• Iu. A. Sirotin National Technical University «Kharkiv Polytechnic Institute», Ukraine
  • T. S. Ierusalimova National Technical University «Kharkiv Polytechnic Institute», Ukraine



three-phase circuit, classical instantaneous power, vector instantaneous power, complex 3-dimensional Fourier series, active and reactive power, complex vector power, apparent power, complex pulsation power, power equation, unbalanced mode


Purpose. To identify the mathematical relationship between the instantaneous powers (classical and vectorial) and integral powers in non-sinusoidal mode and to get complex form of instantaneous powers in 3-phase 4-wire power supply in terms of the spectral approach. Methodology. We have applied the vector approach with one voice allows you to analyze the energy characteristics of 3-phase power supply circuits (for 4-wire and 3-wire circuits) in sinusoidal and non–sinusoidal mode, both the time domain and frequency domain. We have used 3-dimensional representation of the energy waveforms with the complex multi-dimensional Fourier series. Results. For 4-wire network with a non-sinusoidal (regardless of their symmetry) processes, we have developed the mathematical model one-dimensional representations of the complex form for the active (scalar) instantaneous power (IP) and 3-dimensional form (inactive) vectorial IP. It is possible to obtain two dual integral power equations for complex scalar and vector integrated power of non-sinusoidal modes. The power equations generalize generalizes the equations of sinusoidal modes for 4-wire network. Originality. In addition to the classification of energy local regimes in the time domain for the first time we spent the classification of non-sinusoidal modes in the spectral region and showed the value and importance of the classification of regimes based on the instantaneous powers. Practical value. The practical value the obtained equations is the possibility of their use for improving the quality of electricity supply and the quality electricity consumption.

Author Biographies

Iu. A. Sirotin, National Technical University «Kharkiv Polytechnic Institute»

Doctor of Technical Science, Professor

T. S. Ierusalimova, National Technical University «Kharkiv Polytechnic Institute»

Assistant Lecturer


Sirotin Iu.А. Vectorial instantaneous power and energy modes in three-phase circuits. Tekhnichna elektrodynamika – Technical electrodynamics, 2013, no.6, pp. 57-65. (Rus).

Korn G., Korn T. Spravochnik po matematike dlia nauchnykh rabotnikov i inzhenerov [Mathematical handbook for scientists and engineers]. Moscow, Nauka Publ., 1973. 832 p. (Rus).

Sirotin Iu.A. Non-pulsed mode of supply in a three-phase system at asymmetrical voltage. Przeglad Elektrotechniczny, 2013, no.7, pp. 54-58.



How to Cite

Sirotin, I. A., & Ierusalimova, T. S. (2016). INSTANTANEOUS AND INTEGRAL POWER EQUATIONS OF NONSINUSOIDAL 3-PHASE PROCESSES. Electrical Engineering & Electromechanics, (1), 69–73.



Power Stations, Grids and Systems