# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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46

4. Noise parameters

к To G,

kT,, G, =

__________\ 1го51/________

4 (Gs + Gy — Gy) Af

(I %J) + (l»2flJ)

4 (Gs + Gy — Gy) Af

I Hi l\ + I Ip2- l\ iv„ _l у I2

— ^ I Gn ч ' \ l~tinu l-1 э ;

IGsAf

Definition 3.5 leads to Equation (3.12). Using Equations (2.7) and (2.9) leads to !P _ (llGsl) + (|!gJ> + (1ея„1)1^5 + Yy\2

''' “ (l‘M

= 1 + ^ (g„ + Лп |YS + K,|2) (4.11)

In a quite similar way from Figure 4.5 it can be derived tliat

jpe = 1 + (г" + 9n |Zg + Zyj2^ (4-12)

Note that Fe > 1 if Gs > 0 and F, < 1 if Gs < 0. The two expressions for Fc, Equations (4.11) and (4.12), show how the extended noise factor depends on the noise parameters and the source immittance.

In order to examine how the noise factor depends on the source immittance Equation (4.11) can be written as

F, - 1 2 Rn

Gs — ( n n------------

V [Bs - (-By))2 = ~ {F'~ 1) I* - !=■ (4.13)

-i Лп &7\. &n

which is recognized as the equation for a circle (Fc constant) in the 15-plane. The circle has its centre in

and its radius is

(Gs,Bs) = (_Л__с7.-В7) (4.14)

j(F, - l)2 ^G-( Gn _____

V 4'г" V ,e 4 Rn l4'iJj

FJYs) is symmetrical around (Fe, Gs, Bs) = (1 + 2RnGy, 0. — B-,). Since vanishing of the right-hand side of Equation (4.13) always implies two roots for F,,, two extrema for F, exist. These are as follows:

/

Femin — 1+2 ^R„G7 + у R„Gn + (RnGy)2 j (4-16)

for Ysof = -JojRn + G2 - j By Г4.17)

4.1. Noise voltages and currents

47

F,max = 1 + 2 (r^ - + (RnG1)'2'Sj (4.13)

for Ysof' = - \fGn/Rr. + G% - jB., (4.19)

The index SOF stands for Source Optimum with respect to the noise Factor and is used for Gs > 0. SOF' is used when Gs < 0- Note from Equations (4.16) and (4.18) that Fmin > Fmax. In Figure 4.6 the noise factor contours are shown.

Figure 4.6: Contours for constant extended noise factor in the source admittance plane with data from Example 4.1.

Similarly, from Equation (4.12) a local minimum.

Fс min — 1+2 + \]0nrn т (<7n R-t)" ) (1.20)

2soF ~ \j rnlGn + Щ - 3 A-, ( 4.211

and a lora! maximum

— \3nR7 \J9n^n + {9nf j (^-22)

for

fZ

у rni On -г Ji;, - j y\~

(4.ZJ )

48

4. Noise parameters

Equation (4.11), which is illustrated in Figure 4.6, and Equation (4.12) give information on the noise factor as a function of a chosen source immittance. If the minimum noise factor is the design criterion then Equation (4.17) or (4.21) determines the source immittance and Equation (4.16) or (4.20) gives the value of the minimum noise factor. If the source is active the Fcmax gives the smallest noise factor for the stage.

Sometimes the values of Fernin, YsoF, and Rn are used as noise parameters and similarly with Fem{n, ZsoF> and gn. This leads to

It should also be noted that the circles for constant Fe, Fe,n,„, and FCTnax can be replaced by Ta, Ты min, and Тыт„ as seen from Equation (3.13), T„ = (Fe - l)Tti.

Example 4.1 In order to draw the noise factor circles as a function of the source admittance for an amplifier with the following noise parameters

Y-, — 2.0 + j 7.5 rnS the first thing to do is to find Fcmin and FCmax from Equations (4.16) and (4.18) and

(4.24)

Fem.n + ^rlZs - ZsOFl2

Ms

(4.25)

Rn = 25 П Gn = 4.3 mS

the corresponding source admittances from Equations (4.17) and (4.19):

Fc

e min

1 + 2 ^RnGy + \J~RnGn + (RnG-,)2^

1 + 2 I0.025 x 2.0 +

1.8 (2.55 dB)

for

ГТя

\l----------h 2.O'2 - j 7.5 = 14 - j 7.5 mS

V 0.025 J

and

1 + 2 I 0.02-

о A

•4.0 + iU.U2.) X Z.U

n

0.4

■for

Ysof' = - + 2-°2 - J7-5 = -14-j7.5mS

4.1. Noise voltages and currents

49

The centres and radii for the noise factor contours are given by Equations (4.14) and (4.15)

tCs-Ss) ' (w - -B’)

V 4 Rn 1 1 R, Rn

Some results are shown in Table 4.1. It is now possible to draw Figure 4.6. If only passive

Fe = Gs + j BS [mS] Rcf [mSj

3.0 38.0 - j 7.5 35.3

2.5 28.0 — j 7.5 24.2

2.0 18.0 - j 7.-5 11.3

1.8 14.0 - j 7.5 0.0

0.4 -14.0 - j 7.5 0.0

0.0 -22.0 - 7 7.5 17.0

-1.0 -42.0 - jl.o 39.6

Table 4.1: Centres and radii as function of Fe.

source admittances are possible only the right-hand side of the figure is of interest. From this figure the actual noise factors for different source admittances can be found. However, they can also be found directly from Equation (4.11). If Y$ = 20 + j 0 mS then

Fc = 1 + тг~ (<?п -r Rn IYb’ + Kyi2)

Us

= 1 + ^ (4.8 + 0-025 j20 + 2 + j7.5|2) = 1.92 (2.82 dB)

The designer has now to decide whether the amplifier should be noise matched, which gives him an extra 0.27 dB, or if a noise figure of 2.82 dB is sufficient. A noise match should be with as little loss as possible, as any losses degrade the noise factor. In this case a loss of a quarter of a dB ruins as much as is gained. With transmission lines an almost lossless match can be performed. If the input frequency is 600 MHz an 83.7 mm shorted 50 П stub across the 20 mS source admittance and a 30.3 mm SO 0. lineiength to the amplifier will perform the noise match (cr — 1). It should be noted that oniv 3 narrow band match has been performed.

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