In this paper, the performance of the $CO_2-C_2H_6-N_2$ cascade liquefaction cycle with respect to temperature differences in the LNG heat exchangers is analyzed theoretically using HYSYS software and then compared the COP(coefficient of performance) of the cascade liquefaction cycles using $C_3H_8-C_2H_4-C_1H_4$ and $CO_2-N_2O-N_2$. In comparison of COP of three cycles, the cascade liquefaction cycles using $C_3H_8-C_2H_4-C_1H_4$ showed the highest COP. And the liquefaction cycle using $CO_2-C_2H_6-N_2$ and $CO_2-N_2O-N_2$ presented the second and third highest COP, respectively. In case of COP, the $C_3H_8-C_2H_4-C_1H_4$ cascade liquefaction cycle yields better COP. But, in terms of the environment and maintain, it is confirmed that the cascade liquefaction cycle using $CO_2-C_2H_6-N_2$ provides favorable characteristics.