The use of water vapor selective membranes can reduce the energy requirement for extracting water out of humid air by more than 50%. We performed a system analysis of a proposed unit, that uses membranes to separate water vapor from other atmospheric gases. This concentrated vapor can then be condensed specifically, rather than cooling the whole body of air. The driving force for the membrane permeation is maintained with a condenser and a vacuum pump. The pump regulates the total permeate side pressure by removing non-condensable gases that leak into the system. We show that by introducing a low-pressure, recirculated, sweep stream, the total permeate side pressure can be increased without impairing the water vapor permeation. This measure allows energy efficiency even in the presence of leakages, as it significantly lowers the power requirements of the vacuum pump. Such a constructed atmospheric water generator with a power of 62 kW could produce 9.19 m3/day of water (583 MJ/m3) as compared to 4.45 m3/day (1202 MJ/m3) that can be condensed without membranes. Due to the physical barrier the membrane imposes, fresh water generated in this manner is also cleaner and of higher quality than water condensed directly out of the air.