Instantaneous Power in Nonsinusoidal Circuits

The product

$$p(t)=u(t)i(t)$$

is instantaneous terminal power for any waveforms. Its interval average

$$P=\frac{1}{T}\int_0^T u(t)i(t)\,dt$$

is net active power. These statements survive distortion, harmonics, switching, and nonperiodic observation windows. What does not survive unchanged is the unique sinusoidal interpretation $S=P+jQ$.

RMS and apparent power #

Over the same interval,

$$U=\sqrt{\frac1T\int_0^T u^2(t)\,dt},\qquad I=\sqrt{\frac1T\int_0^T i^2(t)\,dt}.$$

$S=UI$ is apparent power and $P/S$ is total power factor. Harmonic distortion can lower that ratio even when the fundamental voltage and current have no phase displacement.

Orthogonal current projection #

Treat waveforms as vectors with inner product

$$\langle x,y\rangle=\frac1T\int_0^T x(t)y(t)\,dt.$$

Project current onto voltage:

$$i_a(t)=\frac{P}{U^2}u(t),\qquad i_r(t)=i(t)-i_a(t).$$

Then $\langle u,i_r\rangle=0$. The projected current $i_a$ is the minimum-RMS current waveform that transfers the same average power for the given voltage, and its equivalent conductance is $G=P/U^2$.

The residual contains every current component that transfers zero net energy over the chosen interval: phase-shifted fundamental current, harmonic current orthogonal to the voltage, and possibly distortion unrelated to energy storage.

Spectral view #

For periodic waveforms in a linear system, average power is the sum of same-frequency voltage-current contributions; different harmonics are orthogonal over an integer number of periods. A window that does not contain coherent periods introduces leakage into both spectral and time-domain averages.

Always state the averaging interval and power definition. Competing nonsinusoidal reactive-power theories answer different engineering questions and should not be mixed by notation alone.

For measured waveforms, align voltage and current channels before multiplying them. A relative timing error creates an apparent phase shift and therefore biases both average power and any residual-current decomposition.