Vpsd = cos(wr+Ø) cos(wst)

=1/2 cos[(wr + ws)t+Ø] + 1/2 cos[(wr - ws)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.

Afront-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

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