Time Constants And Sensitivity - Stanford Research Systems SR865A Operation Manual

44 Basics
next section, we'll see how the SR865A performs the low pass filtering and dc
amplification required at the output of the PSD's.

Time Constants and Sensitivity

Remember, the output of the PSD contains many signals. Most of the output signals have
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.
RC filters
Traditionally, the time constant setting of a lock-in amplifier determines the bandwidth of
an RC lowpass filter. 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/octave 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/octave 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 tradeoff 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 (ENBW) for noise
measurements. The ENBW is not the filter −3 dB pole, it is the effective bandwidth for
Gaussian noise. More about this later.
The digital signal processing in the SR865A 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.
Why are multiple filter stages 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.
SR865A DSP Lock-in Amplifier
Chapter 2