= Power factor
{title2=$P/(V_{\rm rms}I_{\rm rms})$}
The <power factor> is the ratio of <active power> to the product of RMS terminal <electric potential difference> and <electric current>:
$$
\mathrm{pf}=\frac{P}{V_{\rm rms}I_{\rm rms}},
$$
when the denominator is positive. It measures how effectively those terminal magnitudes support net average transfer. With signed absorbed power it can be negative for net generation; for an absorbing load the usual range is zero to one. The <Cauchy-Schwarz inequality> applied to the time average of $v(t)i(t)$ proves $|P|\le V_{\rm rms}I_{\rm rms}$.
For one-frequency sinusoids, $P=V_{\rm rms}I_{\rm rms}\cos\varphi$, so the <power factor> is $\cos\varphi$. The phase-only quantity for the fundamental is also called <displacement power factor>. The primary treatments make the RMS convention explicit.<University Physics Volume 2 by Samuel J. Ling and others>[§15.4, equations15.12–15.13]{ref}<AC Power Flow in Linear Networks by James L. Kirtley Jr.>[Energy and Power, equation62; §7.1 RMS Amplitude]{ref}
A distorted current can give another result. For purely sinusoidal supply potential difference and a zero-mean current with fundamental RMS magnitude $I_1$,
$$
\mathrm{pf}=\frac{I_1}{I_{\rm rms}}\cos\varphi_1.
$$
Only the in-phase fundamental contributes to the average product with that supply waveform. For an original example with fundamental3A in phase and third-harmonic RMS magnitude4A, total RMS current is5A. The <displacement power factor> is one but total <power factor> is0.6. Changing the fundamental phase alone would not remove this harmonic-current contribution.
At fixed absorbed <active power> and supply RMS potential difference, $I_{\rm rms}=P/(V_{\rm rms}\mathrm{pf})$. For an original900W,230V example, changing the factor from0.6 to0.9 changes source current from about6.52 to4.35A. In the same upstream resistance, the corresponding $I^2R$ loss becomes $(0.6/0.9)^2=4/9$ of its former value. This is a stipulated circuit calculation, not an assertion about a particular tariff or installation.
<Power factor correction> addresses the mechanism causing the low ratio. An ideal capacitive branch can offset inductive <reactive power> at one frequency; it does not automatically correct an arbitrary distorted current. State the waveform conditions and whether the reported number is total or displacement factor when comparing measurements.
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