Modeling and Small-Signal Linearization of Three-Phase Rotating Transformation Using Exponential Matrix: Summary, Verification, and Discussion

For the three-phase system, analysis and control are usually carried out in the rotating frame, where the rotating-frame model is required. To obtain the rotating-frame model, for the linear time-invariant (LTI) unit, this paper summarized the unified explicit analytical relation, i.e. the formulas, between the specific transfer functions in the rotating frame and that in the static frame using the exponential matrix, capable of both the three- and four-wire systems; then, for the rotating transformation related nonlinear units, the sample units and the output units, the unified small-signal linearization for both the three- and four-wire systems is presented with the phase-locked loop (PLL) and its discrete-time implementation considered. Corresponding examples with experiment verifications are given to demonstrate the theories.