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Thermal White Noise - Johnson–Nyquist noise

  1. Feb 4, 2014 #1
    I'm trying to measure the thermal white noise generated by chemical batteries.
    So far I've measured the current noise, the voltage noise (V noise) and the bandwidth (delta v).
    From the equation below, I'm trying to solve the equation for the frequency.
    The problem is that there's an exponential expression in it. But it gets worse, the frequency f is in it.


    All I could come up with is to put hf/kt on one side of the equation and I'm stuck with ln (4hfr delta v / V^2). Do you have a solution?
  2. jcsd
  3. Feb 4, 2014 #2
    Looks like you are using the wrong formula for you situation.

    The formulas for thermal noise that we typically use in EE are derived from classical thermodynamics which leads to power spectral density being independent of frequency. Of course this implies that there is an infinite amount of thermal noise power in every resistor when we include all frequencies up to infinity (look up "ultraviolet catastrophe").

    The resolution to this paradox (thanks to Planck) is that when frequencies get very high, classical thermodynamics breaks down because the quantization of energy (E=hv) becomes significant.

    Your formula is a general formula that includes quantum effects, and thus shows a dependency on frequency.
    Last edited: Feb 4, 2014
  4. Feb 4, 2014 #3
    I think you are wrong. The equation calculates the frequency with respect to (1) the current running through an internal resistance, and (2) the bandwidth of the white noise. What you end up with is simply a complex frequency, which is solved using the Lambert W equation. Might as well throw in the towel and call it a simple frequency.
    Last edited by a moderator: Feb 5, 2014
  5. Feb 4, 2014 #4
    And by the way, for the record, Johnson–Nyquist noise is classical thermodynamics.
    Last edited by a moderator: Feb 5, 2014
  6. Feb 4, 2014 #5
    You can not, I repeat, you can not exclude the frequency when calculating white noise. You don't know noise is measured in Hertz?
    Last edited by a moderator: Feb 5, 2014
  7. Feb 4, 2014 #6
    Last edited by a moderator: Feb 5, 2014
  8. Feb 4, 2014 #7


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    noise ( thermal) is measured in K ( Kelvin) frequency is measured in Hz ( Hertz )

    Last edited by a moderator: Feb 5, 2014
  9. Feb 4, 2014 #8


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    Color me confused. How is noise measured in Hertz?

    I thought noise was expressed in power or voltage or current. The amount of power may depend on the bandwidth ( so noise voltage is sometime expressed as volts/root-hertz, etc)

    At very high frequencies you can determine the power spectral density by using hf. Do a unit analysis of hf and you get volts, or eV.
  10. Feb 4, 2014 #9


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    uh huh

    when I am dealing with thermal noise in low noise RF preamplifiers etc its always measured in K

  11. Feb 5, 2014 #10


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    Yes, in his paper Nyquist used classical thermodynamic arguments including the classical equipartition law (along with transmission line theory) to derive
    v_n^2 = 4 R k_B T \delta \nu
    in order to explain the experimental results published by Johnson. This is the classical result. Notice that it is not the equation you posted. Yours has Planck's constant in it, which does not show up in classical physics. Nyquist does state the equation you use at the end of his paper to indicate what happens at very high frequencies and/or low temperatures, when the classical equipartition law no longer holds and quantum effects must be included. Of course the classical result is all that was needed to explain Johnson's data. If you take the limit as [itex]h \rightarrow 0[/itex] (or equivalently assume [itex]h f << k_B T[/itex]) in your expression then you recover the classical result.

    May I ask how you are doing this measurement? Are you really in a regime where it is not true that [itex]h f << k_B T[/itex]? What is the bandwidth of your measurement? etc. If you really want help (as opposed to simply looking to "in your face" people) then we need more information. For example, usually we know [itex]f[/itex] and [itex]\delta \nu[/itex] from the measurement setup, so it isn't clear why you want to solve for [itex]f[/itex] at all ...

    Last edited by a moderator: Feb 5, 2014
  12. Feb 5, 2014 #11


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    I don't doubt it, but technically speaking I think you are referring to Noise Temperature, which of course has units of temperature. We do similar things at my workplace, where we might say, "that amp has 2 dB of noise" when we are actually referring to the noise figure. In any case, the noise performance of amps is one step beyond the noise from a resistor, which I think is all the OP is concerned with.

  13. Feb 5, 2014 #12
    I am speaking about the frequency disturbances caused by thermal radiation. As far as I understand, the frequency generated from current passing the internal resistor in a battery can be found in the unit for frequency in the equation in the link below:

  14. Feb 5, 2014 #13
    Thank you for your reply. I will be as clear as I possible can.
    Please go to the following website regarding 1995 IEEE international frequency control symposium:


    In general, it is accepted that a common DC battery only supplies a DC voltage.
    However, a small frequency somewhere arround 20 - 1000 Hz is generated, due to heat generated in the battery. I call it thermal noise, forget I ever said white noise. You can calculate the current noise with the instrument shown on page 370 (page 4 in the pdf). You can then use the current noise to calculate the voltage noise. Subsequently you use the voltage noise and the measured temperature of the battery, which would be between 300 - 323 Kelvin, to find the bandwidth of the frequency/noise. Finally, you can add this data into the equation in the link below, as shown on page 1:


    But you see, there's a problem, you've got exp / e in the equation. By simplifying the equation, you can get y = ln (x*y -1) / w. You can solve this equation with respect to y, which represents the frequency. You end up with an expression including a product log:


    But the problem doesn't stop here. Now you're stuck with a complex number.
    But really, this is a representation of a complex frequency.

    Concept of complex FrequencyDefinition: A type of frequency that depends on two parameters ; one is the ” σ” which controls the magnitude of the signal and the other is “w”, which controls the rotation of the signal ; is known as “complex frequency”.

    Makes sense, right? A complex number is just a two dimensional number:

    So I think I just need to know, is the frequency in the equation, the frequency which you'd see on an oscillator, generated from the heat from the battery?

    Sincerely, Michael
    Last edited: Feb 5, 2014
  15. Feb 5, 2014 #14
    You are using the wrong equation.

    You are using equation (1) of the NIST document.
    You should be using equation (2) "At low frequencies Johnson noise can be approximated by...", which is the same equation that jasonRF has posted.

    You are dealing with kHz right, not GHz or THz.
    Last edited: Feb 5, 2014
  16. Feb 5, 2014 #15


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    I read the paper. The noise is essentially white except for the lowest frequencies, as indicated by their data and by their statements. They do not calculate the bandwidth from the voltage noise and the temp; instead, the resolution of their spectrum analyzer defines the bandwidth. In fact, "measurement bandwidth" is always what the bandwidth number in Nyquists formula means. This is even stated in the beginning of the paper you link. Note that their voltage noise measurements are presented in spectral density = Volts / sqrt(Hz). To place their measurements in this form they had to know the bandwidth already (after all, they set up the spectrum analyzer, right?).

    You are still mistaken - that equation describes the frequency dependence of thermal noise. Noise is generated at all frequencies! In any case, for parameters relevant to your case: T = 300K and f = 1000 Hz,
    \frac{h f}{k_B T} \approx 10^{-10}
    so surely the classical result is all that matters. Just look at the data in the paper. It is flat with frequency except at the lowest frequencies where other processes are dominating. It is not frequency dependent (edit: it is not frequency depended in the portion of frequency space you seem to care about).

  17. Feb 5, 2014 #16
    Exactly, the equation describes the frequency dependence of thermal noise.
    The noise you put in the equation is a specific one. The frequency I want to calculate using the equation, is the same frequency I measure using an oscilloscope. Please stop saying it's not frequency dependent when the frequency is in the equation, and the frequency is what I want to calculate. In other words, it's a DC voltage signal I'm looking for, with oscillations.

    I've added a graph:


    Here's why it's so important to me:


    The small wave represents the wave/frequency generated from heat from the battery, and the larger wave is the desired wave which I generate in a circuit. The DC voltage with oscillations is introduced into the circuit, and I believe with a feedback/loop, thing's just get worse.
    Last edited: Feb 5, 2014
  18. Feb 5, 2014 #17

    For the third time...you are using the wrong forumla. Use equation (2) in NIST document, not equation (1).

    We don't need to invoke Planck's constant when working with signals in the kHz range.
  19. Feb 5, 2014 #18
    "They do not calculate the bandwidth from the voltage noise and the temp".
    I think you need to read the paper again. They calculate the Voltage noise with an equation which consists of the bandwidth and the temperature. You can rearrange this equation and solve it with respect to frequency. Please look at the following image in the link below!!!


    This frequency F, is this the frequency I would see on an oscillator? Is this the frequency generated from heat from the generator?
  20. Feb 5, 2014 #19
    I think I understand what you're trying to say, but you don't want me to use the equation 1, just because an approximation would be good enough, which would exclude a very important data.
  21. Feb 5, 2014 #20
    Which Boltzmann's constant is being used? Is it 8.617 3324×10−5 eV/K?
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