= Active power
{title2=$P=\langle v(t)i(t)\rangle$}
= Real power
{synonym}
<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> gives
$$
P=\frac1T\int_0^T v(t)i(t)\,dt.
$$
The 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 $\varphi$, $P=VI\cos\varphi$. For a pure <resistor> with constant resistance $R$, the stronger waveform-independent statement follows from $v(t)=Ri(t)$:
$$
P=R\frac1T\int_0^T i(t)^2\,dt=RI_{\rm rms}^2.
$$
This is why the <root mean square> current is the constant-current value giving the same heating in that resistor.<University Physics Volume 2 by Samuel J. Ling and others>[§15.4, RMS definition and average power in a resistor]{ref}
An original harmonic example shows why RMS values alone do not determine <active power>. Let
$$
v(t)=\sqrt2(100)\cos\omega t,\qquad
i(t)=\sqrt2[3\cos\omega t+4\cos3\omega t]
$$
in volts and amperes. Orthogonality over a period gives $I_{\rm rms}=\sqrt{3^2+4^2}=5\,\mathrm A$, 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 gives
$$
P=V_0I_0+\sum_{n\ge1}V_nI_n\cos(\alpha_n-\beta_n),
$$
when 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.<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Energy and Power, equations51–55]{ref}
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