原创 Basic principles of a lock in amplifier

 Basic principles of a lock in amplifier

Operation of a lock-in amplifier relies on the orthogonally of sinusoidal functions. Specifically, when a sinusoidal function of frequency ν is multiplied to another sinusoidal function of frequency μ not equal to ν and integrated over a time much longer than the period of the two functions, then the result is zero. In the case when μ is equal to ν, and the two functions are in phase, the average value is equal to half of the product of the amplitudes ^[6-8].<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />

In essence, a lock-in amplifier takes the input signal, multiplies it by the reference signal (either provided from the internal oscillator or an external source), and integrates it over a specified time, usually on the order of milliseconds to a few seconds. The resulting signal is an essentially DC signal, where the contribution from any signal that is not at the same frequency as the reference signal is attenuated essentially to zero, as well as the out-of-phase component of the signal that has the same frequency as the reference signal (because sine functions are orthogonal to the cosine functions of the same frequency), and this is also why a lock-in is a phase sensitive detector ^[8-10].

For a sine reference signal and an input waveform Uin(t), the DC output signal Uout(t) can be calculated for an analog lock-in amplifier by^[1,2]:<?xml:namespace prefix = v ns = "urn:schemas-microsoft-com:vml" />

where φ is a phase that can be set on the lock-in (set to zero by default).

Practically, many applications of the lock-in only require recovering the signal amplitude rather than relative phase to the reference signal; a lock-in usually measures both in-phase (X) and out-of-phase (Y) components of the signal and can calculate the magnitude (R) from that.

A detail mathematical derivation is as follows ^{[2- 4]}:

Using Fourier’s theorem, any input signal, including the noise accompanying it, can be represented as the sum of many sinewaves of different amplitudes, phases and frequencies. The phase-sensitive detector in the lock-in amplifier multiplies all these components by a signal at the reference frequency ^[17].

Figure 1-2 Orthogonal phase-locked 

来自2008年毕设