Books in black and white
 Books Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics

# 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 .. 23 24 25 26 27 28 < 29 > 30 31 32 33 34 35 .. 85 >> Next Here again
P =0 ~~ ^22^LI ~ , I n 12\ Ct m
P?0 P2 ^ ^ j^,, j2 \\^2\ ) (o.31)
P'S = Pi iT-^ji (1ą┤ą┤12) (5.ąŁ2)
lit
1ąÆąö1
where i refers to the port number;
_ 5i253ir5
"22 ŌĆ£ i - sŌĆ×r5
is the output reflection coefficient of the two-port, where ąō5 is the source reflection coefficient; and
<Ōä¢i2) _ \S:
2ii
(2
(|┬Żs!2) |1 - 5ŌĆ×rsi2|l - 5'2rL|2
where Ti is the load reflection coefficient.
These equations give the following expression for the exchangeable power gain:
r = ____________________PlP2\S2l\2Q - lr.s|2)______________________ . ,
1 - i^l2 + |rs|=(|5n|J - |As|2) - 2Re[rs(5i, - S^s)1
As in the former section the contours for constant 1 j(Je (and thus constant Gc) are found to be circles, as Equation (5.33) can be rewritten as:
5.4. Graphic representation in the refection plane 89
Rl =
l^n - 522A5I2 + (j5n|2 - jA5j2 + PlPtl^l|2^)(PlP2|.g2tPg: - I + I 5>212) (|5ll!2 ŌĆö i^Sp + P\P2\^2\]? J77)2
(5.35)
Equation (5.34) shows that for l/Gc constant the contours for 1/G'e are circles with radii equal to the square root of the expression in Equation (5.35) and with centres, Tec, given by
|S\il2 - |ąöčī'!2 + Pi?2i52i|2
(5.36)
Extrema for 1 jGe are found for = 0. The numerator in Equation (5.35) can be written as
I Sul21 Sul" + P1P2IS21I2 [|┬Żn|* + I -^'2212 - 1 - l^sl"] čā;ŌĆöh !SŌĆÖ2i (4 ypy = 0
When using the stability factor k, which here is
1 + (Ad2 - !5U!2 - IS,2!2
2 ] ąŠ 12 5 21
the extrema for 1/G> are given by
1 li'd
(5.37)
(pipik ┬▒ Vk2 - l) (5.38)
Gt |S2J!
It is seen that extrema only exist when kl > 1. For fc2 < 1 ljGe is unlimited as all the circles pass through one point (for k2 = 1) or two points (for ą║2 < 1) on the unit circle.
For .fc2 > 1 it follows that
1 |.?H ( . /71------\
mm |ŌĆö j = 77;- = 7^. (pi+ vk- - 1 1 (5.39)
L';ej 'ŌĆó-'ŌĆójmu.r i^21| 4
which gives Tsog (for source optimum gain) by inserting Equation (5.39) into Equation (5.35):
2(5-, - 52;ą╗;)
1 + !5U[2 - |522i2 - jAi-p - 2pi/)?!5I252. !\/a
(5.40)
1 + !5ui2 - |522i - JAvj- + 2 mPilSuSziWk* -
2 j 5.*! - 5-A;.!2
x (1 - 522AJ)
and that
11 1 i 5 j 91 /
UICLA
iGTi = = v/;2-l) (5.42,
90
5. Noise measure a ad graphic representations
which gives Tsoc? by inserting Equation (5.42) into Equation (5.36):
p _ __________________2(6U ŌĆö 3j-2As)_____________________ ,r. iąŠ.
iGy 1 + l^il2 - Ud2 - |AsP + 2plŌä¢!5ŌĆÖ12521lv'P^T [0ŌĆś !
1 + !.S'l 1!2 - |522|2 - |ąöą┤;2 - 2Pm\Sl2S2l\^r^
2 157! - 5'22Aj!2 x (5*! - Sn&s) (5.44)
From Equations (5.39) and (5.42) it is seen that
Gemax 5; G^ jam (-3.4.))
This inequality is valid whatever the sign of pip2 or k. It is also easy to see that
the sign of G,rnax and ąĪąĄčéčł is the same. Thus the signs shown in Table 5.4 are
obtained.
P\P2 ą║ SgnG% rnax\ Sgn[6sm;a]
+ 1 > +1 +
+ 1 < - 1 -
- 1 > -r 1 -
- 1 < - 1 +
Table 5.4: Sign for Gtm,,i and Gsmin as a function of čĆą│čĆ2 and k.
When examining Ysoa and ąō50G' it 's seen from Equations (5.40) and (5.44) that
ąō^ŌĆÖą×čüąōą┤ą░čü- = 1 =?Ō¢Ā |ąōą┤ąŠąŠ! |IŌĆÖi'OG'l = 1 (5.46)
This means that either ąō50ąĪ or TsoG' ls passive while the other is active. The special case, when ITsoci = ITsog'I = E requires that R.e[2y = 0. As the exchangeable gain is defined only for ReiZ*-] čä 0 the unit circle is excluded from the region of detimtion. It is also seen that
rSOC[Pi;,2 = ^ = r50CjŌĆ× MO
ą│ą┤ą░čü|ąĀ1ąĀ2=_5 = rsoo'i?!ŌĆ×,ŌĆ× '5'48)
From Equation 5.36 it is seen liiai the centres are 'located on the straight line r
through the origin:
, ,-r , im!i7, - ,, rŌĆ× , ,,
Ō¢Ā i(.e[L qq\ 1ąø.49)
~ ąĀ,.ąō ąĪ- ąĪ ąø*ąō 1
^clŌĆśJil ~ ^22^sJ
Another line of interest is the line q which divides the circles into two parts. This corresponds to a singularity when the denominator in the expressions for rV-v:
5.4. Graphic representation in the reflection plane
91
and Rq (Equations (5.36) and (5.35)) equals zero. In order to locate the line q the quantity Tea ~ Rg, where Rq is on the line r, is investigated when the singularity is approached. The singularity is at
ą│ =
- !.SU|2
Let Ge < 0 from which it follows that |5čå|2 - ]As|2 > ą× ąø P1P2 = +1 or
Ō¢ĀVr
I .?i 112 - |AS|2 < ą× ąø pipo = -1. Coming from one side the point I\r on the line
r is determined bv
lim (ąōąĄąĄ + ą╣ąŠąĄ)Ōä¢
(ISi 112 ŌĆö |A512 }G- TP1P21^21 I2ŌĆö1ŌĆś0 +
where
ąøąŠą┤; =
ąōą│čü + RcpJVo = ---------ŌĆö-------^22--^'----------eJ'f
00 ┬░ \Sn\2 - |AS|2 + '
( ąō (l-SŌĆÖii!2 - i^sl* + -1čé ąąŻą│ą│ąō2)] \
x(1 + \|[ ----------------------------------------------------------------------------\)
In approaching the limit the Taylor approximation y'l + x 1 f ^ x for smaH x is used and
IV = I _ ^ _15ŌĆśi - 522Agl
(|su|2-!a,~I2)c.+pi┬½I*i!2-o+ V.J-b'l 112 - |ąÉčī'|2 + PiP2\S2i\-1j:
1 - 1
1 (l^nl2 ŌĆö !As|2 + Pi Pa 15^1 i" čéąŚčé) (P1P2 i 1' - i + jS^I")
'2 ' !5{t - s;ja;|2
lim
t|Su I* -|A st^Ge+Pi P ~0+
\ 2 _ b'22A".l2 J
1 - j- + _ vv
2 I'b'n - 322┬▒'S\2
_ 1 + i-S'ą│ 1!2 - !-b221^ ;A_<I~ .
2\su - s22^2 ' 11 ~
Approaching the point. ąō7,- from the other side and keeping '┬Ż 0 the same Previous << 1 .. 23 24 25 26 27 28 < 29 > 30 31 32 33 34 35 .. 85 >> Next 