Infinity - John Dunn, Consultant, Ambertec, P.E., P.C.

I usually think I know at least something about this or that blog topic, however blithe that notion may be, but not this time. Instead, I have a question to pose.

Consider some resistance value and call it "R". For that value of resistor, we have an associated noise voltage called Johnson noise of Erms noise = sqrt ( 4kTBR ) where k = Boltzmann's constant, T = temperature in degrees Kelvin, B = bandwidth in Hz and R = resistance in ohms. By comparison however, I have never heard of, nor have I ever read of, anything like Johnson noise being associated with inductance or capacitance. Reactive elements, so far as I have long thought, have always been considered noiseless but I have begun to have my doubts about that.

Please consider an idealized transmission line of infinite length for which the circuit model is an infinitely long cascade of inductance per unit length elements "L" and capacitance per unit length elements "C". 

The characteristic impedance of this model will be Z = sqrt (L/C) where Z will look resistive. This Z will look like an R, because any energy that we send into the cable in the process of measuring the input impedance will go traveling away from the observer and never come back again.

The inductances and the capacitances are all ideal. There are no resistive attributes to either of those reactances. Therefore, neither the L or the C elements should generate any noise, so I have long thought, as they would not have Johnson noise properties.

However, if these lossless and noiseless elements combine in this model to present a resistive input impedance, R, at the one end of that infinitely long transmission line, does that presented R exhibit Johnson noise?

If so where is that Johnson noise coming from given that all of the circuit elements in this model are supposedly lossless and noiseless?

I'm starting to think that L and C elements must make noise after all, but I've never read, or heard, of any such thing. Have you??

Comments invited.

Comments

[George Storm]

As you well know:
A lossless transmission line works as a transformer. Left long enough, the noise properties are exactly those of the transformed impedance.
But of course we can never leave an infinite transmission line long enough for things to settle. In reality, you will only ever see what has happening a finite distance away, which is the effects of the stored energy at the time the line was built. If this energy was thermal, and the line was created rapidly enough that there were no reflections, you would see the noise corresponding to the characteristic impedance of the line (and the temperature at the time at which the reactive components became lossless*).
*Remember, Johnson noise is usually derived based on Boltzmann's stored thermal energy. We could of course buck this one with a lumped transmission line by (for example) initially creating an ideal vacuum-spaced capacitor and then reducing its value by stretching - but that would mean we are adding mechanical energy.

[Mark Oude Alink]

It might have to do with the notion of 'infinity', which may have some mathematical subtleties regarding limits etc. For an infinite transmission line, steady-state is never reached over the whole line, so can we actually still talk about impedances (and associated noise)? You might have to perform a Laplace-analysis here (instead of Fourier), and then you may get the proper results [although I would not know how to handle noise in that case].

Consider any finite-length transmission line: the wave will reflect back at the end and interfere with the wave going forward. In steady-state, for which 'impedance' is defined, the voltage and current waveforms depend on the length of the transmission line.

The only way not to get reflection, is to terminate the line with its characteristic impedance, which in this case would be a resistor. This is then where the noise comes from.

It is absolutely definitely true that an ideal capacitor or inductor does not add noise. Always. The noise of any element is Real(Z(f)).

[George Storm]

True that isolated ideal capacitors and inductors do not add time-dependent noise. But they do have random stored thermal energy (average=kT/2, where T the temperature at the time of isolation from the environment). When you connect them in a transmission line the stored charges/fluxes will start to interact, and you will see this as thermal noise.

It's possibly easier to visualise this for the situation of an ideal gas. At any instant the individual molecules have mean energy of kT/2.N (N the number of degrees of freedom); if you isolate a molecule it simply has constant veocity, etc. Once you allow molecules to interact the energy will change on each collision - the energy is now thermal (or noisy) with a bandwidth depending on the collison rate.

[rajasekhar]

Interesting Question. Some how when we assume idea elements I could see many other case like this only..

1.Imagine RC low pass filter, whose o/p integrated noise is KT/C, means with zero resistance also there is some noise (only with cap). Here we should watch out for KVL,KCL violations because of infinite speed charging.

2. Take an Inductive degenerated LNA, whose I/p impedance is pure resistive again, but here we cant use noise 4KT*real(Zin).

Thanks,
Raj.