What Does A Lock-In Measure - Stanford Research Systems SR865A Operation Manual

40 Basics
R measures the signal amplitude and does not depend upon the phase between the signal
and lock-in reference.
A dual-phase lock-in, such as the SR865A, has two PSD's, with reference oscillators 90°
apart, and can measure X, Y and R directly. In addition, the phase θ between the signal
and lock-in reference, can be measured according to

What Does a Lock-in Measure?

So what exactly does the SR865A measure? Fourier's theorem basically states that any
input signal can be represented as the sum of many sine waves of differing amplitudes,
frequencies and phases. This is generally considered as representing the signal in the
"frequency domain". Normal oscilloscopes display the signal in the "time domain".
Except in the case of clean sine waves, the time domain representation does not convey
very much information about the various frequencies that make up the signal.
What does the SR865A measure?
The SR865A multiplies the input signal by a pure sine wave at the reference frequency.
All components of the input signal are multiplied by the reference simultaneously.
Mathematically speaking, sine waves of differing frequencies are orthogonal, i.e. the
average of the product of two sine waves is zero unless the frequencies are exactly the
same. In the SR865A, the product of this multiplication yields a dc output signal
proportional to the component of the signal whose frequency is exactly locked to the
reference frequency. The low pass filter which follows the multiplier provides the
averaging which removes the products of the reference with components at all other
frequencies.
The SR865A, because it multiplies the signal with a pure sine wave, measures the single
Fourier (sine) component of the signal at the reference frequency. Let's take a look at an
example. Suppose the input signal is a simple square wave at frequency f. The square
wave is actually composed of many sine waves at multiples of f with carefully related
amplitudes and phases. A 2V pk–pk square wave can be expressed as
where ω = 2πf. The SR865A, locked to f will single out the first component. The
measured signal will be 1.273 sin(ωt), not the 2V pk–pk that you'd measure on a scope.
In the general case, the input consists of signal plus noise. Noise is represented as varying
signals at all frequencies. The ideal lock-in only responds to noise at the reference
frequency. Noise at other frequencies is removed by the low pass filter following the
multiplier. This "bandwidth narrowing" is the primary advantage that a lock-in amplifier
provides. Only inputs at the reference frequency result in an output.
RMS or Peak?
Lock-in amplifiers as a general rule display the input signal in Volts RMS. When the
SR865A displays a magnitude of 1V (rms), the component of the input signal at the
reference frequency is a sine wave with an amplitude of 1 Vrms or 2.8 V pk–pk.
SR865A DSP Lock-in Amplifier
−1
θ = tan
(Y/X)
S(t) = 1.273 sin(ωt) + 0.4244 sin(3ωt) + 0.2546 sin(5ωt) + ...
Chapter 2