Lab 7: Johnson Noise (~2 Weeks)

• Johnson Noise Lab: The goal of the lab is to make a plot of output noise power times BW vs. R. The slope will be used to determine Boltzmann's constant, k. We will also compare the results with the expected Johnson noise from the resistors and the amplifier.
  • Here is the writeup for the Physics 122 Johnson noise experiment. It gives an overview of the subject and theory. The LabVIEW version of the experiment is similar, but our circuit, VI and procedures are different. See below.
  • Reference for the procedure and circuit we will use:
    • Melissinos and Napolitano (M&N), Experiments in Modern Physics, 2nd. Ed., Sec. 3.6, "Measurements of Johnson Noise."
    • Links near the bottom of the page give leads to the history of Boltzmann's entropy formula and connections with statistical mechanics, information theory and cosmology.
  • Here is a checklist for the writeup for the experiment.Ignore checklist item 10. You will be using matplotlib and python to do your least squares regression measurment and get the errors on the values. A complete writeup in the form of a brief technical report is required. The point scaling for this list can be found here in the grade checklist.
  • Our circuit (available here with additional notes) uses a different low-noise op-amps from the ones indicated in the reference. We use the LT1793 op-amp for the first stage and the AD797 for the second stage. An LF411 is used to make a 2-pole Butterworth low-pass filter (see Horowitz and Hill, Secs. 5.06 and 5.07). Note that the LT1793 has low voltage noise and extremely low current noise specifications. This will allow us to use relatively large resistance values where the thermal noise from the resistor will greatly exceed the amplifier contribution.
  • The Johnson Noise Test Fixture (JNTF): for this experiment, you will use a pre-built circuit for the amplifier and filter for better performance and to save time wiring (the circuit diagram is here). It has +15 V, -15 V and ground connections (wires color-coded to match the binding posts on the circuit test boards), a BNC output jack and a threaded SMA input connector for the calibration signal or for an external resistor. A multiple positiion switch allows you to connect one of the following 0.5% metal film resistors to the input. Note the labels differ from the nominal (and actual) values in some cases! The nominal values are given below.

    Label

    Short

    300

    1K

    3K

    6K

    10K

    30K

    60K

    100K

    500K

    1M

    Ext

    Resistor Nominal Value

    0 Ohms

    301

    1.00 K

    3.16 K

    6.03 K

    10.0 K

    24.3 K

    60.4 K

    100. K

    499 K

    1.00 M

    SMA Input

  • Accurately measured values of resistance are given in a chart linked here.
  • Key points for measurement (see our circuit diagram):
    • Measure and plot g(f), the gain of the amplifier as a function of frequency, using the function generator, voltage divider and accurate wide-band digital multimeter (DMM). It is sufficient to check for midband flatness plus a few points near the corner frequency and beyond rather than to do a detailed measurement of the entire function. Be sure the high frequency corner frequency is as expected (16 kHz), the low frequency is still flat in the range 100 Hz to 1 kHz and that the gain is down by an order of magnitude by the Nyquist critical frequency (100 kHz). Use the provided approx. 1000:1 voltage divider (in a shielded box) between the signal generator output and the JNTF external input. Use shielded cables for connections to prevent pickup of interference. Measure the voltage divider voltage ratio and note the accuracy of your measurement. Check that the midband gain G agrees with the stated value for your JNTF in the chart available in the lab.
    • Connect the circuit output to the NI ADC using a nonreferenced single-ended (NRSE) input with voltage range from -0.5 V to 0.5 V (use analog input 0)..
    • The power spectrum will be measured using the VI on SmartSite (Scope_Spect_mod_nw_mod.vi). It also displays a histogram and waveform plots of the sampled noise waveform. The number of points should be 8192 and the sampling frequency should be 200000. The spectrum should extend from 0-100 kHz in bins 0-4096
    • Start with the100 kilohm 0.5% metal film resistor as the noise source. The spectrum should have the same shape as your amplifier gain vs. frequency curve if the signal is dominated by Johnson noise (either from the amplifier or the resistor). There may be some additional pickup of signals from 60 Hz AC (and harmonics). These should show up as peaks at the low end. But there should be a region which is fairly flat within the amplifier bandwidth (perhaps from approx. 1 kHz to 8 kHz – calculate which channels correspond to this frequency interval). If there is excessive pickup, check the wiring.
      • The sum of the squares of the amplitudes of the FFT components within these limits (f₁ to f₂) will correspond to a measurement of <V²>, the mean squared output voltage in that bandwidth. For an explanation, refer to these notes.
      • For each resistor value, you will need to sum the channels of interest for a single measurement. Repeat at least N=10 times to get a more accurate mean value for <V²> and an estimate of its standard deviation, sigma. The error in the mean value of <V²> will be sigma/(square root of N).
      • Repeat for other resistor values available in the JNTF.
      • Find and plot <V²> vs. R with errors and fit to a straight line (two parameter fit). (See example in SmartSite wiki on linear least squares fit with errors.)
        • This linear fit corresponds to the function <V²> = (e_A² + 4kTR)G² Δf , where
          • e_A² represents the noise contribution from the amplifier (amplifier current noise component should be negligible with LT1793)
          • G² is the average of the square of the gain of the amplifier g(f) over the frequency range used. If you use the range 1 kHz - 8 kHz, this should be reasonably constant.
          • Δf=(f₂-f₁) where we are taking the discrete form of this integral, g²(f) df.
          • Be sure to record the ambient temperature T from the thermometer in the lab. Assume a +/-.1C on the measurment that we took in class. The measurements were either 24.5 or 23.0 depending on the day you were there.
          • The parameters e_A² and Boltzmann's constant k can be found from the fitted parameters. Find the errors in these quantities from the fit covariance matrix (using propagation of errors as appropriate).
          • Compare with the accepted value of Boltzmann's constant.
          • Compare e_A with what you expect from the op-amp specifications in a non-inverting amplifier configuration (see Horowitz and Hill, p. 447).
  • Reference for the procedure and circuit: Melissinos and Napolitano (M&N), Experiments in Modern Physics, 2nd. Ed., Sec. 3.6, "Measurements of Johnson Noise."