= Root mean square phasor
{title2=$V_{\mathrm{rms}}$}
= RMS phasor
{c}
{synonym}
A <root mean square phasor> represents a single-frequency sinusoid by a complex amplitude whose magnitude is its <root mean square> value. With the convention $v(t)=V_{\rm pk}\cos(\omega t+\phi)$ and $\omega>0$, define
$$
\underline V=\frac{V_{\rm pk}}{\sqrt2}e^{i\phi},\qquad
v(t)=\sqrt2\,\operatorname{Re}(\underline V e^{i\omega t}).
$$
Indeed, the period average of $v^2$ is $V_{\rm pk}^2/2$, so $|\underline V|=V_{\rm rms}$. The factor $\sqrt2$ in reconstruction distinguishes this convention from the peak-amplitude <phasor> convention in the parent article. The DOE handbook derives the same effective value by comparing the heating from <alternating current> and <direct current>.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 7, AC Generation Analysis, printed pp. 4–7, especially equations 7-2 and 7-5 and Figure 4]{ref}
Using <RMS phasors> for both <electric potential difference> and <electric current>, the passive-sign convention gives <complex electrical power>
$$
S=\underline V\,\underline I^{\,*}=P+iQ.
$$
To see the normalization directly, multiplying the two reconstructed sinusoids and averaging over a period leaves $P=\operatorname{Re}(\underline V\underline I^{\,*})$. Peak-amplitude <phasors> would instead require the prefactor $1/2$. This <peak-versus-RMS normalization in complex electrical power> must be stated before combining a source model with measured amplitudes.
For an original example, let $\underline V=10e^{i0}\,\mathrm V$ and $\underline I=2e^{-i\pi/3}\,\mathrm A$. The <electric current> lags the <electric potential difference> by $60^\circ$, giving
$$
S=20e^{i\pi/3}\,\mathrm{VA},\qquad P=10\,\mathrm W,\qquad Q=10\sqrt3\,\mathrm{var}.
$$
The corresponding time-domain peaks are $10\sqrt2\,\mathrm V$ and $2\sqrt2\,\mathrm A$. Treating those peaks as <RMS phasor> magnitudes would falsely double the computed <complex electrical power>. This representation assumes a sinusoid at one common <frequency>; a nonsinusoidal signal can have a <root mean square> value without being represented by one <RMS phasor>.
Back to article page
