# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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3. A < 0 Л В2 - 4AC < 0 There are no reai solutions lor

The centres are located on a straight line, v, whose equation is found from the complex Equation (5.55) by eliminating Mr The result is

5.4. Graphic representation in the reflection plane

97

Figure 5.8: Contours for constant extended noise measure in the source reflection coefficient

5. Noise measure and graphic representations

Figure 5.9: Contours for constant extended noise measure in the source reflection coefficient plane when A < 0 and the .•пггечпмгМтд function of extended noi;v rn? r-tlon”, iho iine

5.4. Graphic representation in the reflection plane

\

99

{P1P2T,(|S31|J + PiP2[\Su^ - |Д5Г1)зш^ + Tj Im[.S'i - 522AJ]}Re[rCAH

+ {PU^flSzi!* + pift[|Su|2 - 1^5!']) cos9-, - Tp Re[5,, - 5;2Д}]} ЫГсл/]

У

= T>(Iin[5n - 5-22A}] cos'^-y 4- - 5о2Л$]sin *?7) (0.60)

It should be noted that this line generally does not go through the origin. Figures

5.8 and 5.9 show two examples of constant extended noise measure circles, one

corresponding to А > 0 and the other corresponding to .4 < 0.

5.5 References

[1] Haus, H. A. к Adler, R. B.: "Circuit, theory of linear noisy networks'*, Technology Press and Wiley, 1059.

[2] Engbcrg, J. к Gawler, G.: "‘Significance of the noise measure for cascaded stages'’. Proc. IEEE Trans, on Circuit Theory, vol. CT-16, pp. 259 - 260, May 1969.

[3] Fukui, H.: ‘"Available power gain, noi.se figure and noise measure of two-ports and their graphical representations”, Proc. IEEE Trans, on Circuit Theory, vol. CT-13. pp. 137

- 14*2, June 1966.

[4] Engberg, .Т.: '"Simultaneous input power match and noise optimization using feedback'1, R69ELS-79, Electronics Laboratory, General Electric., Syracuse, NY. 1969.

6

Noise of embedded networks

Noise parameters of a transistor or another active element are often given in datasheets, but what happens if bias network, feedback elements, or other external elements are added to the original network? This is the subject of this chapter where lumped elements, distributed elements, and combinations of two-ports are analyzed. Then an example is given of a two-port with feedback via two transformers in such a way that the rules analyzed in section 6.1 do not apply. Only three-poles are considered, as most active elements have three poles which are the common reference terminal, the input and the output terminals. Therefore formulae for transformation of noise parameters from common emitter or source to common collector or drain and to common base or gate are given. Finally, some thoughts on computer aided design of linear circuits with noise are expressed.

6.1 Lumped embedding

Consider a three-pole network as in Figure 5.1 to which linear and lumped one-ports such as resistors, capacitors, and inductors are added by a number of successive parallel and series connections. It will be convenient to define "‘embedded'’ and “embedding’’ networks. The original network is called the embedded network (denoted by 1, 2, and 3 in Figure 6.1), and the linear one-ports are called the embedding network. The resulting network is also a three-pole network (denoted by l'; 2', and 3'). A practical way to obtain the new signal parameters of such a network can be explained as follows. Add the first set of parallel admittances to che Y parameters of the embedded network. Transform the resulting Y parameters to Z parameters and add the first set of series impedances. Then transform the resulting Z parameters back again to У parameters and add the second set of parallel elements. Continue this procedure until complete ll.2i.

The noise parameters of the composite network are computed in a similar way.

101

102

6. Noise of embedded networks

ZA2

^-4 2

Zai

Га

Vbi

Embedded three-pole

Zc i

“L

tz

Zb\

Zc 2 З'С’ 2

T

i2'

3'

Figure 6.1: Three-pole embedded in lumped one-ports.

Like signal parameters, noise parameters have several forms as shown in Chapter 4. The most useful forms of this application are the equivalent П and T noise two-ports introduced by Rothe and Dahlke [3] and examined in Chapter 4. The transformation procedure is as follows. First, compute the noise parameters (in П form which corresponds to Y parameters) of the network consisting of the embedded network (with its noise parameters in П form) and the first set of parallel elements. Transform the resulting noise parameters to T form, and combine the noise parameters (in Y form) of the resulting network with those of the first, set of added series elements. Then transform the resulting aoise parameters back to П form and add the effect of the second set of parallel elements, continuing this procedure until complete.

Since the admittance of a parallel element and the impedance of a series element may be zero, it is possible to compute the signal and noise parameters of а таЛЛ\ет general class of embedding networks (leedback, biasing, stray elements etc.).

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