= Peak-versus-RMS normalization in complex electrical power
<Peak-versus-RMS normalization in complex electrical power> determines the coefficient multiplying the product of voltage and conjugate current <phasors>. A <peak amplitude phasor> reconstructs a sinusoid directly, while a <root mean square phasor> requires a factor of $\sqrt2$. The DOE effective-value derivation relates these amplitudes by comparing heating from AC and DC currents.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 7, AC Generation Analysis, printed pp. 4–7, equations 7-2 and 7-5]{ref}
To track any combination, write
$$
v(t)=a_V\operatorname{Re}(Ve^{i\omega t}),\qquad
i(t)=a_I\operatorname{Re}(Ie^{i\omega t}),
$$
where each reconstruction scale is $1$ for a <peak phasor> or $\sqrt2$ for an <RMS phasor>. A period average of the product leaves the <real power>
$$
P=\frac{a_Va_I}{2}\operatorname{Re}(VI^*).
$$
Using the same positive-frequency and passive-sign convention, the corresponding <complex electrical power> is
$$
S=\frac{a_Va_I}{2}VI^*=P+iQ.
$$
Thus the coefficient is $1/2$ for two peak-normalized amplitudes, $1$ for two RMS-normalized amplitudes, and $1/\sqrt2$ for a mixed pair. The last case is why merely recognizing a product as a power calculation is not enough to choose its normalization.
For an original example, let the <electric potential difference> be represented by peak amplitude $V=100\,\mathrm V$ and the <electric current> by RMS amplitude $I=2e^{-i\pi/6}\,\mathrm A$, at the same <frequency>. Then
$$
S=\frac{200}{\sqrt2}e^{i\pi/6}\,\mathrm{VA},\qquad
P=50\sqrt6\,\mathrm W\approx122.47\,\mathrm W,\qquad
Q=50\sqrt2\,\mathrm{var}\approx70.71\,\mathrm{var}.
$$
Multiplying the two numbers as though both were <RMS phasors> overestimates the <complex electrical power> by $\sqrt2$. Inserting a peak-pair factor $1/2$ instead underestimates it by $1/\sqrt2$. The time-domain reconstruction, $v(t)=100\cos\omega t$ and $i(t)=2\sqrt2\cos(\omega t-\pi/6)$, independently verifies the period average.
The <mixed peak and RMS gain-calibration error> also matters when source and receiver amplitude reports are combined: matching units does not establish matching amplitude conventions. This calculation concerns one sinusoidal component. A nonsinusoidal waveform requires its actual period average or an appropriately normalized harmonic decomposition, not one unspecified phasor product.
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