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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 .. 21 22 23 24 25 26 < 27 > 28 29 30 31 32 33 .. 85 >> Next ^ (G‚Äû + Rn\Ys + Y,\2)
1 + To (4'8 + ¬∞-025l10 + 2 + -?12!2) = 2-200
Fe = 1 +
R',1 = 1 +
Re,2 = 1 +
Re,3 = 1 +
1 + ‚Äî (4-8 + 0.024|10 + 5 + j9|2) = 2.2
14
Fe, –∑ = 1 +‚Äî (9.6 + 0.00625| 10 + 8 + jl4|2) = 2.285
fe - 1
Me =
2.200 - 1 Me,I = 1 _ j_ = 1-600
1 4.0
2.214 - 1 Me2 = -------‚Äî = 1.349
* ‚Äú To 9 9–Ø.–ª -- 1
–ú–µ.–∑ = ' - x" = 1.428
1 10
As 0 < Mix < < –ú–µ–ª the best amplifier is the one which consists of amplifier
stages with the numbers 2-3-1.
If each amplifier in front of its input terminal has a network which transforms the 10 mS of the source or the output admittance of the former amplifier to the optimum admittance for minimum noise factor then the order of the amplifiers is changed. This is demonstrated in the following:
Femi n = 1+2 + yj RnOn + (RnG^)2J
= 1 + 2 (0.025 x 2.0 -f \/a025 x 4.8 + (0.025 x 2.0}–õ = 1.80
V ' J
5.3. Graphic representation in the admittance plane
79
Xe,2 min ‚Äî 1 + 2 f 0.024 x 5.0 -+- y0.024 x 4.S -r (0.024 x 5.0)2J ‚Äî 1.96
= 1 + 2 [0.00625 x 3.0 + yj0.00625 x 9.6 + (0.00625 x 8.0)2
= 1.60
1.96 - 1
Mcmln,2 = ‚Äî--------j‚Äî = 1.067
1.60 - 1
Memin, 3 = ‚Äî-‚Äî = 0.66l
1 ‚Äî TT)
As 0 < 3 < Memin,i = –ú–ª—Ç 1–õ.2 then one of the two equal combinations 3 -
1-2 or 3-2-1 gives the best combined amplifier.
The two examined amplifier chains can be compared by computing their overall noise factors or noise temperatures. The results are:
F. 3 - 1 F, i - 1
Ft, 2 31 II +
= 2.214 +
T ‚ñÝL –µ–µ.2 31 = (Fe, 23 1
^,31 2 = Ff.'3 21
Te-,3 1 2 = T..' 32 1
Gbi 2 GenGCt3 2.285 - 1 2.200 - 1
10 10 x 10
= 2.355
1.96 - 1 1.80 - 1
1.60 + --------------- + -------------- = 1.704
10 10 x 10
It is interesting to note that the combination with matching networks generates about half as much noise power as the combination without matching networks.
5.3 Graphic representation in the admittance plane
In order to examine graphically *:he behaviour of the extended noise measure it is necessary to look at the involved parts, the extended noise factor and the exchangeable power gain. The extended noise factor is shown in Appendix –° to be a hyperboloid of two sheets. The exchangeable power gain is examined before the extended noise measure is investigated. Fukui  was the first, to be interested in this subject and the present work is an enlargement of Fukui –∑ work.
80
5. Noise measure and graphic representations
5.3.1 Graphic representation of the exchangeable power gain
The exchangeable power gain for a two-port can be expressed as a function of the two-port‚Äôs –£ parameters and the source admittance as:
Ge ‚Äî
I‚Äôi'i‚Äôir G‚Äôs
Re |(–õ—É + –£^'—è'–ù–£–ø + Ys)'j
where = –£—Ü–£–≥–≥ ‚Äî ^12^21 and for any index the admittance Y = G + j B. Since the extended noise measure equation contains the exchangeable power gain as 1 /Ge the contours for 1/Ge as a function of Ys are found by rewriting Equation (5.12) as follows:
/ IK,, I2 RpTY,‚Äû–ö,, 1 V2
Gs ~
(5.12)
, RefYis^i] ‚Äû
+ ------------ - U11
2 GcG-¬±2 2G22 f Imfy^^i
+ \BS -
\ 2 G22
- Bu
= Rr
(5.13)
where
–ª
2G2
GSG*22 \^4GcG22 2 G‚Äô22
)
and it is seen that when Rq > 0, the contours for constant 1/Ge (and thus constant Ge) are circles in the –£—É-plane. Extrema exist for 1/Ge if Rq = 0, or when
2 G'i 1G22 ‚Äî Rs [i'i -_> –£ 21 ]
i ty^G'^G^ ~ 4GuG22Re[yi2l'2i] ‚Äî (1—Ç[–£12–£21])2^ (5.15)
is real. The latter requires that
*G2nGh - 4C?u<?22Re(vir<xi (–¢–ø^–£–ª])' > 0 (5.16)
This inequality is equivalent to k2 > 1. where –∫ = 1/C (G ~ Linvill‚Äôs factor) is a common stability factor
2G11G22 - Refy^ynj , _ .
= -, —É —É ‚Äù [----- v-1‚Äô)
M 12-^ 211
(Unconditional stability requires –∫ > 1.) When k2 > 1. extrema given by Equation (5.15) exist for
1 –ì.
Ysog = i ‚ÄîV^GjjG^ - 4 GiiG22Re[yi2>'2ij - (Im[–£121 ])2
2 G22
/–¢—Ç[–£12–£21) n ,–æ—á
+ -M‚Äî7TF------------I f0-18-1
5.3. Graphic representation in the admittance p/ane
‚ñÝSI
Here the index SOG stands for Source Optimum with respect to (he exchangeable Gain. It is easy to see that, in the expression for the extrema of 1 /‚ÄôGe, the two extreme values have the same sign (which is the sign of k). The proof is:
\/-iG2uG?l2 - 4G,1Go2Re[yn'V;il - (–¢—Ç[–£12–£–ª1!2
< |2 G‚ÄûC22 - –ò–µ[–£12–£21]|
or - (1—Ç[–£12–£21!)2 < (Re[y12y21])2 q. e. d.
When k2 < 1. R?g > 0 for all values of 1 /–°–µ; so there are no extrema for 1/G‚Äû. which thus is unlimited. The circles for constant 1/Ge given by Equation (5.13) all cross each other at the two points:
\
* -
jy^2il2 (r. Re[y2y,,;
/
(5.19)
It should be noted that 1/GV; is not defined –ì–æ–≥ Gs ‚Äî 0 where the two points are situated. Figure 5.2 shows the contours for 1 jGt as a function ol‚Äô Y\$ (a) with extrema, and (b) without extrema.
1/GS as a function of –£5 is symmetrical around
~,Gs,Bs) = C2GllG¬´ - ^4,0,^^-–î–ø) (5.20)
Ge J \ I'21 I' ^ G22 /
Keeping Bs = Im[li2y2i j / (2 G22) - –í –∏ = –í–≥–∞. which is the Bs value for all the centres of the contour circles, then ^4- (Gs) can be given in terms of the value of Gcg for the centre of a contour circle and the radius, Rq, of that circle can also be expressed in terms of Gcg- The centres satisfy the equation for a straight line: Previous << 1 .. 21 22 23 24 25 26 < 27 > 28 29 30 31 32 33 .. 85 >> Next 