This paper describes a mathematical model of engines operating with an ideal Stirling cycle and subject to limited heat transfer, internal thermal losses, and mechanical friction losses. The goal of the paper is to describe the fundamental effects of these imperfections on the performance of an otherwise ideal Stirling engine. These are defects suffered by all real Stirling engines, but the intent here is not to predict the actual performance of specific engines. Rather, the aim is to obtain some mathematical insight into the nature of the theoretical dependence of engine performance upon these imperfections.