General Instantaneous Dynamic Phasor

Dynamic phasor has now been widely used in the fields of ac power and signal, such as power electronics, power systems engineering, and instrumentation. It is, however, broadly misunderstood by users that the dynamic phasor is valid for an averaged period of time. This misbelief stems from the definition of Fourier series coefficients for a variable where the time sliding integration is taken over a period to get its average. In this article, it is verified that the dynamic phasor is valid instantaneously for any time and that averaging over a period has nothing to do with the dynamics of phasor variables. An extended phasor transformation is newly proposed as a generalized instantaneous dynamic phasor. This instantly valid dynamic phasor is thoroughly verified by mathematics, simulation, and experiments. The instantaneous dynamic phasor is found to be valid for an arbitrary and even time-varying frequency, which is not necessarily the same as the source frequency. The transfer function of the phasor space system is found to be a frequency shift form of the original one of the real space system for the fixed frequency dynamic phasor.