Analytical Investigation of the Properties of Transients in Unbalanced Three-Phase Four-Wire Networks

In this paper, the theoretical properties of transients in three-phase networks, including the fourth wire and unbalanced source, are systematically investigated by resorting to several analytical tools. First, a Cartesian space vector is introduced to provide the geometrical three-dimensional representation of a three-phase voltage/current transient. It is shown that a voltage/current transient can be always represented as a three-dimensional trajectory leaving one plane (corresponding to the previous steady state) towards another plane (corresponding to the new steady state at the end of the transient). The quantity driving the trajectory from one plane to another is the Park/Clarke zero component. Second, the Clarke transformation is used to study the original three-phase circuit as a superposition of two independent transients: the transient of the Clarke space vectors and the transient of the Clarke zero components. Third, the Park transformation is used to evaluate the dq0 components of transient voltages/currents. Since the Park transformation is related to the Clarke transformation through a simple frequency shift, the effects on the frequency content of the steady-state and transient waveforms are put into evidence. Three case studies are presented and solved to prove the theoretical results.