# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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1.1 Johnson Noise

Quickly passing W. Schottky’s 1918 paper [10] on the theory of shot noise, the next major development was Nyquist’s and Johnson's papers [11,12,13] in 1927-28. Here Johnson showed experimentally and N’yquist theoretically the thermal noise from a one-port. An outline of Nyquist's proof is given here.

In Figure 1.1 everything is assumed ideal. The (long) coaxial transmission line is lossless and not radiating, and the switches are lossless and open at the beginning of the experiment. The two resistances have the same surrounding temperature and as they remain in thermodynamic equilibrium the noise power ,V2l transferred from

2 to 1 must equal the noise power .Vis transferred the other way. Then the two switches are closed simultaneously. The two noise powers are perfectly reiiert.ed at the ends of the iine and the energy thus trapped i:i the transmission line will form oscillations at the fundamental mode and its harmonics with voltage node at each

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1. Some milestones in the development of noise theory

R

N12

Л-21

Transmission line of length i and characteristic impedance Zq = R

Figure 1.1: Nyquist’s theoretical model.

end. The frequencies of oscillations are

fi = -77, h='2fu /з = 3/,, ...

i- I

where с is the speed of light and I the length of the transmission line. This length is imagined to be large (I —» 00).

At every frequency there are two degrees of freedom (~ electrical and magnetic energy), and from the (classic) theory of thermodynamics it is known that each degree of freedom has the energy of 7 ■' T where к = 1.38 X 10-2:5 J K-1 is Boltzmann’s constant and T the absolute temperature in kelvins.

Д/ = = determines the number of frequencies between

fm and fn to be 11 - m = (/„ - The energy in the frequency band from

fm to fn is then

[En}f;l = 2- ifcT(n-m) = kT(fn - fm)^ [J]

This, trapped energy must be equal to the noise energy delivered from the two resistances in the time г it takes for the power to be transferred from one end to the other. Therefore

f t? l/n — 9 г,vl- Г7!

where [N\*p is the thermal noise power from one resistor and r = l~. This determines

[N]fo = к T (/„ - U = к T Д/ [W]

In the quantum mechanical theory Nyquist suggested that the energy kT is replaced by h //(exp rX — 1) where h = 6.626 x 10_зл Jo is Planck’s constant and thus the expression for thermal noise in the frequency band Д/ is

N л r = f

i hf , 'bJ •М/ exp - 1

Today most authors agree that the zero-point energy term [\kT) should be included in the quantum mechanical expression of the energy: soo. [14,15,16].

1.2. Receiver noise

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1.2 Receiver noise

In the thirties and early forties noise in receivers was the great subject of interest. It took some time to separate the noise from the source (the antenna) from the receiver noise itself. One early attempt was made by Burgess [17], who introduced a К factor which was dependent only on the source resistance, the resistance of the input network and the equivalent noise resistance of the first valve.

A figure of merit for receiver noise - the noise factor - was introduced by D. 0. North [18] and independently by K. Franz [19]. Two years later II. T. Friis [20] wrote a paper on the noise figure (which today is called noise factor) and a lot of articles emerged discussing the definitions of North and Friis - a rivalry on which Okwit [21] has written an interesting article.

The definitions were expressed a little differently, but they were all on the familiar noise factor. Franz did not call his definition anything, but he clearly used the concept of available power. North also introduced the “operating noise factor” which multiplied by the noise standard temperature, To equals the modern operating noise temperature, Top. Friis’s definition was very stringent. He used available power and available gain and he also derived a formula for the noise factor of networks in cascade.

One more thing that was discussed was the value of the standard noise temperature. Values from 288.39 to 300 К had been proposed - see [21] and [22, pp. 54-55] - until 290 К was chosen as the standard noise temperature by IRE in 1962 [23].

1.3 Linear two- and multi-ports

In 19.55 Rothe and Dahlke [24.25] enlarged the well-known (voltage and/or current based) small-signal parameters (four complex numbers) to include noise by adding four more numbers (two real and one complex'. They also facilitated noise computations by replacing two partly correlated noise sources with two uncorreLated noise sources and a correlation immittance. The four noise quantities are called noise parameters and they exist in many ‘orms. Later, noise power wave based noise parameters were developed by Penfieid, Mevs and others as explained in Chapter 1.

This theory was further developed to a linear noisy network theory in 1959 by Haus and Adler, collected in [26]. They introduced the noise measure axid showed that the mminiuiu noise measure was invariant by embedding in xioiseiess components. Also they introduced the extended noise factor for negative sources. In 1967 Bosma [27] introduced the characteristic noise temperature which can be related to the noise measure. It is, however, seldom used - perhaps because the characteristic noise temperature is negative for ordinary amplifiers.

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