# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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s'm,-, T + T'l

T _ f"P 1 'r~’ D~!"r 1 -v / I СГЛ

n: imUcTj - ------- ——:---------------------- \t.uj)

COS ^ 2 17 Sin

Before drawing the circles for constant Tee (or constant F*) it is noteworthy to from Equation (1.58) that the centres all are located on tiie line m given by

. - sin - . ---

m: ImUc"/-! =----------------de;irr! !4./U!

cos

The lines m and n intersect at the point

58

4. Noise parameters

It should also be noted that

|Гтл| = > I (4.72)

- -‘-i

Tee as a function of the source reflection coefficient, I\- = |Fcl exp[7 v?s] is shown in Figure 4.10. When the magnitude of this reflection coefficient is kept constant, which corresponds to following a circle around the origin with radius |Г^| in Figure 4.10, Tee as a function of the phase is

2|Г5|Т7 , , ,

Teei^s) = Pi Y_-jrj|2------------------P1 COS^S ^

This can be rewritten as

Tteivs) = Tm - Ta cos(y>s + ¥>-,) (4.74)

where

_ Г„ + \TS\2T0 171 Pl l _ |Tsj2 (4-'5)

2 irs| T\

T,j- = (-^’6)

It is seen that Tcc(ips) is sinusoidal with a mean value Tm given by Ta and Tp, and an amplitude Ta given by T7 and thus by the magnitude of the cross-correlation

coefficient of the noise waves. An example oi T„{<Ps) ls shown in Figure 4.11. If

ips = — Vi then Tce is minimum for a passive source immittance and maximum for an active source immittance.

When looking at Tee as a function of |Гз| with 95 = — y-,, which corresponds

to the line m in Figure 4.10, Equation (4.54) can be written as

T МГ M - n ^ ~ (a rj)

^ee([lS|J — Pi ^ _ |F'J" "

and Illustrated in Figure 4.12.

From Figure 4.12 or Equation (4.54) it is seen that Tee -* ±.70 for JI’s'l — 1,

because (| 65I2) —► 0 as seen from Equation (4.42). Also

lim Tee(|rs|) = PiTa

|Г5|-гО

lim Т<5(!Г5|) = -PiTj

|i j —-со

Naming the halfplane, which contains Che unit circle, as halfplane 1 and the other as halfnlane 2,

Pi = +1 : Halfplane 1: T^emjn < Tte < oo (In the unit circle)

— oo < Tf , p < - T ]

4.2. Noise power waves

•59

(a)

Figure 4.1i: An example of the extended effective noise temperature Tee(^s) as a function of the phase of the source reflection coefficient.

(a1) Passive source (Re[Zi] > 0 , (Ts! < ^ oc < 0 , -Г.'' > L).

fb) Active source (Hel7/Л > f.) . jP^-l > li or fRel/Ti' <" 0 . <" IV

60

4. Noise parameters

source source

(a)

source source

(b)

Figure 4.12: An example of the extended effective noise temperature X*»( 11'jj) as a function of the magnitude of the source reflection coefficient for ips — — cp-(.

(a) ReiZi) > D.

(b) Re(Zi) < 0.

4.3. Transformations between sets of noise parameters

61

Halfplane 2: - Tp < Tee < T x ее max

Halfplane 1: - OG < -L ее < T -1 ei max

Та < Tee < ОС

Halfplane 2: T1 -^esmin < < Т,з

(In the unit circle)

Example 4.3 In Figure 4.10 the circles for constant Tce in the source reflection plane

are drawn. These are constructed for a transistor with the following T noise parameters:

Ta = 550 К T;3 = 200 К TTeJ<^ = 225e“J,-8K

First Teemm and Титлах are found from Equations (4.60) and (4.64) to be 475 К and

— 125 К respectively. Then for Tee > 475 К and Tes < —125 К the centres and radii of the circles of constant Tee are found from Equations (4.58) and (4.59). Some of the results are shown in Table 4.3. The two lines n and m are found as n goes through the origin and given by

Fmn = ^(Fsot + Г sot' )

= i (0.333 + 3.000) e~J 13 = 1.667 e"jls

and the line m is perpendicular to n in Tmtl.

Tes I'ct Rt

700 0.250 e-j1-8 0.479

.550 0.300 18 0.300

475 0.333 e-J 13 0.000

-125 3.000 e-J13 0.000

-150 'i.O'ilO p~‘1-3 2.500

-250 — 4.500 f~J 1,8 6.021

-350 — l.oOOe--' LS 2.S72

Table 4.3: Centres and radii as functions cf noise temperature.

4.3 Transformations between sets of noise parameters

There are almost an infinite number of ways in which a set of noise parameters ran be defined. So far in this book five sets have been defined. One further set of noise

62

4. Noise parameters

parametexs ms introduced by tiachtold and Strutt [11J and it uses Fsmin and Г5. From Equation (4.24) one has

R

Fc ~ Femin + yr- i^s -Ysof|2 (4.78)

Gs

As

2s’ - Z, 1 - YsZ;

1 s =

Zs -r Z" 1 -f- YsZ^

where Zs is the source impedance and Z\ is the reference impedance for the input port, one gets

1 - rs

Ys

and

1 sZ* -f Z\

IV - V I* - 4(R.e[Z1])2|r50F - Г5|2

1 s i0F| \rsz; + z^PsofZ; + z,p Gs = Uys + Ys) = l_r:’

2 2 у Fs Zj + r-sZl + Z[

From this it follows that

7 FfP (4.80)

)x 5 T ^11“

Equations (4.79), (4.80), and (4.78) lead to

4R e[Zl]Rn [rs-rSOF|2

enm \Tsof z; + Ztf 1 - \TS\3 [ ]

Here Fe is expressed by -Fem;„, T$oF, and Rn. It is, however, convenient to introduce a new noise parameter, Qnc as

n = 4Яе[^]Дп Qnc \TsofZ; + Zi\*

and F~ can be written as

4.3.1 Transformation formulae From Gn, R~, and K7

(4.85)

(4.86)

To Femin, li.n, ana Уsof•

Fr^;~ - 1 + 2 i + </H.„G- + I R-G-,)2 I

V v ........... ” J

Rn = Дп Ysof = \/Gn/Rn + G;f - j B-,

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