Thermal fluctuations in complex networks

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Abstract
This paper is concerned with the thermal fluctuations generated in networks containing reactance as well as resistance. Earlier work is reviewed, and it emerges that Moullin and Ellis were unable to interpret Nyquist's equation for a network at uniform temperature by their method of representation. An alternative method is suggested in which each element of resistance is replaced by a fluctuation generator in series with the element of resistance. The fluctuation voltage appearing between any two points in a linear network with any temperature distribution is then evaluated. This general expression is shown to be in agreement with Nyquist's result if the temperature is uniform. It is also shown that the relevant values of resistance in these formulae are the equivalent power-loss resistances and not the metallic resistances.There follows a description of experimental tests which are regarded as satisfactory verification of the method of representation adopted and of the values of resistance used.The noise/signal ratio existing between any two points in the generalized network is then discussed, and is shown to be incapable of exceeding the value obtaining in the arm in which the signal is introduced. General recommendations for optimum noise/signal ratio are given.A few special cases are then considered. It is shown that in radio reception a resonant circuit is preferable to a bandpass filter as a coupling between aerial and amplifier. In the former case an analysis of the optimum coupling between aerial and circuit is given which takes account of the fluctuations generated in the amplifier itself. A simple practical method of obtaining the optimum conditions is outlined.Fluctuations in long cables are briefly discussed. It is shown that temperature conditions in the vicinity of the receiving end only need be considered. The use of transformers or resonant circuits to couple the cable to an amplifier is governed by the same recommendations as for aerialamplifier coupling.

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