Revolving Coordinate Systems as a Tool to Analyze Dynamics and Distortion in Electric and Power-Electronic Systems Graphically

Stationary or quasi-stationary conditions are desirable and dominant in most systems involving electric energy. E.g., voltages of an electric power grid are often nearly sinusoidal quantities with nearly fixed frequency or nearly dc. Three main effects cause short-time or small deviation from such conditions: Dynamic reaction to changes of the operating conditions, modification of set-points and superimposed distortion caused by the operation of power-electronic converters. These deviations are hard to visualize in practical situations, because they are either small or non-periodical with respect to the fundamental of the dominant quantity or both. Their assessment normally bases on lumped characteristics, e.g. the total harmonic distortion (THD) or mean squared deviation. However, these characteristics do not contain relevant information on the reason for deviations and they change slowly, so that they have low relevance for the control of, e.g., power converters. Many of such systems are three-phase systems, allowing for a concise representation by space-vector quantities. In such a case a representation in a revolving coordinate system, using e.g. the fundamental of the dominant quantity as a reference, increases interpretability and assessment of all kinds of deviations relevantly. This paper treats a suitable methodology in detail. The dynamic reaction of an electric machine, the action of a power-electronic compensator and the stator-flux track curve of an inverter-fed induction machine act as examples to demonstrate the application and the way to interpret the result. Adaptable extensions to quasi-stationary and dynamic conditions are suggested.