Time-varying phasor

In communication theory, time-varying phasors are used for analyzing narrow-band signals, whose signal bandwidths in the frequency domain are considerably smaller than the carrier frequency.<ref name=":0">Venkatasubramanian, V. (1994). "Tools for dynamic analysis of the general large power system using time-varying phasors". International Journal of Electrical Power & Energy Systems. 16 (6): 365–376. doi:10.1016/0142-0615(94)90023-X. S2CID 109676900.</ref><ref name=":1">Jeltsema, Dimitri (2015). Camlibel, M. Kanat; Julius, A. Agung; Pasumarthy, Ramkrishna; Scherpen, Jacquelien M.A. (eds.). "Time-Varying Phasors and Their Application to Power Analysis". Mathematical Control Theory I. Lecture Notes in Control and Information Sciences. Cham: Springer International Publishing. 461: 51–72. doi:10.1007/978-3-319-20988-3_4. ISBN 978-3-319-20988-3.</ref> Time-varying phasors are mostly used for analysis of frequency domain of band-pass systems.<ref name=":1" /><ref name=":0" /> The method uses classical impulse response.<ref name=":0" />

In electrical power system, phasors are used for transient analysis of the power system keeping the quasi-stationary conditions.<ref name=":0" /><ref name=":2">Venkatasubramanian, V.; Schattler, H.; Zaborszky, J. (November 1995). "Fast time-varying phasor analysis in the balanced three-phase large electric power system". IEEE Transactions on Automatic Control. 40 (11): 1975–1982. doi:10.1109/9.471228. ISSN 1558-2523.</ref><ref name=":3">Belikov, J.; Levron, Y. (December 2018). "Uses and Misuses of Quasi-Static Time-Varying Phasor Models in Power Systems". IEEE Transactions on Power Delivery. 33 (6): 3263–3266. doi:10.1109/TPWRD.2018.2852950. ISSN 1937-4208. S2CID 53741086.</ref> They were introduced to facilitate the computation and analysis of power systems in stationary operation.<ref name=":2" /> Time-varying phasors are used in dynamic analysis of a large power system.<ref name=":0" /><ref name=":4">"Comparison of time-varying phasor and dq0 dynamic models for large transmission networks". ResearchGate. Retrieved 2021-01-28.</ref> The phasor representation of sinusoidal voltages and currents is generalized to arbitrary waveforms.<ref name=":1" /> This mathematical transformation eliminates the 60 Hertz (Hz) carrier which is the only time-varying element in the stationary case.<ref name=":2" /> The longer usage of time-varying phasors in large power systems since 1920s have created many misconceptions. One of the misuses suggest that quasi-stationary models are always accurate, but only when the system dynamics are slow as compared to nominal system frequency which is usually 60 Hz.<ref name=":3" />

The concern to study time-varying phasors is raised to understand in-depth the fast amplitude and phase variations of emerging electrical power generator technologies.<ref name=":3" /> This is because current and voltage signals of latest machines may have harmonic components and they can damage the entire transmission system which is coupled with the machine.<ref name=":2" /><ref name=":3" /> However, if we employ quasi-static model, we can accurately model AC signals by using time-varying phasors as opposed to traditional quasi-static model which supports constant voltage and current signals throughout the network.<ref name=":4" />