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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
Download (direct link): noisetheoryoflinearandnonlinear1995.pdf
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result as above is obtained. Also when |5ц}" — |^5Г < 0 A pip-i — 4*1 or I'S'nJ2 - |As!2 > 0Л P1P2 = -1 the result is the same when the limit ь approached through positive as well .15 negative value».
92
5. Noise measure and graphic representations
One important thing about the location of Г9Г is that it is located just midway between TsoG and Г^од/ which is seen from Equations (5.40) and (5.44):
Г?г = ^ (Гsog + FsOG')
i + i*11 i2 - ifei" - i^si2
(5.51)
215,-, - 522AJ|2
Another characteristic of Г?1. is that it is always located on or outride the unit circle. This is seen from
|Г,г| > 1
|Г„|2 > 1
(1 - |522|2 + |5„|2 - |AS|2)2 > 4 |5*, - 522AJ|2
(1 - |522|2)2 + (|S,i|2 - |As|2)2
f2(l - |522|2)([5,i|2 - |A5|2) > 4(1 - |S22|2)(j5u|2 - |AS|2)
+ 4 |51252i|2 (1 — |й’гз12 - |Sn|2 + |As|2)2 > 4|5,252i|2
k2 >1 q. e. d.
For k2 < 1 it follows that Rq > 0 for all values of Gc. It can be shown that different circles corresponding to different constant Gc values all intersect each other in two points on the unit circle for к2 < 1 and in one point on the unit circle for к2 = 1. In these points Gc is not defined as Ge is not defined on the unit circle. The two (one) points are given by
Г5 = exp
1 1 1 + (■S'li!2 - I‘-^2212 - |As!2Ai
(5.52)
5,', — 522AJ - |5,, - o22^si i
It remains to be shown that the numerical value of the argument to the arccos expression in Equation (5.52) is less than or equal to one.
11 + |Si,!- — |^s
; 2|5г,-522а:]
11 . i.v22;2 + ],<?,,i2 - !Дд|2у2 < -i \s', 5„a:;|2
(1 - !522i2)2 + (i5„|2 - iAsi2)2 + 2(1 - |522|2)(J5'n!2 - |As!2) < 4(1 - |Sm|2)(|S„|2 - |As|s)
1 .t I C n i2
т ** 1^12^211
5.4. Graphic representation in the reflection plane
93
Figure 5.6: Contours for constant exchangeable power gain in the source reflection coefficient plane for лГ > i and the corresponding function of exchangeable pow^r gain alon^ the line </. A third part of the curve (bottom, left) is outside the figure - compare with Figure 5.7.
94
5. Noise measure and graphic representations
Figure 5.7: Contours for constant exchangeable power gain in the source re.lection coefficient plane for k~ < I and the corresponding function of exchangeable power gain along the 'me q
5.4. Graphic representation in the reflection plane
95
(j - 1^22!' - I^Tlj2 + jAsl‘)2 < 4 |5'l25,21 f2
к2 к 1 q. e. d.
The equal sign is valid, when k2 = 1. and that, corresponds to one common point on
the unit circle where all constant G, circles intersect. When k2 < 1 the G, circles
intersect in two points on. the unit circle.
5.4.2 Graphic representation of the extended noise measure
Inserting Fe from Equation (4.55) and Gc from Equation (5.12) into Equation (5.7) the expression for the noise measure is
,, + jrs|2Tj - '2 T^RelTj-e-’^-'])
Me = -J)-------------------------- (°"53)
where
D = To ({|52i|2 — P1P2U - |-?2г!')}
- |Tb'i‘ii J21|" +PtP2(№ir - 1^51“)} + 2 7JiJJ2Re[rj(5"fJ - 522Д5)])
As Me са-n be illustrated as circles for constant Me in the admittance plane
it follows from the theory of analytic functions that Me also can be illustrated as
circles in the source reflection coefficient plane. Equation (5.53) can be written as a set of circles:
|Ts - Гсд./Г = R\j (5.54)
where
г - PiiSaipJy ^ + pip2MeTo(S~1 - S22A5) ^ .--4
C'rl „ iC..I2T.. 1 IJ'T.dC-.II I - n./IC 12 Л (0.00)
Hit -
Pii52i|2rfl + MsTo\\S21\2 + PIP2(\SU[2 - |As!2)i
\P1\S21 \2Ъ e~J-'1 + PiPiM'Totfh - S22l\'s)\
{Pi!52ii2T 'r> + MeT0{\S2i\2 + PiP2(!5n| 1 - iA,s!*))}2
мл„\ J52! Г - PxPii 1 - 15-2-’I2)] - PI! ^ 2 ! j ^
Pi\S'2i\2Tp + ~ PipA'Sh]2 - !A,i!2)
The extrema are found by setting R\t — 0 in Equation (5.561. Doing this ?.iul rearranging the equation the following expression is obtained:
where
|.J22!” ~i“ /•*!/^2 liI 2 i i" ;v?2l!"J j—— i!
96
5, Noise measure and graphic representations
В - -р2'Го{Та(|5ц|“ - |As!2) + Tp(l - I622I2)
+ PiP2|.?2i|J(X - 2>) - 2T7Re[(6,iI - ■S,22AJ)eJV'']} (5.59)
c = -\Sn\\TaTp -T*) (5.60)
From Equation (4.56) it is seen that С < 0 which is important to remember in the following:
1. Л > 0
When A > 0 then - 4 .4 С > 0. This means that there exist two extrema for Mc. It is also seen that \B\ < \/32 - 4 A С from which it follows that the signs of the extrema are different. As /1 is positive it is seen from Equation (5.57) that R\, is negative between the two roots and thus Merntn > 0 and M. < 0. This yields
- В 4- VB2 - 4 A С
min = ----------------------- > 0 (5-61)
- в - Vb2 - 4 л с
iV/e7nax = ---------------:-------------- < 0 (5.62)
I A
2. 4 < 0 Л В2 - 4 AC > 0
Here it is seen that two extrema exist as B2 — 4 AC > 0. As A < 0 and
С < 0 it follows that \B\ > V B2 -4AC and thus the signs for the extrema
are the same. It is also seen that the range for Me is between the extrema.
Therefore
M-
■ В - \/Bl -4 AC
2A
■ В 4- \Г~Вг - 4 A С
24
(5.63)
(5.64)
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