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Vpsd = cos(w
r
+Ø) cos(w
s
t)
= 1/2 cos[(w
r
+ w
s
)t+Ø] +
1/2 cos[(w
r
- w
s
)t+Ø]
The sum frequency component is attenuated by
the low pass filter, and only those difference
frequency components within the low pass filter's
narrow bandwidth will pass through to the dc
amplifier. Since the low pass filter can have time
constants up to 100 seconds, the lock-in can reject
noise which is more than .0025 Hz away from the
reference frequency input.
For signals which are in phase with the reference,
the phase control is usually adjusted for zero
phase difference between the signal and the
reference. This can be done by maximizing the
output signal. A more sensitive technique would
be to adjust the phase to null the signal. This
places the reference oscillator at 90 degrees with
respect to the signal. The phase control can now
be shifted by 90 degrees to maximize the signal.
Alternatively, since the phase control is well
calibrated, the phase of the signal can be
measured by adding 90 degrees to the phase
setting which nulls the signal.
Understanding the Specifications
The table below lists some specifications for the
SR510 lock-in amplifier. Also listed are the error
contributions due to each of these items. The
specifications will allow a measurement with a 2%
accuracy to be made in one minute.
We have chosen a reference frequency of 5 kHz
so as to be in a relatively quiet part of the noise
spectrum. This frequency is high enough to avoid
low frequency '1/f' noise as well as line noise. The
frequency is low enough to avoid phase shifts and
amplitude errors due to the RC time constant of
the source impedance and the cable capacitance.
The full-scale sensitivity of 100 nV matches the
expected signal from our sample. The sensitivity
is calibrated to 1%. The instrument's output
stability also affects the measurement accuracy.
For the required dynamic reserve, the output
stability is 0.1%/°C. For a 10°C temperature
change we can expect a 1% error.
A front-end noise of 7 nV/√Hz will manifest itself
as a 1.2 nVrms noise after a 10 second low-pass
filter since the equivalent noise bandwidth of a
single pole filter is 1/4RC. The output will converge
exponentially to the final value with a 10 second
time constant. If we wait 50 seconds, the output
will have come to within 0.7% of its final value.
The dynamic reserve of 60 dB is required by our
expectation that the noise will be a thousand times
larger than the signal. Additional dynamic reserve
is available by using the bandpass and notch
filters.
A phase-shift error of the PLL tracking circuits will
cause a measurement error equal to the cosine of
the phase shift error. The SR510’s 1° phase
accuracy will not make a significant contribution to
the measurement error.
Specifications for the Example Measurement
Specification Value Error
Full Scale Sensitivity 100 nV
Dynamic Reserve 60 dB
Reference Frequency 5 kHz
Gain Accuracy 1% 1%
Output Stability 0.1%/°C 1%
Front-End Noise < 7 nV/√Hz 1.2%
Output Time Constant > 10 S 0.7%
Total RMS Error 2%
Shielding and Ground Loops
In order to achieve the 2% accuracy given in this
measurement example, we will have to be careful
to minimize the various noise sources which can
be found in the laboratory. (See Appendix A for a
brief discussion on noise sources and shielding)
While intrinsic noise (Johnson noise, 1/f noise and
alike) is not a problem in this measurement, other
noise sources could be a problem. These noise
sources can be reduced by proper shielding.
There are two methods for connecting the lock-in
to the experiment: the first method is more
convenient, but the second eliminates spurious
pick-up more effectively.
In the first method, the lock-in uses the 'A' input in
a 'quasi-differential' mode. Here, the lock-in
detects the signal as the voltage between the
center and outer conductors of the A input. The
lock-in does not force A's shield to ground, rather it
is connected to the lock-in's ground via a 10½
resistor. Because the lock-in must sense the
shield voltage (in order to avoid the large ground
loop noise between the experiment and the lock-
in) any noise pickup on the shield will appear as
noise to the lock-in. For a low impedance source