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April 30, 2015


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.


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).


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