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# Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825 Previous << 1 .. 16 17 18 19 20 21 < 22 > 23 24 25 26 27 28 .. 85 >> Next s'm,-, T + T'l
T _ f"P 1 'r~ŌĆÖ D~!"r 1 -v / I ąĪąōąø
n: imUcTj - ------- ŌĆöŌĆö:---------------------- \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
|ąōčéą╗| = > I (4.72)
- -ŌĆś-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 |ąō^| in Figure 4.10, Tee as a function of the phase is
2|ąō5|ąó7 , , ,
Teei^s) = Pi Y_-jrj|2------------------P1 COS^S ^
This can be rewritten as
Tteivs) = Tm - Ta cos(y>s + ┬ź>-,) (4.74)
where
_ ąōŌĆ× + \TS\2T0 171 Pl l _ |Tsj2 (4-'5)
2 irs| T\
T,j- = (-^ŌĆÖ6)
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 TŌĆ×{<Ps) ls shown in Figure 4.11. If
ips = ŌĆö 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 |ąōąĘ| with 95 = ŌĆö y-,, which corresponds
to the line m in Figure 4.10, Equation (4.54) can be written as
T ą£ąō M - n ^ ~ (a rj)
^ee([lS|J ŌĆö Pi ^ _ |F'J" "
and Illustrated in Figure 4.12.
From Figure 4.12 or Equation (4.54) it is seen that Tee -* ┬▒.70 for JIŌĆÖs'l ŌĆö 1,
because (| 65I2) ŌĆöŌ¢║ 0 as seen from Equation (4.42). Also
lim Tee(|rs|) = PiTa
|ąō5|-ą│ą×
lim ąó<5(!ąō5|) = -PiTj
|i j ŌĆö-čüąŠ
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)
ŌĆö oo < Tf , p < - T ]
4.2. Noise power waves
ŌĆó59
(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 , -ąō.'' > L).
fb) Active source (Hel7/ąø > 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 ŌĆö ŌĆö cp-(.
(a) ReiZi) > D.
(b) Re(Zi) < 0.
4.3. Transformations between sets of noise parameters
61
Halfplane 2: - Tp < Tee < T x ąĄąĄ max
Halfplane 1: - OG < -L ąĄąĄ < T -1 ei max
ąóą░ < Tee < ą×ąĪ
Halfplane 2: T1 -^esmin < < ąó,ąĘ
(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 ąÜ T;3 = 200 ąÜ TTeJ<^ = 225eŌĆ£J,-8K
First Teemm and ąóąĖčéą╗ą░čģ are found from Equations (4.60) and (4.64) to be 475 ąÜ and
ŌĆö 125 ąÜ respectively. Then for Tee > 475 ąÜ and Tes < ŌĆö125 ąÜ 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 + ąō 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~ŌĆś1-3 2.500
-250 ŌĆö 4.500 f~J 1,8 6.021
-350 ŌĆö 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 ąō5. From Equation (4.24) one has
R
Fc ~ Femin + yr- i^s -Ysof|2 (4.78)
Gs
As
2sŌĆÖ - 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 - ąō5|2
1 s i0F| \rsz; + z^PsofZ; + z,p Gs = Uys + Ys) = l_r:ŌĆÖ
2 2 čā Fs Zj + r-sZl + Z[
From this it follows that
7 FfP (4.80)
)x 5 T ^11ŌĆ£
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;ŌĆ×, T\$oF, and Rn. It is, however, convenient to introduce a new noise parameter, Qnc as
n = 4ą»ąĄ[^]ąöą┐ 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 ąŻsofŌĆó
Fr^;~ - 1 + 2 i + </H.ŌĆ×G- + I R-G-,)2 I
V v ........... ŌĆØ J
Rn = ąöą┐ Ysof = \/Gn/Rn + G;f - j B-, Previous << 1 .. 16 17 18 19 20 21 < 22 > 23 24 25 26 27 28 .. 85 >> Next 