# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Kg; I2 + \IorA2 + KoR'.l2 + l/,R''!2

\ IoGs\2

B' T B"

B'D" + В "D‘ + YSB'B"

В' + B"

B'D" 4- B"D' + YSB'B"

S' + B"

(6.52)

=

B'D" + B"D' + YsB'B" Ig" ysb" + i:;(o' + в") + d" - d' B'D" + B"D' + YsB'B" '

YsB' 4- Y"(B' + B") + D' - D"

r _ _____ 7 4____\______' ____________ jp

°K ~ B'D" + B"D' + YsB'B" R"' are the output short circuit currents from each noise generator. The identity can now be expressed as

- (Gn + Rn\Ys + Y,\2) = J- (с’п + G'r

n, \YSB" + YIt(B' + В") + D" - D'|2

~Rn Щ, + B„T2

\YSB' + Y”(B' + B") 4- D' - D"|:

+ K

i S' + B" j'

From this identity the results for parallel connection are д = f6..53)

j В' + B"I2 R'R"\B'Y- B"Y” + D" - D'?

6.54)

ЩАВ"Р + Я"!-й'!2

_ R!nB"'{Yl(B' + B") 4- D" - O'j 4- fi"S'-[K"(S' f B") + O' - D"}

(6.55)

In this way it is possible to compute the noise parameters of a known transistor in parallel with a feedback two-port. If this two-port consists of a piece of transmission

6.3. Noise parameters of interconnected two-ports

119

line in series with a feedback element the noise parameters of the feedback two-port should be computed first. It is therefore necessary to investigate the noise parameters of two-ports in cascade.

6.3.2 Two-ports in cascade

The same procedure as above is used for calculating the noise parameters for the equivalent fwo-port of two two-ports in cascade. Again, if more two-ports are in cascade, they are taken two at a time. Let the small signal and noise parameters of the first two-port be primed, and let the parameters of the second be double-primed. The resulting parameters are unprimed.

Figure 6.10: Two noisy two-ports in cascade and equivalent circuit.

Computing the noise currents just after the second-stage noise two-port (and thus before the noise free equivalent of the second stage, which does not add more noise power but only amplifies it with the same amplification), gives

1 0 (j V —

YSB' + D I

yTWT

ilo,

in,

•foG" L-. R”

}SB;

Ig«

(y’

\ '

D'

t-ir

л. + C;'-A (Г-' YSB' + D'J

6. JSoise of embedded networks

From these equations the following identity can be derived:

— (Gn + Д„|У3 + Y,\2) = ±- {G'n + R'n\Ys + >;'|3 + G''\B'YS + Щ2

4- R"\(R'Y. + D‘\Y" + A'Is + C’\2\

.11 V - • / 7 ^ I j

From this identity the results for cascade connection are

Rn = К + G”\B'[2 + S!'n\B'Y:; + Af (6.56)

Yy = ~ [д'пУ^ + G"£'*D' + if" (в'У" + A')* (d%" + C')] (6.57)

Gn = G'n + д;|У7'|2 + G';\nf + K\D'Y; + C'\2 - Rn\Yy\2 (6.58)

Equations (6.56) - (6.58) can be used to compute the noise parameters of a feedback element which includes two-ports such as transmission lines used to connect one or two lumped feedback elements. The total feedback element can then be connected in parallel with a transistor and the noise parameters are computed by Equations (6.53) - (6.55).

6.3.3 Albinsson’s method of interconnected two-ports

This method is divided into two parts. In the network of two-ports the noise parameters of each two-port in the first part is transformed to the input side of the resulting network as the noise parameters i?n,:, Gni; and Ylt{. This part is dependent of the network configuration and must be performed individually for the network considered. As the network is linear the superposition principle is valid and the second part is to perform this superposition.

In order to investigate this second part it should be kept in mind that the noise parameters used by Albinsson are the П noise parameters from [3] which are based on Figure 4.2. The noise voltage en is the summation of the transferred noise voltages enii and noise current in is the summation of the transferred noise currents in<i, thus

(6.59)

(6.60)

As the voltages from the different two-ports are uncorrelated it follows from Equations (2.13) and (2.6) that

I

Rn = Y.R^‘ (6-61)

- 2-/tn-' У~\

6.3. Noise parameters of interconnected two-ports

121

From Equations (4.9) and (4.5) it follows that

v _ (‘"O

- J&F) {л2)

Thus

£>=i( |e„,i|2> ~ ^'b3)

Tut* uncorreiatea part of the current in Figure 4.2, inl, determines Gn. From

4 = in, l + (6.64)

it follows that

I I

4i + = ^(inU + (6.65)

i=i t=i

and thus

I

i-П 1 = ^ Tl, ~ V^)] (6.66)

j=i

and, as noise from the different two-ports is uncorrelated,

I

Gn = J2(Gn,i + Д-vlKv - i;|2) (6.67)

1=1

This is another way to compute the noise parameters Rn (Equation (6.61)), K, (Equation (6.62)) and Gn (Equation (6.67)) of interconnected two-ports.

6.3.4 Matrix formulation

Hillbrand and Russer [7] have made a matrix formulation of parallel, series and cascade connections of linear, noisy two-ports. The noisy two-ports are represented either by an admittance representation with a noise free part and two noise current sources, or by an impedance representation with two noise voltage sources, or by a chain representation with a noise current source and a noise voltage source both at the input side. These representations are shown in Figure d.ll.

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