Using Differential Geometry to Revisit the Paradoxes of the Instantaneous Frequency
For researchers in power systems and signal processing, this work provides a unified explanation for known IF paradoxes, but the contribution is incremental as it reframes existing concepts.
The paper proposes a geometric framework based on differential geometry to reinterpret the five paradoxes of instantaneous frequency in three-phase power systems, demonstrating its effectiveness through examples and an IEEE 39-bus system model.
This paper proposes a general framework to interpret the concept of Instantaneous Frequency (IF) in three-phase systems. The paper first recalls the conventional frequency-domain analysis based on the Fourier transform as well as the definition of IF which is based on the concept of analytic signals. The link between analytic signals and Clarke transform of three-phase voltages of an ac power system is also shown. Then the well-known five paradoxes of the IF are stated. In the second part of the paper, an approach based on a geometric interpretation of the frequency is proposed. This approach serves to revisit the five IF paradoxes and explain them through a common framework. The case study illustrates the features of the proposed framework based on a variety of examples and on a detailed model of the IEEE 39-bus system.