 Active power

Active power, also called real power, is the time average of instantaneous power transferred through a terminal pair. For periodic electric potential difference $v(t)$ and electric current $i(t)$ of common period $T$, the passive sign convention givesThe net energy entering during each period is $PT$. Positive active power denotes net absorption under this convention; it can represent useful conversion to other forms of energy as well as heating.

For sinusoids of RMS magnitudes $V$ and $I$ with relative phase $\phi$, $P=VI\cos \phi$. For a pure resistor with constant resistance $R$, the stronger waveform-independent statement follows from $v(t)=Ri(t)$:This is why the root mean square current is the constant-current value giving the same heating in that resistor.^[ref]

An original harmonic example shows why RMS values alone do not determine active power. Letin volts and amperes. Orthogonality over a period gives $I_{\mathrm{rms}}=\sqrt{3^2+4^2}=5\mathrm{A}$, but the third harmonic averages to zero when multiplied by the fundamental potential difference. Consequently $P=100(3)=300\mathrm{W}$, whereas apparent power is $100(5)=500\mathrm{VA}$. The power factor is0.6 even though the fundamental current is exactly in phase with the fundamental potential difference.

More generally, Fourier analysis of real periodic waveforms with RMS harmonic magnitudes giveswhen the expansion and average are justified. Cross terms of different harmonic orders vanish. A single fundamental-frequency complex electrical power therefore need not capture every harmonic contribution to the total average.

For an ideal inductor or capacitor in periodic steady operation, the stored energy returns to its starting value each cycle, so its net active power is zero despite nonzero exchange during the cycle. Real components can also have losses. The primary MIT treatment explicitly separates resistor dissipation from periodic inductive and capacitive energy exchange.^[ref]