# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Here again

P =0 ~~ ^22^LI ~ , I n 12\ Ct m

P?0 P2 ^ ^ j^,, j2 \\^2\ ) (o.31)

P'S = Pi iT-^ji (1дд12) (5.Э2)

lit

1ВД1

where i refers to the port number;

_ 5i253ir5

"22 “ i - s„r5

is the output reflection coefficient of the two-port, where Г5 is the source reflection coefficient; and

<№i2) _ \S:

2ii

(2

(|£s!2) |1 - 5„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 — 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 - |Дь'!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 [|£n|* + I -^'2212 - 1 - l^sl"] у;—h !S’2i (4 ypy = 0

When using the stability factor k, which here is

1 + (Ad2 - !5U!2 - IS,2!2

2 ] о 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 к2 < 1) on the unit circle.

For .fc2 > 1 it follows that

1 |.?H ( . /71------\

mm |— j = 77;- = 7^. (pi+ vk- - 1 1 (5.39)

L';ej '•-'•jmu.r i^21| 4

which gives Tsog (for source optimum gain) by inserting Equation (5.39) into Equation (5.35):

2(5-, - 52;л;)

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 — 3j-2As)_____________________ ,r. iо.

iGy 1 + l^il2 - Ud2 - |AsP + 2pl№!5’12521lv'P^T [0‘ !

1 + !.S'l 1!2 - |522|2 - |Дд;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 Сетш is the same. Thus the signs shown in Table 5.4 are

obtained.

P\P2 к 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 ргр2 and k.

When examining Ysoa and Г50G' it 's seen from Equations (5.40) and (5.44) that

Г^’ОсГдас- = 1 =?■ |Гдоо! |I’i'OG'l = 1 (5.46)

This means that either Г50С 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*-] ф 0 the unit circle is excluded from the region of detimtion. It is also seen that

rSOC[Pi;,2 = ^ = r50Cj„ MO

гдас|Р1Р2=_5 = rsoo'i?!„,„ '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, - ,, r„ , ,,

■ i(.e[L qq\ 1Л.49)

~ Р,.Г С- С Л*Г 1

^cl‘Jil ~ ^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

г =

- !.SU|2

Let Ge < 0 from which it follows that |5ц|2 - ]As|2 > О Л P1P2 = +1 or

■Vr

I .?i 112 - |AS|2 < О Л pipo = -1. Coming from one side the point I\r on the line

r is determined bv

lim (Гее + йое)№

(ISi 112 — |A512 }G- TP1P21^21 I2—1‘0 +

where

Лод; =

Ггс + RcpJVo = ---------—-------^22--^'----------eJ'f

00 ° \Sn\2 - |AS|2 + '

( Г (l-S’ii!2 - i^sl* + -1т УггГ2)] \

x(1 + \|[ ----------------------------------------------------------------------------\)

In approaching the limit the Taylor approximation y'l + x 1 f ^ x for smaH x is used and

IV = I _ ^ _15‘i - 522Agl

(|su|2-!a,~I2)c.+pi«I*i!2-o+ V.J-b'l 112 - |Аь'|2 + PiP2\S2i\-1j:

1 - 1

1 (l^nl2 — !As|2 + Pi Pa 15^1 i" тЗт) (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'г 1!2 - !-b221^ ;A_<I~ .

2\su - s22^2 ' 11 ~

Approaching the point. Г7,- from the other side and keeping '£ 0 the same

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