Time Constants And Dc Gain - Stanford Research Systems SR810 Manual

SR810 Basics

TIME CONSTANTS and DC GAIN

Remember, the output of the PSD contains many
signals. Most of the
frequencies which are either the sum or difference
between an input signal frequency and the
reference frequency. Only the component of the
input signal whose frequency is exactly equal to
the reference frequency will result in a DC output.
The low pass filter at the PSD output removes all
of the unwanted AC signals, both the 2F (sum of
the signal and the reference) and the noise
components. This filter is what makes the lock-in
such a narrow band detector.
Time Constants
Lock-in amplifiers have traditionally set the low
pass filter bandwidth by setting the time constant.
The time constant is simply 1/2πf where f is the -
3 dB frequency of the filter. The low pass filters
are simple 6 dB/oct roll off, RC type filters. A 1
second time constant referred to a filter whose -
3 dB point occurred at 0.16 Hz and rolled off at
6 dB/oct beyond 0.16 Hz. Typically, there are two
successive filters so that the overall filter can roll
off at either 6 dB or 12 dB per octave. The time
constant referred to the -3 dB point of each filter
alone (not the combined filter).
The notion of time constant arises from the fact
that the actual output is supposed to be a DC
signal. In fact, when there is noise at the input,
there is noise on the output. By increasing the time
constant, the output becomes more steady and
easier to measure reliably. The trade off comes
when real changes in the input signal take many
time constants to be reflected at the output. This is
because a single RC filter requires about 5 time
constants to settle to its final value. The time
constant reflects how slowly the output responds,
and thus the degree of output smoothing.
The time constant also determines the equivalent
noise bandwidth
measurements. The ENBW is NOT the filter -3 dB
pole, it is the effective bandwidth for Gaussian
noise. More about this later.
Digital Filters vs Analog Filters
The SR810 improves on analog filters in many
ways. First, analog lock-ins provide at most, two
stages of filtering with a maximum roll off of
12 dB/oct. This limitation is usually due to space
output signals have
(ENBW) for noise
and expense. Each filter needs to have many
different time constant settings. The different
settings require
switches to select them, all of which is costly and
space consuming.
The digital signal processor in the SR810 handles
all of the low pass filtering. Each PSD can be
followed by up to four filter stages for up to
24 dB/oct of roll off. Since the filters are digital, the
SR810 is not limited to just two stages of filtering.
Why is the increased roll off desirable? Consider
an example where the reference is at 1 kHz and a
large noise signal is at 1.05 kHz. The PSD noise
outputs are at 50 Hz (difference) and 2.05 kHz
(sum). Clearly the 50 Hz component is the more
difficult to low pass filter. If the noise signal is
80 dB above the full scale signal and we would
like to measure the signal to 1 % (-40 dB), then
the 50 Hz component needs to be reduced by
120 dB. To do this in two stages would require a
time constant of at least 3 seconds. To accomplish
the same attenuation in four stages only requires
100 ms of time constant. In the second case, the
output will respond 30 times faster and the
experiment will take less time.
Synchronous Filters
Another advantage of digital filtering is the ability
to do synchronous filtering. Even if the input signal
has no noise, the PSD output always contains a
component at 2F (sum frequency of signal and
reference) whose amplitude equals or exceeds the
desired DC output depending upon the phase. At
low frequencies, the time constant required to
attenuate the 2F component can be quite long. For
example, at 1 Hz, the 2F output is at 2 Hz and to
attenuate the 2 Hz by 60 dB in two stages requires
a time constant of 3 seconds.
A synchronous filter, on the other hand, operates
totally differently. The PSD output is averaged
over a complete cycle of the reference frequency.
The result is that all components at multiples of
the reference (2F included) are notched out
completely. In the case of a clean signal, almost
no additional filtering would be required. This is
increasingly useful the lower the reference
frequency. Imagine what the time constant would
need to be at 0.001 Hz!
3-8
different components and