At present, the demand for high cooling capacity Gifford–McMahon (GM) refrigerators is increasing to cool the high-temperature superconducting magnets in cryogenic temperature range. Therefore, it is necessary to enhance the cooling capacity of existing refrigerators, which needs a thorough understanding of related thermodynamic phenomena. In this paper, thermodynamic processes of a GM refrigerator are analyzed and its performance is compared with Carnot cycle, and Brayton cycle with and without work recovery. Subsequently, a mathematical model has been formulated for a pneumatically driven GM refrigerator by applying the fundamental principles of thermodynamics and mechanics. The model computes the influence of geometrical and operating parameters upon its refrigeration performance. Subsequently, the impact of valve opening intervals upon internal thermodynamic and dynamic processes is evaluated. The dynamic characteristics of the displacer motion, transient variation of mass flow, and pressure distribution are studied by the model for different values of idling angles. It is found that an increase in the idling angle of the rotary valve of GM refrigerator reduces its cooling capacity and enhances its specific cooling capacity. An experimental study has also been undertaken to corroborate the mathematical results.