This paper presents the new cascade liquefaction cycles using $CO_2-C_2H_6-N_2$ and $CO_2-N_2O-N_2$. The performance of the cascade liquefaction cycles with respect to temperature differences in the LNG heat exchangers is analyzed using HYSYS software and then compared the performance of these cycles with phillips optimized cascade liquefaction cycle. The coefficient of performance of the new liquefaction cycles considered in this study decreases with the temperature differences in the LNG heat exchangers, but the compressor work, expander work and heat capacity in the LNG heat exchanger increases, respectively. From the comparison of performance of three cycles, the cascade liquefaction cycles using $CO_2-C_2H_6-N_2$ showed the highest COP. And the cycles using $CO_2-C_2H_6-N_2$ and $CO_2-N_2O-N_2$ presented the second and third highest COP, respectively. In the view of performance, the optimized cascade liquefaction cycle using $C_3H_8-C_2H_4-C_1H_4$ yields much better COP. But, in the environment view, it is found that the cascade liquefaction cycle using $CO_2-C_2H_6-N_2$ shows favorable characteristics.