I have two questions:
Why does a lock-in amplifier multiply the incoming signal s(t) with both a cosine and a sine reference and not just a single one?
What problem may eventually arise, if you multiply the signal only with either cosine or sine?
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Trig theory to extract the phase & magnitude.
If you only care about phase...
Say you have a signal \$Acos(\omega t + \phi)\$ and you want to extract \$\phi\$. You can use an oscillator of the same frequency to extract this info BUT the issue is the phase.
\$V_{sig} = Acos(\omega t + \phi)\$
\$V_{osc} = cos(\omega t)\$
\$V_{sig}V_{osc} = Acos(\omega t + \phi) Cos(\omega t)\$
By the double angle identity:
\$ = \frac{1}{2}Acos(\phi) + \frac{1}{2}ACos(2 \omega t + \phi)\$
a DC term relating to the phase can be realised as well as a component at twice teh freqency of the carrier. The phase can then be extracted by a moving average filter at the carrier frequency of a simple low pass filter.
\$V_{sig}V_{osc}Filt = \frac{1}{2}Acos(\phi)\$
If you care about phase and magnitude
To clearly extract the phase and magnitude then two oscillators, in quadrature, are required.
\$V_{sig} = Asin(\omega t +\phi)\$
\$V_{osc0} = Xsin(\omega t)\$
\$V_{osc90} = Xcos(\omega t)\$
\$V_0 = Xsin(\omega t)Asin(\omega t +\phi) = \frac{XA}{2}(cos(\phi) - cos(2\omega t + \phi))\$ \$V_{90} = Xcos(\omega t)Asin(\omega t +\phi) = \frac{XA}{2}(sin(\phi) + sin(2\omega t + \phi))\$
Filter these signals to remove the twice carrier component
\$V_{0f} = \frac{XA}{2}(cos(\phi) ) \$
\$V_{90f} = \frac{XA}{2}(sin(\phi) ) \$
via trig:
\$\phi = atan( \frac{V_{90f}}{V_{0f}} )\$ \$A = \frac{2}{X}\sqrt{V_{0f}^2 + V_{90f}^2 } \$