= Displacement power factor
{title2=$\cos\varphi_1$}
The <displacement power factor> is $\cos\varphi_1$, where $\varphi_1$ is the phase difference between the fundamental components of terminal <electric potential difference> and <electric current>. It describes phase displacement at that frequency. For purely sinusoidal signals it equals total <power factor>; with distorted waveforms those quantities can differ. The one-frequency phase relation follows from the RMS average-power expression.<University Physics Volume 2 by Samuel J. Ling and others>[§15.4, equations15.12–15.13]{ref}
For a purely sinusoidal potential difference of RMS magnitude $V$ and zero-mean periodic current, orthogonality of harmonics gives
$$
P=VI_1\cos\varphi_1,\qquad
\mathrm{pf}=\frac{I_1}{I_{\rm rms}}\cos\varphi_1.
$$
The ratio $I_1/I_{\rm rms}$ is the <current distortion factor> under these assumptions. A displacement factor near one does not establish that the whole current waveform closely follows the supply sinusoid.
For an original example, let the supply be100V RMS. Let the fundamental current be4A RMS at a lagging phase with cosine0.8, and add a3A RMS third harmonic. Total RMS current is5A, <active power>320W and <apparent power>500VA. The displacement factor is0.8 while total <power factor> is0.64.
Now cancel only the fundamental quadrature current, whose magnitude is $4\sqrt{1-0.8^2}=2.4\,\mathrm A$. The remaining in-phase fundamental is3.2A and the third harmonic is still3A. RMS source current is $\sqrt{3.2^2+3^2}\approx4.39\,\mathrm A$, so total factor is about0.73 even though the displacement factor has become one. This illustrates a limitation of phase-only <power factor correction>, not a universal model of a correcting device.
The cosine also loses lead/lag sign: phases $+\varphi_1$ and $-\varphi_1$ have the same displacement factor but opposite sinusoidal <reactive power>. Retain the phase or lead/lag convention when that distinction matters. If the potential difference itself contains harmonics, total <active power> needs the contributions of the corresponding harmonic pairs; this simple fundamental-only reduction is no longer sufficient.
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