Real-time detection of the voltage sag sources' relative location requires fast and accurate methods. Therefore, in this paper, the transient period of voltage sags is used with useful detection information, which is not considered in the literature. In this context, this work firstly analyses the main positive-sequence phasor-based (PB) and instantaneous-based (IB) methods within both transient and steady-state periods of voltage sags caused by network faults and transformer energizing. Secondly, new methods are proposed using five different modifiers, applied in the transient period of voltage sags, i.e., half and one cycle time windows, to achieve a faster and more accurate response. These modifiers use the PB/IB criteria obtained from the existing methods, such as power, impedance, and current, and are applied as: The mean of the criterion changes, the first largest peak (FLP) on the criterion changes, the mean of the zero-mean criteria during a sag, the FLP of the zero-mean criteria during a sag, and the Trend (slope) of criteria's trajectories versus time. Voltage sag source detection methods are evaluated by applying 1992 simulated voltage sag events in a Brazilian regional power network. The results reveal that the proposed modifiers, used in the new methods, improve the ineffective existing methods by taking half/one cycle within a transient period of voltage sags. The modifiers also show an accuracy equal to other existing enhanced methods due to employing them within the transient period, thus evidencing their appropriateness. Correspondingly, a selection is made amongst the new modified methods in order to choose the most accurate time window (half or one cycle) for the methods. The selected modified methods are also tested by applying field measurements in a Slovenian power network to confirm their effectiveness in the transient short periods. According to a recommendation of the fastest and most accurate new methods in this study, an important application can be using the recommended methods as the directional function in the relays, along with an accurate voltage sag/fault inception time detection algorithm in real-time.