# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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1 _ 2 C22 2G11G22 ~ Re[y12y2i]

/-< — \y !2 + ,y |2 (5.21)

Il21| I '211

and the radii are expressed as foilows:

Ro =

r,2 , !i iJ (r RefVnlj,] ,

(jcg + —TT^i---------I Gu----------—-------) fo.22)

• G'22 V 2<

Figure 0-3 show's three cases of -U/7,)l

u............. ............ ~ '\BS-Bcr,

illustrates the behaviour of ^-(У3).

5.3.2 Graphic representation of the extended noise measure

The extended noise measure wa,s defined in Equation. (5.7). Inserting Fe from V,-quation (1.11) and Ge from Equation ‘ 5.12) into Equation (5.7) leads to

|V2ii2(G, + R.n[Ys + r,|s)

M- -

|y>i|2Gs - Re[(I’u + Уч'П'Лу 4- V22ys)]

82

5. Noise measure and graphic representations

(a) 4Ol.Gl2 - 4GilG22He[V'12Y2i] - (1гп[У12*л])2 > 0

(b) AG\^G\t - 4GuG.2,Refy12y21] - (1т[У12У21))2 < 0 Figure 5.2- Contours for constant I !G« -i-s :i function of (a) with extrema, crnd (b) without

extrema.

5.3. Graphic representation in the admittance plane

S3

Condition (a) from Figure 5.2 and 2G\\G22 - > 0

Condition (a) from Figure 5.2 and 2 G11G22 — ^[^12^21] < 0

Condition (b) from Figure 5 2 and 2G\-\G-22 ~~ HeiVw^oij <-' П Figure 5.3: l/Ge as a function of Gs for Bs = Гт(У'хoi 21 ]/(2 G22) ~~ B\i and G22 > 0.

84

5. Noise measure and graphic representations

Extrema of Ms(Ys) can be found bv setting the partial derivatives with respect to Gs and Bs equal to zero. This yields two equations of fourth order in Gs and Bs with no explicit analytical solutions. As the expressions for both F, and 1/G, are represented by circles, the theory of linear transformations of analytic functions shows that Equation (5.23) must also represent circles and can be written as

(Gs - Gcm)2 + (Bs - ВсмУ = R\i (5.24)

-1

\Yn I'2 Rn + M,G22 x {\Y21\2G-,Rn - iiVe|y2i|2

+M,GnG22 — ^ Л/гЯ-е[У[2121]J (5.25)

-1

|У21|2ДП + MtG22

x l\Yn\2ByRn + M.B^Gn - \ MeIm[y12r2ill (5.26)

L ^ J

r2 = WnV

M (\Y21\2Rn + MM22)2

x У1212 + l^2i|2 — 4G11G22 + 2Re[yi2y2i])

+ Me (д„Де[У12У21(Уц - У7)*] - G22Rn\Yix - У7|2

-G22Gn - \Y21\2GyRn)

-|Уц| гСпЕп\ (5 27)

j

Equation (5.24) is valid when R\t > 0. By examining Equations (5.24) - (5.27)

and by setting the expression in curly brackets m Equation (5,27) equal to A M} +

В Me + С where С < 0, the following conclusions can be drawn:

!. (Gcm(A[,), ВсмШ*)) 's °n a straight line in the У5 plane:

1

Bf-■ M it/ , ) , n 11- 1 , f , ,

) i 211" 1 12* 21 j ~ ^'22\^7 ~ ^11 I

where

Gcm =

В cm =

x|(2Gj2(B7 - B\i) 4- Itn(yi2V2i])Gcm + С-Дп^У^У^]

+ 2 С221т[У1'У-,] - В, (|У21|2 + Re[y12y2i])} (5.28)

5.4. Graphic representation in the reflection plane

85

2. From Equation (5.25), or the expression:

ЩСсм) = ~\Y21\2Rn + A-\Yn\2Rn

(.722 ^ 22

I ^ 21 12 ~ '2G11G22 + R-e[^12^3l] + 2GyG22 - .

л l^il2 - 2GuG22 + RellVi^i] - '2 GCmG22 '

it is seen that Ms(Gcm) >s monotonic with one pole.

3. Since С < 0, then for .4 > 0 or

:^n|2 + ibil2 — 4G11G22 + ^RefKuVoi] > 0 (5.30)

extrema will exist for Me. The signs for the extrema are different because

— 4 AC > 0 and thus |i?j < V B2 — 4 AG. Note that the condition of

Equation (5.30) is the same as that for R2G > 0 when 1/G. = 1.

4. When A < 0, j-B[ > \/ В - - 4 A С whenever B2 — 4 .4 С > 0, so that

the signs for the extrema are the same.

5. Calling the extrema jWel and M„2 where M, \ > Me2, the range for M5 is

M. > Me 1 > 0 and Me < Me j < 0 for A > 0

Me 2 < Me < Mel where Sgn[iW*i] = sgn[Me 2] for .4 < 0

Figures 5.4 and 5.5 indicate the behaviour of the noise neasure. The extreme

values are found from Equation (5.27) by setting R\, = 0. and the correspondins

optimum source admittance, Ysom, from Equations (5.25) and (5.26).

5.4 Graphic representation in the reflection plane

As in the former section the exchangeable gain and the extended noise measure can be illustrated as functions of the source reflection coefficient. As these quantities are all traced as circles in t.he source admittance plane it. follows from the theory of linear transformations of analytic functions that they should also be generalized circles (a circle, a straight line, or a point) in the plane of the source reflection coefficient. This section examines the behaviour of Gs and Me as functions of the source reflection coefficient.

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5. Noise measure and graphic representations

Figure 5.4: Contours for constant extended noise measure in the source admitt.anced plane for A > 0 and t,ho corresponding function of r-xtpnded nois^ measure v^rqns sminv COQuUClanCc ’Л'ПбП Us — iJrj\f ■

5.4. Graphic representation in the reflection plane

87

Figure 5.5: Contours for constant extended noise measure in ^he source admittariced conductance when Bs — Всм■

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.5, Noise measure and graphic representations

5.4.1 Graphic representation of the exchangeable power gain

The exchangeable power gain is defined as the ratio of the exchangeable power at the output to the exchangeable power from the source by Equation (3.2) where

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