A Signal Theory of Instantaneous and Average Power

In this paper the standard formulas for instantaneous and average power in sinusoidal systems are generalized to non-sinusoidal systems by appealing to the Hilbert Transform and Bedrosian’s Theorem for a general class of non-sinusoidal currents and voltages. It is shown that a complex representation of real instantaneous power is the sum of complex Hermitian power and complex non-Hermitian power. Hermitian power is a complex representation of baseband power and non-Hermitian power is a complex representation of passband power. These two complex powers determine baseband and passband, active and non-active, real and reactive, positive and negative components of power. A formula is derived for the frequency of variation of instantaneous power in non-sinusoidal cases. When real instantaneous power is defined for a circuit with a single time-varying equivalent impedance, active and non-active components of power may be associated with real and reactive circuit components. The spectral representation of average power decomposes into readily identifiable and interpretable components of average power. A simple block diagram shows how components of instantaneous power may be extracted from metered voltage and current. When Hermitian complex power is plotted on the complex plane, the resulting Lissajous-like figure provides a visual display of baseband power. A Lissajous-like figure and its path length are proposed as a measure of nonstationarity of real valued instantaneous power. Complex Hermitian power may be spectrum analyzed for the spectrum of real baseband power.