# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

**Download**(direct link)

**:**

**32**> 33 34 35 36 37 38 .. 85 >> Next

The formulae, which are derived in [i], are given below. The unprimed entities are the small signal and noise pnrnmetprs of the three-port without new embedding, and the primed entities are the parameters with a set oi embedding demerits.

Parallel embedding:

Y,\ = Yu + Ya 4- YB

(6.1)

6.1. Lumped embedding

10.3

(6.3)

У22 + У в + Ус Е,,

К

(6.5)

(6.6)

Б, DVE,

(6.7)

where

Dy = |УВ - Ун!2

By = Gb + Gc + 1^21 |2Дп

я, = (Уц + У-i + Yi\)Gb + (Уп + У 4 + Ya)Gc

+ (Уц — У В ^21 Rn + (У-t + У В + V-,; У м ! ‘ 1(п

Ly = | i's — У'21I*" (G а + Gn) -f- |Уп f У4 + Y2\V Gb

+ |Уп + У а + Ув^Сс + |(Уц — У-,) Ув + (Дл + У в + У-^Ул!2 Rn

Series embedding:

(6.12)

(6.8)

(6.9)

16.10)

(6.11)

(6ЛЗ)

Lz \H: \

Ds D, £■

£г = Rb + Rc -f \^2\\~9n

H, — (Z11 -f Za — Z21) Rb + (2ц + Za x Zg)Rc

104

6. Noise of embedded networks

Lz — !Zg + Z2\\2{Ra + rn) 4- |Zu f Za - Z2\\2 Rb + \Zn + ZA + Zb\2Rc +- i(Z~, - Zu)Zb + (Za + Zb + Z,t) Z2^\гдп

Transformation from Z to Y parameters:

(6.15)

(6.16) (6.17)

Yn = Zn

П 2 = — Z\2

Y21 = -Z21

Y22 = Zn

Rn = rn + gn\Z^\2

Gn = rn

\z-,\2 + rn!gv.

y7 = z;

|z7!2 + Tnl3n

where = Z\\Z22 ~~ ^12^21

Transformation from Y to Z parameters:

IKr + Orj'Rr,

П / R

(6.18)

(6.19)

(6.20) (6.21)

Zn = ~ (6.22)

Ду

Z12 = (6.23)

Zn = -r^- (6.24)

LAY

Z22 = ~ (6.25)

A у

gn = Gn + flniv;!2 (6.26)

(6.27)

V'-

It is interesting to note that the formulae work both ways: embedding and deembeddins. As an example take an encapsulated transistor whose small signal

6.1. Lumped embedding

105

parameters and noise parameters are measured. The effect of (the known) strav capacitors and stray inductors can be removed by adding the negative of the strav elements. This is done such that the stray elements closest to the terminals of the encapsulated transistor are removed first and then the second set of stray elements until finally the small signal and noise parameters of the transistor chip are obtained.

Transistor

■47/IS ^ 0.5 mS 30 fi

Figure 6.2: Transistor with bias and feedback and equivalent circuit. Capacitors are considered as shorts at the working frequency.

Example 6.1 A transistor amplifier stage as seen in Figure 6.2 consists от a transistor with known Y and noise parameters and also bias and load resistors and finally a series feedback resistor. The data for the transistor are

I'll = 10 4- j 2.1 mS

Vis = 0.5 — j 0.866 mS

Yii = 125 - j 29 mS

y22 = 1 + i 3 mS ,J r-

R., = 25 0 = 4.'? mS

y. - ‘> -4- 7 7 5 mS

These data give Fmin - 1.3 and i 'sof — 14 — j 7.5mS. in order to compute the small signal and noise parameters of the stage, one has to apply Equations (6.22) - (6.28) to get the transistor data into Z form. Then Equations (6.8) - (6.14) add the series feedback resistor (whiie Z4 = Zc = 0 + jO). Equations (6.15)- (6.21) transform to

106

6. Noise of embedded networks

the Y parameters and Equations (6.1) - (6.7) add the bias and collector resistor (with Yb — 0). The final result, computed with a Fortran program where the equations used are shown as subroutines in Appendix D, is:

Yn = 1.586 4- j 1.548 mS

Yu = 0.4448 - j 1.017 mS

Yu = 25.16 - j 2.935 mS

- -* ex. 0.2837 -{- j 1.469 — С .UlU

Rn = 62.81 0

G'„ = 4.735 mS

K, = •3.442 + j 3.432 mS

= 2.606 and Ysof = 9.340 — 3.432 mS. The main part

of this noise degregation is due to the series feedback. The bias and collector resistors raise the noise factor only slightly - from 1.800 to 1.807.

Figure 6.3: Feedback elements for simultaneous input power match and noise optimization.

Example 6.2 An optimization procedure for simultaneous input power match and noise optimization using only one series and one shunt feedback element as shown in Figure 6.3 has been developed. Since lossless feedback does not change the value of the minimum noise measure (does not add noise), only lossless elements are considered. The fact that М,.тЫ is constant for lossless feedback is a convenient check on results of computer programs. What dees change, of course, is the value of the source admittance, YsoM: minimum noise measure.

For each pair of iossiess feedback elements a computer program - using the subroutines in Appendix D - gives the values for the primed signal and noise parameters from Equations (6.1) - (6.28). The minimum noise measure, Меты, and the corresponding source admittance, YsoM- from Equations (5.27), (5.?5) and (5.26) are also computed.

6.1. Lumped embedding

107

-1 0 1

ImfV'/l mS

Figure 6.4: Transducer gain Gt (full lines) and load admittance Gl, Bl (dotted lines) versus lossless feedback for unity VSWR and minimum noise measure.

When the source admittance, Ys = Ysom <s determined, input power matching occurs when the load admittance, Yl, 's chosen to satisfy the relation:

>L - —>22 + -----------777

Ун — > я

This equation allows choosing the load admittance for input power match and minimum noise measure (V^oa/). Then, with the source and load admittances equal to Yso\f and YiQ\f respectively, the transducer gain, Cj, of the stage is computed.

**32**> 33 34 35 36 37 38 .. 85 >> Next