= Harmonic active power sum
{title2=$P=V_0I_0+\sum_{n\ge1}V_nI_n\cos(\alpha_n-\phi_n)$}
The <harmonic active power sum> expresses average electrical <power> as the sum of contributions from matching components of periodic terminal <electric potential difference> and <electric current>. With a common <fundamental frequency> and <root mean square> magnitudes for nonzero <harmonic components>, write
$$
v(t)=V_0+\sqrt2\sum_{n\ge1}V_n\cos(n\omega t+\alpha_n),\qquad
i(t)=I_0+\sqrt2\sum_{n\ge1}I_n\cos(n\omega t+\phi_n).
$$
Here $V_0,I_0$ are the signed <DC components>. The <passive sign convention> makes the average product absorbed <active power>. Averaging over a complete fundamental period gives
$$
\boxed{P=\langle vi\rangle=V_0I_0+\sum_{n\ge1}V_nI_n\cos(\alpha_n-\phi_n).}
$$
For finite sums this follows from the cosine-product identity: products of different integer <harmonic components> average to zero, and matching components give half the cosine of their relative phase. For square-integrable periodic waveforms, the corresponding <Fourier series> identity follows by continuity of the inner product in the mean-square norm. The <orthogonality> is a property of the averaging interval and basis, not an assumption that the physical sources are independent.
For an original example, set $V_0=10\,\mathrm V$, $I_0=2\,\mathrm A$, $V_1=100\,\mathrm V$, $I_1=3\,\mathrm A$, and fundamental relative phase $\pi/3$. Add third-harmonic magnitudes $V_3=20\,\mathrm V$, $I_3=1\,\mathrm A$ with relative phase $\pi$. Then
$$
P=20+100(3)\cos(\pi/3)+20(1)\cos\pi
=150\,\mathrm W.
$$
The third-harmonic contribution is $-20\,\mathrm W$, even though the total absorbed <active power> is positive. The <root mean square> values satisfy $V_{\rm rms}^2=10500\,\mathrm{V^2}$ and $I_{\rm rms}^2=14\,\mathrm{A^2}$, so <apparent power> is about $383.41\,\mathrm{VA}$ and total <power factor> about $0.3912$.
For a purely sinusoidal supply with no <DC component>, only $V_1$ survives, giving $P=V_1I_1\cos(\alpha_1-\phi_1)$ even if the <electric current> is distorted. Current in other <harmonic components> can still increase conductor heating. The usual one-frequency <complex electrical power> construction is derived in the primary MIT treatment; the sum above applies that average-product reasoning separately to orthogonal components.<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Energy and Power, equations 45–63; §7.1, RMS amplitudes]{ref}
A proposed measurement records the two waveforms synchronously and compares their directly averaged product with the resolved sum. A non-integer-period window, unresolved <harmonic components>, channel timing error or changing waveform can spoil the simple finite-record equality. The formula assumes a shared periodic model; multiplying separately measured RMS magnitudes gives <apparent power> instead.
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