This document presents a theoretical analysis of a cascade refrigeration system for cooling applications. It discusses two types of refrigerants used in cascade cycles that have different boiling points and run in independent cycles joined by a heat exchanger. It describes the process, includes assumptions about compression and heat transfer, and presents graphs on how varying cascade condenser temperature and condensing temperature affect COP. It concludes the R744-R717 refrigerant pair has the highest COP for cooling applications compared to other refrigerant pairs.