The phase-locked amplifier is actually an analog Fourier converter. The output of the phase-locked amplifier is a DC voltage, which is proportional to the signal amplitude of a specific frequency (parameter input frequency) in the input signal. The other frequency components in the input signal will not make any contribution to the output voltage.
The two sinusoidal signals have a frequency of 1 Hz and a phase difference of 90 degrees. The result obtained by multiplying them by a multiplier is a sinusoidal signal with a DC offset.
If it is a 1Hz and 1.1Hz signal multiplied, the result obtained by multiplying with a multiplier is a modulation signal with a sinusoidal profile and a DC offset of 0.
Only signals that are completely consistent with the frequency of the reference signal can get a DC offset at the output of the multiplier. Other signals are AC signals at the output. If a low-pass filter is added to the output of the multiplier, all the AC signal components are filtered out, and the remaining DC components are simply proportional to the amplitude of the signal components of a specific frequency in the input signal.