Fault classification and voltage sag parameter computation using voltage ellipses in Clarke's domain are presented in this paper. The voltage sag parameters computed were the point-on-wave of initiation and recovery, residual voltage, and phase-angle jump. A change in the phase reference in Clarke's transformation was proposed to simplify the fault classification procedure. The point-on-wave of initiation and recovery were defined based on the reference circle in Clarke's domain, as described by the system during normal operation. The residual voltage and phase-angle jump were computed using the expressions of $\alpha$ and $\beta$ components in Clarke's domain and trigonometric theorems. The proposed fault classification and voltage sag characteristics computation were applied to real data provided by the Department of Energy and the Electric Power Research Institute. From the results obtained, it is concluded that the zero-sequence component must be taken in consideration to compute the residual voltage and the phase-angle jump correctly, and to use these results to improve the accuracy of impedance-based fault location methods. The proposed fault classification algorithm is capable of classifying events and computing the residual voltage and phase-angle jump. Additionally, the voltage sag duration of short events and multistage events was successfully computed. This work is an incremental contribution within the field of voltage sag analysis in the time domain using Clarke's transformation.