the in phase signals (V. ) from the p DSodo for t v/ith the
X U S G C
time constant T = RC calculated in ohms x farads, The time
constant thus determines the net gain of the filter» The signal
bandwidth of the operational amplifier with an RC filter in
the feedback loop is the same as the simple R.C. low pass filter
( F ig c 2o3b) i.e. where f = '1 (equation 16 ) 0 This upper 2tcRC
15 2
value is a result of defining the bandwidth, Af, as equal to
that sinusoidal frequency at which the circuits response is
1/ / 2 of its maximum value (A f and f are therefore the same upper
value)o However consider a white noise source generating a
noise voltage per unit bandwidth which gives an rms voltage at
the output of the filter and refer to equation 6,
It can be seen that to find the noise output of the RC filter,
the square of the noise voltage must be integrated over the 152.
range of frequencies of interest. The result is an equivalent
noise bandwidth o f :
f = 1 (2 2)
upper ^
This is significantly different from the signal bandwidth 1 .
2tcRC
appears in the equations for shot and Johnson noise- This
argument is true for v/hite noisee I? the noise is not white,
the relation between noise and bandwidth is different but in
general reducing A f reduces the noise at the output-
Not only is the bandwidth of the system determined by
the time constant but also the response time is affected- The
response time is defined as the time it takes for the output
signal to reach 99% of its final value- It maybe shown that
Tr = 4-6RC (23)
Combining equations (22 and (23):
I, = 1o15 A/ 1 (24-)
£ 7 “ ^ £ f
If the electrical bandwidth is reduced in an attempt to reduce
random noise the response time increases (equation 2h)0
Unfortunately not only does this extra time required cause
difficulties but also the trade of time for noise is not linear-
X
The noise only decreases as the (time) 2 a consequence of equivalent
noise bandwidth- (i-e- equations (3) to (3) where the
magnitude of the various noise sources are proportional to the
square root of the bandwidth)- To illustrate this point consider
that going from a time constant of 1 second to one of 10 0 seconds
narrows A f by a factor of 10 0, but this only decreases the noise
X
by (100) 2 i-e- 10- Thus it is necessary to wait 100 times
larger to improve this signal-to-noise ratio by a factor of 10-
Moreover the increase in response time will introduce errors
and inconveniences which effectively limit the extent by which
A f may be profitably reduced- These difficulties are outlined
below:
(1) Analysis time - the extra time necessary for the amplifier
to give a readout becomes inconvenient when carrying out multiple
readings such as statistical checks on a large number of samples
and standardso
(2) Sample consumption - A longer analysis time will mean
more sample solution consumed which may be a disadvantage for
some applicationso
(3) Drift Errors - Instrumental drifts resulting from source
spectral radiance drift, amplifier drift etc« have a greater
effect on analytical accuracy the longer it takes to complete
a measuremento
(A) Spectrum-scanning - For recording line spectra, the mono-
-1
7
chromator scan speed r(non sec ) should be approximately :
where S is the spectral bandpass of the monochromator0 In the
was used routinely i^hereas the optimum should have been C& o008 nm
accuracy is required..
The ideal lock-in amplifier is inherently linear, the
main result of this being that unwanted signals are a.c, and
consequently give random fluctuations about the d«,c„ level given
by the wanted signalo By contrast the noise passed by a tuned
amplifier into a conventional polarity sensitive rectifier gives
an aoCo output superimposed on a d«Co offset„ This d0c„ output
is an error which depends in magnitude upon the mean value of
the noise and is subject to changes which do not depend on the
signal level..
D Building of the lock-in amplifier
The basic principles of the lock-in amplifier circuit
‘have already been outlined in section 2o3°'\°Co The circuit
(23)
present work S was 0„25 nm and in fact a scan speed of 0„03 nm sec
— 1 A —1
sec (&if = 0*032 sec )„ However the Jarrell-Ash monochromator
- 1
3 3
was taken from the design by Caplan and Stern * This design
was specifically titled 'An Inexpensive Lock-In Amplifier' and
so it has proved to be* All the components were inexpensive
although it was necessary to build a tv/in power supply (£17)
to power the amplifier„ The total cost of the electronics
(including the preamplifier) was less than £10 0* The design
for the power supply was taken from the R.S* Components Ltd*
catalogue and R*S, voltage regulatorswere usedo
Two relatively minor modifications were made to the
lock-in amplifier circuitry., Firstly a modification to the input
amplifier. This amplifier as designed had too low an input
impedance to be compatible id.th the photomultiplier tube or the
preamplifier which was subsequently built <> Secondly a new output
amplifier (differential low-pass filter) was.built to be used as
supplementary to the design output amplifier* This new differential
low-pass filter was temperature stabilised to give minimum
drift with ambient temperature, thus permitting longer time
constants to be’ used at the output*
The main problem encountered during the building of the
lock-in amplifier was with high frequency oscillation caused by
instability of the integrated circuit operational amplifiers*
(This is a common problem with such amplifiers and there are
standard remedies based usually on trial and error with different
values of circuit components )* In this section the circuitry
in its final form is reproduced in sections as the description
of its features continues* Any differences between the Caplan
and Stern design and this circuitry are pointed out where
necessary* The preamplifier, described below, was not part of
the original design but was necessary to match the lock-in
amplifier to the photomultiplier tube*