The transient component of the fault current in doubly-fed induction generators (DFIGs) contains a decaying AC component, which causes synchronization measurement errors in digital relay protection devices. Therefore, it is necessary to accurately analyze this transient component, offset the measurement errors, and prevent the misoperation of protection equipment. Existing studies are applicable when faults occur near the generator terminals. When faults occur at remote locations, however, calculation errors increase as the fault distance from the generator terminals increases. To reveal the evolution characteristics of DFIG fault transient currents under remote faults, this study proposes a method for calculating DFIG fault transient currents in the complex frequency domain. First, a second-order dynamic equation for voltage response is established when fault impedance exists in the fault circuit. Then, under the scenario where the crowbar protection circuit is activated, the calculation formulas for fault currents in both the complex frequency domain and time domain are derived. These formulas can analyze the amplitude, frequency, and decay information of each transient component of the DFIG fault current. For synchronization phasor measurement errors, a full-wave Fourier continuous integral model is established to quantify the measurement errors caused by transient AC components in DFIGs. A full-order electromagnetic transient simulation model is used for verification. The results show that the proposed method can accurately calculate the DFIG transient fault current when the fault circuit contains fault impedance. Compared with existing methods, the peak calculation error is reduced from over 9% to less than 3%. In addition, compared with the full-order electromagnetic transient "black-box" simulation, the proposed calculation method can realize the analytical calculation of each transient and steady-state component. Furthermore, the proposed Fourier continuous integral model can quantify synchronization phasor measurement errors. This provides theoretical support for the relay protection and control operation of systems integrated with DFIGs.