A general model for an irreversible solar-driven Brayton multi-step heat engine is presented. The model incorporates an arbitrary number of turbines (N-t) and compressors (N-c) and the corresponding reheating and intercooling processes; thus, the solar-driven Ericsson cycle is a particular case where N-t,N-c. For the solar collector, we assume linear heat losses, and for the Brayton multi-step cycle, we consider irreversibilities arising from the non-ideal behavior of turbines and compressors, pressure drops in the heat input and heat release, heat leakage through the plant to the surroundings, and non-ideal couplings of the working fluid with the external heat reservoirs. We obtain the collector temperatures at which maximum overall efficiency eta(max) is reached as a function of the thermal plant pressure ratio, and a detailed comparison for several plant configurations is given. This maximum efficiency is obtained in two cases: when only internal irreversibilities are considered and when both internal and external irreversibilities (which corresponds to the fully irreversible realistic situation) are simultaneously taken into account. Differences between both situations are stressed in detail. In the fully irreversible realistic case, it is possible to perform an additional optimization with respect to the pressure ratio, eta*(max). In particular, this double optimization leads to a valuable increase in efficiency (between 34% and 65%) for a plant with two turbines and two compressors compared to the simple solar-driven one-turbine one-compressor Brayton engine. Copyright (c) 2012 John Wiley & Sons, Ltd.