Lock-in Amplifier - Basic Principles

Basic Principles

Operation of a lock-in amplifier relies on the orthogonality of sinusoidal functions. Specifically, when a sinusoidal function of frequency ν is multiplied by another sinusoidal function of frequency μ not equal to ν and integrated over a time much longer than the period of the two functions, 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.

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 a DC signal, where the contribution from any signal that is not at the same frequency as the reference signal is attenuated close 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.

For a sine reference signal and an input waveform, the DC output signal can be calculated for an analog lock-in amplifier by:

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. For these purposes, lock-in usually measures two components and there are two outputs: and, where - phase difference between the signal and reference. These two quantities represent the signal as a vector relative to the lock-in reference oscillator. X is called the 'in-phase' component and Y the 'quadrature' component. By computing the magnitude (R) of the signal vector, the phase dependency is removed:

.