# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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femin = 1 + 2 ^RnGy + RnGn + (RnG-,}~)

Ysof = ^gJrZVgI, - j

and Д„.

The noise parameters as functions of normalized length and with the attenuation constant a as parameter are shown in Figure 6.8(a), (b) and (c). One obvious use of this figure is to choose a length of the transmission line which makes the real part of Ysof = 20 mS. Then the imaginary part can be removed by a stub and the transmission line two-port combination has optimum noise performance for У5 = 20 mS. Another possibility is to choose the imaginary part of Ysof = 0 S and then add a quarter wave transmission line transformer. This also gives a match with optimum noise figure, but, of course, both cases are narrow band matchings.

6.2.1 The equivalent noise two-port of a lossy transmission line

The results above can be used to derive the equivalent noise two-port of a lossy transmission line. Consider two two-ports in cascade. The first is the transmission lino and the second is a lossless line of length zero. The noise parameters of the second two-port are all zero. The noise parameters of the transmission line are then derived from the Equations (6.29) - (6.32) letting Rn, Gn and Y-, be equal to zero. Thus for a lossy transmission line the parameters are

Да = ~ (е2Ы - e~2al"j = ^sinh2a/ (6.45)

Gn

1 ! e + e~i“ - 2

J 1

G n — С r~l — rj

to

^rnt.h2a/ —

Zo V ’ sinh2a/y

— I coth2a-/ - . —- I i 0.40 1

Z0 V. smh 2ai J

0 (6.47i

1 _|_ ,,-20,1 _ .>\

116

6. Noise of embedded networks

From these results it is easy to find Fm;n and Ysof from Equations (4.84) and (4.86):

У sc

1 + 2 1 + 2

RnG~, + \jRnG„ + (R„G7)2

„2 cl

+ e~2al - 2

V 4

I + e--ai - -?Л 4- —

' ' 16 '

-2c*/ __

“J

(6.49)

t + g2<' jB'

Itn

4 (e2al + e~2al - 2) { (e2al + e~2al - 2)2

j . i .. I т . / i J 0

Z^e2al _ е-2с1)2 Z2(e2*1 - в~Ы)2

Zo

(6.50)

It has been shown above that the minimum noise factor is obtained for a source admittance equal to the characteristic admittance for the transmission line and that the noise factor rises exponentially with the loss factor a and with the length I of the transmission line.

6.3 Noise parameters of interconnected two-ports

A network consisting of linear two-ports each with known small signal and noise parameters can often be replaced by one two-port with small signal and noise parameters derived from the individual parameters. The small signal parameters are computed by general circuit theory and for the noise parameters two methods exist.

One of these methods is to use the definition for the extended noise factor for a two-port, Equation (3.6), or for the extended effective noise temperature, Equation (3.4). In these equations the output noise power density is computed from contributions from the noise parameters, the soiuce and perhaps some one-ports. The same result is then computed from the unknown noise parameters of the total ■■ iгг-11iт The two results must be identically equal to each oth^r fur all values of the source immittance, and by chousing four values of this source immittance the four resulting equations may be solved for the new noise parameters.

The other method, introduced by Albinsson [6], consists of two parts. The first part aims at transforming the noise parameters of each individual two-port one at

6.3. Noise parameters of interconnected two-ports

117

a time to the input of the overall two-port, replacing the other individual two-ports with their noise free equivalents. The second part is to combine these individual two-ports to a noise two-port for the resulting network and from this extract the resulting noise parameters.

The first of these two methods is used for the case of two parallel connected two-ports and for two two-ports in cascade. The first part of Albinsson's method is dependent on the circuit configuration, but the second part is very general and will be examined in this section.

6.3.1 Two-ports in parallel

Equations have been derived for two two-ports in parallel. Each two-port is described by the chain or ABCD parameters and by the Д„, Gn and K, noise parameters. If more two-ports are connected in parallel they can be combined by taking two at a time until complete. The parameters from one two-port are all primed and from the other double-primed. The resulting parameters are unprimed.

v:,-

x. GsIjVr

R

Gni^-У

Figure b.t): 'two noisy two-ports in parallel and equivalent, circuit.

ts in Figure 6.У nave the same excess noise factor which with

parameters from the combined circuit is expressed as

1

Gs

(lt„ + Rnu's + i7j‘)

(6-51)

The excess noise factor can also be expressed as the ratio of the noise power density at the output from noise generators belonging to the two two-ports, to the output.

US

6. Noise of embedded networks

noise power density from the source. As it is the ratio of power densities and as they are proportional to the square of the output short circuit current, the excess noise factor can also be expressed as

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