What Does A Lock-In Measure - Stanford Research Systems SR810 Manual

WHAT DOES A LOCK-IN MEASURE?

So what exactly does the SR810 measure?
Fourier's theorem basically states that any input
signal can be represented as the sum of many,
many sine waves
frequencies and phases.
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,
representation does not convey very much
information about the various frequencies which
make up the signal.
What does the SR810 measure?
The SR810 multiplies the 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 SR810, the product of this
multiplication yields
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 SR810, 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
S(t) = 1.273sin(ωt) + 0.4244sin(3ωt) +
0.2546sin(5ωt) + ...
where ω = 2πf. The SR810, locked to f will single
out the first component. The measured signal will
of differing amplitudes,
This is generally
the time domain
a DC output signal
be 1.273sin(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 frequencies 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 SR810
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.
Thus, in the previous example with a 2 V pk-pk
square wave input, the SR810 would detect the
first sine component, 1.273sin(ωt). The measured
and displayed magnitude would be 0.90 V (rms)
(1/√2 x 1.273).
Degrees or Radians?
In this discussion, frequencies have been referred
to as f (Hz) and w (2πf radians/sec). This is
because people measure frequencies in cycles
per second and math works best in radians. For
purposes of measurement,
measured in a lock-in amplifier are in Hz. The
equations used to explain the actual calculations
are sometimes written using w to simplify the
expressions.
Phase is always reported in degrees. Once again,
this is more by custom than by choice. Equations
written as sin(ωt + θ) are written as if θ is in
radians mostly for simplicity. Lock-in amplifiers
always manipulate
degrees.
3-3
SR810 Basics
frequencies
and measure phase
as
in