A $CO_2$ cycle shows large throttling loss during the expansion process. The application of an expander into the $CO_2$ cycle can reduce the throttling loss and then improve system performance. In this study, the performance of a transcritical $CO_2$ cycle with an expander was analytically investigated in order to improve the cooling performance of the system. The expander was applied to the single-stage and two-stage compression cycles. The performance was analyzed with the variations of compressor frequency, outdoor temperature, and expander efficiency. The single-stage and two-stage compression cycles with the expander showed COP improvement of 25% and 32%, respectively, over the single-stage cycle with an EEV.