# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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4- e-aâ€˜ + e-2*1 - 2aÂ»tyttU - ^ (& + BL) (6.32)

-I J /q 4 '

Equations (6.29* - (6.32) are derived iu the following. With reference to Figure 6.6

the problem can be stated as follows: tiiui the (new) noise parameters al reference plane Ð› expressed by the (old and known) noise parameters at reference plane Ð’ and tile transmission Une constants i, z/q i r<jcii dim positive) ann 7 â€” or -+â– ] ti. Ð³ irst.

Rn

?n "tT

6.2. Reference plane transformation of noise parameters

111

Figure 6.6: Noise two-port with preceding transmission line.

_

Ð¯Ð¿ ~ Yo

j. Ðº V-'

Ñƒ-y = 0-< + J Ðš = Ñ‚Ñ‚"

Vo

â€¢ , Vs

Ð³/5 = as + ] i>s = Ñ‚Ð³~

j 0

The unknown normalized noise parameters are primed:

Ð›: On and yi, = g!y+jb'y

The unknown noise factor at reference plane A can be expressed by

F = 1 + â€” \ g'n + r'Jj/s + y'\ ) (6.33)

gs \ 'â– 1 /

Using the definition of the extended noise factor as the total exchangeable noise power at the output divided by that part of it which originates from the source at standard temperature To, the noise factor at reference plane D can be expressed as

c g's + gr. + x 1Ð›1!2 ;r

Ð³ - ---------~r,---------------- \b.S-i)

9s

Here use is made of the fact that the one-port to the left of Ð’ in Figure 6.6, consisting of a source admittance and a transmission line at the temperature To, generates noise as the real part of Yi at To- The denominator g" represents that part of the exchangeable noise power at the output which is generated in the source.

, / . ,i ys cosh - I -I- sinh -;/

Vs = Os + Ð¯Ñ‡ = ----;â€”Jâ€”;-------------â€”â€”j -

coir, -n r !)S ?!nh 'â– >

jigs + I)2 + t>l}e7rA - [(j/5 - Ð³)- + + j [ % COS 2,31 - 2( ijI 4- - L) sin 2 Jl]

[(</S + l)2 + 6|]e2â€œ' + [(gs ~ I)2 + 62-]e-2"' - 46Ð°- sin 2,3/ - 2(05 -f- Ð¦ - 1 j cos 261}

(6.35)

112

6. Noise o[ embedded networks

A

Ys J-o .

h

To . 7 = a + i 3 . Vj

h

h

Yi

(a) ' (b)

Figure 6.7: Generator with shorted transmission line and equivalent circuit.

The only remaining unknown in Equation (6.34) is now gg. Some of the noise power generated in the source is dissipated in the lossy transmission line, and as one only has to consider the noise generated in the source all other noise sources are eliminated. The relation between the short circuit noise current at Ð’ (see Figure 6.7) and g" is

(j/2|2) = 4 Ðº T'o Af yâ€™sYg (6.36)

where Ðº = 1.38 X 10-23 J K_1 (Boltzmann's constant) a.nd Af is a frequency increment in Hz. The currents Ð” and /j are related by

h - h cosh 7/ (6.37)

Now Ii can be calculated from Figure 6.7(b), where Ð£Ð³ represents the admittance

of the shorted transmission line looking into it at A:

j/i = z-r- = cotli"//

*0

e2al â€” e~2al â€” j 2 sin 2f)l

- e2al + e-2al _ n C0S 2/3/ ^'38^

(|/i|J> = -Ð³-Ð©^-prikToAfgsYo (6.39)

Ws + 3/i iâ€œ

From Equations (6.36) - (6.39) the following expression for can be calculated:

4 !Â«ii2

-I- f

-2at 4. Ð¾ câ€žs1 9/ j. (2

where

I i'll

\VS + 2/i J2

(e2at _ g-2ai)2 + 4 sin2

[(] <?( e2o^4- Ð² 2 ÐŸÐžÐ¯ 2/5/) -t- P *<>ÐŸ2 _L Ð“^{ e-2al_ 0 cqc 2:31) 2sin2 6Pr

6.2. Reference plane transformation of noise parameters

Equation (6.34) is now completely determined by the known noise |i.irninctcrs Ð³., g and jfy and the transmission line constants oti and 31, as y's and its rc;il p;lrt g> are expressed in Equation (6.35), and g'~ is expressed in Equation (6.40). in terms of only al, /3/ and ys-

With Equations (6.35) and (6.40) this expression for the noise factor must, for all ys = gs + jbs, he identically equal with the noise factor expression in Equation (6.33), and it is thus possible to determine the new noise parameters in terms of the old ones and the transmission line data by choosing four values of ys an(i then

solving the four identity equations for the new normalized noise parameters. The

results are:

r'n = ~ {e2aâ€˜ - + gn [ew + e'2â€œ' - 2 cos 2,3/]

+ rn [(5; + 6â€˜) (Ðµ2Ð« + - 2 cos 2pi)

|2?7 (e2*1 - e-2a!) - 4 6. sin 23/

+e2al + e~2al -2cos2;3/]} (6.41)

Ð™ = 37 {." + - 2 + [Â«" - <-â€œ!

n

+ rn{(g2-rb2 + l)

+ 2gje2Â°â€˜ - (6-42)

b'y = 2W S'n 2/3/ + Tn K3"' + ^ ~ !)sin2;3/

- 267 cos 2,3/] j (6.43)

g'n = ije2Â»' - e-2-*' + srâ€ž [e2â€œ< + Â«"*â€¢* + 2 cos 2^]

Ð“I â€¢-> ,r>\ ( '\-v/ , - - Ð›.

^rrn \\!7y -t- o~ j j^e--' 4* e 4- Ð³ cos 2ptJ + 2<?, (e2al - *-Ð³Ð«) + 4b-, sm 2-Â»

+ e2al -f e~2câ€œ - 2 cos 2,3/ j j- - râ€™n (j' r 61,) (6.44)

Example 6.4 For the purpose of illustrating the effect of a transmission line on noise performance this example has been computed. For an active element with the following

114

6. Noise o[ embedded networks

I) A '/A

Figure 6.8: Noise parameters from Example 6.4 as functions of relerence plane.

6.2. Reference plane transformation of noise parameters

115

noise parameters:

Rn = 25 Cl, Gn â€” 4.8 mS, and VI, = 2 + jl2mS

the noise performance of a preceding transmission line of a length up to half a wavelength (A/2) and three different attenuation constants (Â« = 0.0, 0.3 and 0.6 Np/A) has been computed and expressed by Equations (4.84) - (4.86) as

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