In the conventional vapor-compression heat pumps, the heating capacity and the heating load usually vary in opposite directions, which results in a mismatch of the heating capacity and the heating load at off-design conditions. Air (reversed Brayton) cycle is a potential substitute for the conventional vapor-compression cycles. This paper proved that in theory the air heat pump cycle can make the heating capacity in line with the heating load at a stable level of heating COP (coefficient of performance). A thermodynamic model for the air heat pump cycle with practical compressor and expander was developed. The optimal heating COP and the corresponding pressure ratio were derived from the model. Then the cycle performance was analytically expressed under the optimal COP conditions. The heating capacity under different operating conditions was found in line with the heating load. Comparisons between the air heat pump cycle and two typical vapor-compression heat pump cycles were numerically done for further verification. It also turned out that the energy efficiency of air heat pump is comparable to the transcritical CO2 heat pump, particularly at large temperature difference. (C) 2014 Elsevier Ltd. All rights reserved.