# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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N' -= kT = 1.38 x 10"23 x 290 = 4.00 x 10-21 WHz'1

’'The term immittance is used when it is not necessary fo distinguish between impedancc and admittance.

‘Л — 6.626 0755 x 10“'4 ± 0.60 ppm according to Handbook of Chemistry ind Phvsics. 70th ed., 1989.

Jк = 1,380 658 X ± S.5 ppm according to Handbook of Chemistry a.nd Physics, 70th ed..

1989.

9

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2. Noise in one-ports

1.00 0.7.3 0.50 0.25 0.00

106 109 1012 1015

Figure 2.1: Reduction of thermal noise as a function of frequency for temperatures uf 295, 77 and 4 К (room temperature and boiling points of nitrogen and helium).

~ -'204 dB|^yjj -i. Due to the "nice" numerical value of iV' the corresponding temperature, which is close to normal room temperature, is called the standard noise temperature [3,4] and denoted To.

To = 290 К (2.3)

It may also be noted that kT0/q = 0.0250 V, where q =1.602xl0-19 С is the magnitude of the electronic charge.

Please note that N is used to denote noise power [W] and N' is used lor noise power density [W Hz-1].

The noise in a one-port can be represented by either a Thevenin voltage source or a Norton current source as shown in Figure 2.2. Loading these equivalent circuits

Figure 2.2: One-port, equivalent circuits.

with the conjugate of their imiiiittances to obtain power match tile source delivers the available power to the load. If 1:: source generates only thermal noise this

:>u\ver is ««pressed as

.v = kTAf = = [W]

where ('•••) denotes the ensemble average over processes with identical statistical

2.2. Definitions of noise quantities

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properties.11 ( tej'J) and { jij2) are the magnitude squares of the voltage and current measured in the frequency band Д/ with a “true” RMS-meter and R and G are defined in Figure 2.2. From this expression the mean square noise voltage is given by

(M2) = 4 к T R Д/ [V-] (2.4)

and the mean square noise current by

(|г|2> = 4^-ГСД/ [A2] (2.5)

2.2 Definitions of noise quantities

In noise analysis it is (|e|2) (and (|ij2)) which is of interest in noise calculations, as (e) is equal to zero. As (|e|”) is dependent on the bandwidth Д/ this must also be specified. Sometimes the quantities (|е|2)/Д/ [V2Hz_1] or \f\ j e |2) / Д / [V Hz-?] are used5 but mostly one of the following representations is preferred.

Since noise may have other origins than thermal effects and since it is convenient to have equivalent representations for all types of noise, Equations (2.4) and (2.5) are modified in such a way that they are valid for all kinds of noise. This can be done in two ways. One possibility is to keep the temperature T fixed and then change the values of the resistance R and of the conductance G until the equations are fulfilled. The values for R and G obtained in this way are called the equivalent noise resistance Rn and the equivalent noise conductance Gn respectively. The other possibility is to keep the resistance and the conductance at their physical values and then select the temperature T such that the equations are fulfilled. This change of temperature could also be performed in Equation (2.2). The thus obtained temperature is called the noise temperature of the one-port.

In order to be able to characterize one-ports with negative real parts of their impedance or admittance the available power according to custom is replaced by the exchangeable power1’ as introduced by Hans and Adler [5]. Thus the thermal available noise power is replaced by the exchangeable noise power. ;Ve. in Equation (2.2).

2.2.1 The equivalent, noise resistance

Definition 2.1 The equivalent noise resistance of a one-port /?,, is defined as

/ил 4 к io Д/

4See Appendix A.

5The latter is mostly used in dB relative to 1

12

2. Noise in one-ports

where (je|2) [V2] is the mean square (open circuited) noise voltage

at the terminals of the one-port in the frequency band Д/ [Hz], fc =

1.3807 x 10-23 JK-1 is Boltzmann’s constant and TQ = 290 К is the standard noise temperature.

It is seen that the chosen fixed temperature in Equation (2.4) is T0 regardless of the physical temperature of the one-port. A change in the mean square noise voltage with temperature determines the variation of Rn with temperature. Other noise contributions than the thermal noise also change (increase) the equivalent noise resistance. A variation of Rn with frequency / occurs quite often, and Л/ should be chosen so narrow that (|е|2)/Д/ is independent of the value of Д/7 The quantity (|е|2)/Д/ can be regarded as the mean square noise voltage density. It is seen that R„ > 0 even if the one-port lias a negative real part of its resistance.

A one-port characterized by Д„ generates as much noise as a metallic conductor (which only generates thermal noise) with the resistance Rn at standard noise temperature. Rn does not give any information about the ohmic resistance of the one-port. The equivalent diagram of the one-port consists of a series connection of its impedance, a noise voltage generator, the stochastic noise voltage which is determined by

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