First: read the great discussion of Lock-in amplification in the SRS catalog (available in the lab). You should also work through the LabView tutorial.

This experiment will be quite fast-paced, but worth it. The analysis section of your report will be correspondingly smaller, although the theory section
should be in depth.

Answer the following pre-lab questions individually:

1. Show that the product of two sine waves (with arbitrary phase and frequency) can be written as the sum of a sine wave at the difference frequency, plus a sine wave at the sum frequency:

   

   Find A₃, A₄, etc. in terms of A₁, A₂, etc.

Remember that we model our reference (the applied signal, which in this case is proportional to the applied current) as

and the input (the desired signal, the line noise, and the random noise) as

where is the phase of the i'th noise component.  When we multiply the input and the reference,

2. What happens to the signal component?
3. What happens to the 60 Hz component?
4. What happens to the noise component which is at the same frequency as the reference?

Next, average the multiplied signal over several cycles.  When we do so,

5. What happens to the non-DC components?
6. What happens to the DC noise component?
7. What happens to the DC signal component?

Finally, you will implement a bare-bones Lock-in amplifier on your own.  (You may work on this as a group).  See if you can get to step three by Thursday.

Download the following zip file or the vi file.

Experiments/SupC/SuperCond Demo Start.vi -or-
Experiments/SupC/SuperCond Demo Start.zip

Start LabView and open the "Supercond Demo Start.vi" program.  It is very similar to what you made in the "hello LabView" class.

1. First, refamiliarize yourself with the program.  The major change is to the waveform graph. Remember that the output of the sine function box is an array of some thousand points. By default, the waveform chart box just graphs those from zero to one thousand (or whatever the size of the array was).  In order to make your graph read in sensible units, you have to tell it where the array starts (x_o) and how much space each array element represents (dx). There is a Bundle box sitting upstream of the graph on the diagram which accomplishes this.  The other change is minor and deals with how many points we generate each time.
2. Your program displays a sine wave as a function of time.  Add a new graph which displays its input as a function of frequency.  (You will need the Amplitude and Phase Spectrum.vi under Analysis/Measurement).
3. Make the units of the graphs right: the time series should be in seconds, the frequency series in hertz.  For the frequency graph, you can use the df output of the Amplitude Spectrum box to scale your graph correctly.

   If you get stuck, peek at the "Supercond Demo Hint.vi."  It has steps two and three.  Don't just copy stuff out of there, though; get a hint and redo it yourself.  The point is to learn LabView, after all.

   Experiments/SupC/SuperCond Demo Hint.vi   - or -
   Experiments/SupC/SuperCond Demo Hint.zip

4. Let's consider this pair of graphs to be your reference signal.  Add another pair of graphs (time and frequency) to display the sum of a sine at 17 Hz, a background of uniform noise, and a sine at 60 Hz. The noise component should be generated with the Analysis/Signal Generation/Uniform White Noise.vi. Make it possible to adjust the amplitude of each component.  The frequency display for the "input signal" should be a flat background with two peaks.  Now your front panel should look like this:

   

   Remember the cardinal rule of LabView: Bugs = 1/Neatness.

5. Multiply your reference and your input (the noisy signal), then take the average (using the Analysis/Probability/Mean.vi).  Scale it by the appropriate factor to recover your signal amplitude. (As a check, set the noise and line amplitudes to zero... you should get back exactly your input signal amplitude.) 
6. You can graph the result (one point for each trip through the while loop) on a Waveform Chart (as opposed to the Waveform Graph we have been using).  Now you should have something like this:

   

The lower right hand graph shows the correct input amplitude, of 2.0V, salvaged from noise of three times greater amplitude.  Notice the 10:1 reduction in noise amplitude. Hooray!