In order to accurately analyze the performance of the saturated iron core type superconducting fault current limiter, a three-stage mathematical model of the current limiter is established according to the requirements of parameter design. The static characteristics in different working areas and the impedance changes during short-circuit current limiting are studied. In the single machine infinite bus system, the fault process is subdivided into different stages for the change of the current limiter impedance to analyze its impact on transient stability. Through PSCAD simulation, it is found that the critical clearing time of the fixed inductance model and the current limiter model are 0.31s and 0.21s respectively. The results show that the current limiter has better current limiting effect and improves the transient stability of the system, and the model can obtain more accurate critical clearing time. This work is supported by National Natural Science Foundation of China (No. 51707121) and Shanghai Science and Technology Commission Project (No. 18DZ1203502).