= Lock-in quadratures
{title2=$X=A\cos\phi,\quad Y=A\sin\phi$}
Let the input <electric potential difference> be $s(t)=A\cos(\omega t+\phi)$, with peak <amplitude> $A\geq0$, <angular frequency> $\omega>0$, and <phase difference> $\phi$ relative to $\cos\omega t$. For an observation <interval> $[0,T]$ containing an <integer> number $N\geq1$ of cycles, each lasting one <period of a function>, $T=2\pi N/\omega$, define the <lock-in quadratures> in a peak convention:
$$
X=\frac2T\int_0^T s(t)\cos\omega t\,dt,\qquad
Y=-\frac2T\int_0^T s(t)\sin\omega t\,dt.
$$
These <integrals> average the products with dimensionless references $2\cos\omega t$ and $-2\sin\omega t$; $X$ and $Y$ have the input's units. The minus sign makes positive $\phi$ give positive $Y$. The <trigonometric functions> give
$$
\begin{aligned}
2s(t)\cos\omega t&=A\cos\phi+A\cos(2\omega t+\phi),\\
-2s(t)\sin\omega t&=A\sin\phi-A\sin(2\omega t+\phi).
\end{aligned}
$$
The oscillating terms have zero <integral> over the stated <interval>, proving $X=A\cos\phi$ and $Y=A\sin\phi$. Their <orthogonality> turns an oscillating <signal> into two constant coordinates.
Represent those coordinates as a <vector>, or as the <complex number> $Z=X+\mathrm iY$. Its <modulus> and <complex argument> recover the peak <amplitude> and <phase difference>:
$$
R=\sqrt{X^2+Y^2}=A,\qquad
\phi=\operatorname{atan2}(Y,X)\pmod{2\pi},\qquad R>0.
$$
Here $\operatorname{atan2}(Y,X)$ selects the <angle> with the correct signs of both coordinates, rather than applying the <arctangent> to $Y/X$ and losing the quadrant. When $X=Y=0$, the <amplitude> is zero and the <phase difference> is undefined. When $\phi=\pi/2$, $X=0$ despite a nonzero <signal>; measuring $Y$ prevents this blind spot. Rotating both references by a fixed <angle> rotates the coordinate <vector> but preserves $R$.
The factors of two above deliberately recover peak <amplitude>. The SR830 instead reports the selected sinusoidal component in <root mean square> units: in a calibrated, zero-offset display, $R_{\mathrm{rms}}=R/\sqrt2$. For $R=0.4\,\mathrm V$ this is $0.282843\,\mathrm V$, while the peak-to-peak span of that component is $0.8\,\mathrm V$. Display normalization, reference sign, gain and <angle> units must be checked for the particular instrument.<Model SR830 DSP Lock-In Amplifier manual by Stanford Research Systems>[Model SR830 DSP Lock-In Amplifier manual, printed p. 3-3, “RMS or Peak?” and “Degrees or Radians?”; pp. 3-13–3-14, output scaling and display offsets]{ref}
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