## Sunday, December 13, 2009

### Noise II

One type of electronic noise that is inescapable is Johnson-Nyquist noise.  Roughly a hundred years ago, physicists studying electricity noticed that sensitive measurements showed noise (apparently random time-varying fluctuations in the voltage or current).  They found that the power spectral density of (for example) the voltage noise, SV, was larger when the system in question was more resistive, and that higher temperatures seemed to make the problem even worse.  Bert Johnson, then at Bell Labs, did a very careful study of this phenomenon in 1927, and showed that this noise appeared to result from statistical fluctuations in the electron "gas".  This allowed him to do systematic measurements (with different resistances at fixed temperature, and a fixed resistance at varying temperatures) and determine Boltzmann's constant (though he ends up off by ~ 10% or so).  Read the original paper if you want to get a good look at how a careful experimentalist worked eighty years ago.

Very shortly thereafter, Harry Nyquist came up with a very elegant explanation for the precise magnitude of the noise.  Imagine a resistor, and think of the electrons in that resistor as a gas at some temperature, T.  All the time the electrons are bopping around; at one instant there might be an excess of electrons at one end of the resistor, while later there might be a deficit.  This all averages out, since the resistor is overall neutral, but in an open circuit configuration these fluctuations would lead to a fluctuating voltage across the resistor.  Nyquist said, imagine a 1d electromagnetic cavity (transmission line), terminated at each end by such a resistor.  If the whole system is in thermal equilibrium, we can figure out the energy content of the modes (of various frequencies) of the cavity - it's the black body radiation problem that we know how to solve.  Now, any energy in the cavity must come from these fluctuations in the resistors.  On the other hand, since the whole system is in steady state and no energy is building up anywhere, the energy in the cavity is also being absorbed by the resistors.  This is an example of what we now call the fluctuation-dissipation theorem:  the fluctuations (open-circuit voltage or short-circuit current) in the circuit are proportional to how dissipative the circuit is (the resistance).  Nyquist ran the numbers and found the result we now take for granted.  For open-circuit voltage fluctuations, SV = 4 kBTR V2/Hz, independent of frequency (ignoring quantum effects).  For short-circuit current fluctuations, SI = 4 kBT / R A2/Hz.

Johnson-Nyquist noise is an unavoidable consequence of thermodynamic equilibrium.  It's a reason many people cool their amplifiers or measurement electronics.  It can also be useful.  Noise thermometry (here, for example) has become an excellent way of measuring the electronic temperature in many experiments.

CarlBrannen said...

I had the fluctuation dissipation theorem in a graduate statistical mechanics (sort of) class taught by Joe Weber.

He assigned us the almost impossible homework problem of showing that the energy absorbed by a short dipole antenna in a black body region with temperature T, matched that emitted by a 50 ohm resistor (at temperature T). They're connected by a 50 ohm coaxial cable. Thus the power out of and into the antenna matched and no violation of the laws of thermodynamics ensues.

Then in class, he solved it in three lines by happening to know the total cross section of an antenna is always 4 pi or something like that.

Anonymous said...

Shouldn't it be a 377 Ohm resistor?
http://en.wikipedia.org/wiki/Impedance_of_free_space

CarlBrannen said...

Anonymous, I'm guessing that Weber, who had a PhD in electrical engineering, knew his stuff. (I certainly don't know anything about antennas.) I remember 50 ohms, but maybe it was 75 ohms as that is what wikipedia is using to illustrate a short dipole antenna.

His calculation invovled assuming that the antenna cross section was equal to lambda^2 / 4 pi, after integration over all directions, a fact which is well known to EEs. Lambda is wavelength, see wiki article on antenna aperture.

The students were stuck on computing aperture from integrating over the antenna pattern, all done from first principles.

fashion accessories said...

As we know that according to fluctuation dissipation theorem, the response of a system in thermodynamic equilibrium to a small applied force is the same as its response to a spontaneous fluctuation. So, the linear response relaxation of a system from a prepared non-equilibrium state to its statistical fluctuation properties in equilibrium. Often the linear response takes the form of one or more exponential decays.