maximum power output and the corresponding thermal efficiency have been studied. There is a significant improvement in the thermal efficiency (above 15%) of a Brayton cycle with isothermal heat addition over the conventional one. It is seen that the effect of the isobaric side effectiveness is rather pronounced for the power output and the corresponding thermal efficiency. The effect of the turbine efficiency is found to be more than that of the compressor on both power output and thermal efficiency. Also, it is seen that there are optimal values of the various heat capacitance rates between the different reservoirs and the heat engine.