# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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4.1.1 The equivalent noise two-port

In order to describe a noisy linear two-port, the small-signal equations are enlarged to cover the noise contributions as well. In Figure 4.1 the noisy two-purl and three examples of equivalent circuits are shown. In Figure 4.1(a) the noisy two-port is characterized by hatching. In Figures 4.1(b), (c), and (d) the noisy two-port is

41

42

4. Noise parameters

replaced by a noise free but otherwise unchanged two-port and two partially correlated noise generators. This is correct as long as the two-port is linear. Physically the noise sources inside the two-port can contribute to either the input side, the output side, or both. Mathematically two linear equations exist between the input and output sides and the noise appears in both equations.

(a)

Ð³

+ +

V, v2

- -

Figure 4.1: Noisy linear two-ports.

The circuit equations corresponding to Figures 4.1(b), (c), and (d), where the Vâ€™s and I's are complex Fourier series coefficients of voltages and currents, are

' h ' Yu Yn ' V,

. Ð Y21 y2 n

>1 _ ' Zu Z 2 1. â– h

. ^ - . z31 -Z22 J h

' Vi ' A Ð’ 1 Vo

. h Ð¡ n I Ð£ . Ð“ n j

+

4

+ ^Ð°

. â‚b .

J

(4.1)

The noise vectors each represent two noise generators. As the noise is generated inside the two-port by various physical processes, the contributions to the two noise generators are more or iess correlated. It is therefore necessary to know the correlation between the two generators. The correlation between two stochastic variables Ð›" and Y is determined by the complex correlation coefficient

.. _ W>

where jyl < 1. In Figure 4.1(b) the admittance matrix has been used and in Figure 4.1(c) the impedance matTix has been used. Rothe and Dahlke chose Figure 4.1(d) corresponding to the chain or ABCD matrix and thus both noise generators are at

4.1. Noise voltages and currents

43

the input side which is practical in noise theory as the noise quantities are referred to the input side.

Figure 4.2: Partition of the noise current generator into a correlated and an uncorrelated part.

As shown in Figure 4.2(b) the noise current generator is partitioned into two parts, Â»â€ž = !n] 4- iâ€ž2, where one part (in2) is partially correlated with the noise voltage generator and the other part (z^) is uncorrelated. Thus (inii'2) = 0. This means that in2 is proportional to eâ€ž with the complex proportionality factor K, which is called the correlation admittance. This leads to

'nâ€˜2 â€” ^7 en (â€˜t-3)

<k!2) = { ii'nll2) + ( |in2|2)

(eni'n) = (enin\) + (Ðµ*Ð³Ðª) = (eni'ni) (4.4)

Inserting Equation (4.4) into Equation (4.2), it follows that

Re [7] 4- j Im [7]

(en **.)

V ( ienj~/ ( i Ð³ ri M )

(gn^2) Ð³/(Ð«2)(1Ñ‡Ð˜)

(ib2^ (I'nj2)

(4.6)

' (<â€˜Ð¿Ð³) = Ð£'Ðž â€¢ (<K,en)

~ V 1^1 / ' \ 1 ^ Ð› 2 ! )

( 1.8}

4. Noise parameters

Suppose that eâ€ž, iâ€ž and 7 are known then

2>

(Ð«2) = (1 - Ð«2)(Ð«2)

Equations (4.3) and (4.6) determine Ð£7 by

and thus

11 U m y* = 7 111Ð›Ð§-Ð›

V (leÂ«!2)

(j-y + j By

= ReM l/S - 7lm[7],/(b|2|

len|2)

<kl2}

(4.9)

(4.10)

\ i

*nl

I

Figure 4.3: Equivalent noise two-port.

The circuit in Figure 4.3 can be replaced by the circuit in Figure 4.4 where Ð£7 and â€” Yy are noise free which is shown by their noise temperatures of 0 K. The noise generators are conveniently indicated with Rn and Gn by Equations (2.4) and (2.5) replacing the bandwidth dependent eâ€ž and inl. The two circuits are identical as their open circuit voltages and short circuit currents are the same. The equivalent noise two-port - such as the one in Figure 4.4 - are placed in front of the noise free small signal two-port. When small signal analysis is performed, the noise voltage generator is short circuited, en = 0, the noise current generator is open circuited, in 1 = 0, and Ð£7 and â€” Ð£7 cancel each other, therefore the noise two-port will have no cffect on the small signal analysis.

From the two noise generators in Equation (4.1) given by (|eâ€ž|2), (|in|2) and their correlation coefficient 7, the four noise parameters Rn, Gn and Yy = G-,+j B-, shown in Figure 4,4 are determined.

If the noise voltage generator instead is partitioned - with the noise current generator - into a correlated and an uncorrelated part, the resulting equivalent noise two-port is shown in Figure 4.5. Again the four noise parameters rn, gn and Z-, â€” R~, Ñ‚ j X-f are determined by (je^2), ( |tn|2) and 7. Note that in general

z-, / i/*V

4.1. Noise voltages and currents

45

Ys

T = T0

Â±

T = 0 |Ki

e

Ð±'Ð»

T

V,

T = 0 [K]

Figure 4.4: Equivalent noise two-port in ÐŸ form.

Zs T = To

Z7

T = 0 [K]

"<y

-z,

\) T = 0 [K]

3n

Figure 4.5: Equivalent noise two-port in T form.

Two sets of noise parameters have now been defined. These are Rn, Gâ€ž, and Y7 which are most convenient to use in connection with Y parameters and rn, and Z-, which are convenient with Z parameters.

4.1.2 Y and Z noise parameters

From Figure 4.4 the noise factor can be derived. The amplifier following the noise two-port is noise free so that the noise ratios on the output and input side of the amplifier are the same and therefore the (noise free) amplifier does not contribute to the noise factor. To find the noise factor, the exchangeable noise power densities from the source at T0 and from the two-port at the output terminals are found. This is done by finding the current in a short circuit over the output terminals from each of the three noise sources Gs, Rn and G\,,. The noise power density is then the sum of the three power densities which are each found by squaring the short circuit current and dividing by Af and by four times the output conductance. This noise power density is also the noise power density of the source admittance at either To (the contribution from the source) or Tâ€ž (the contribution from the two-Ñ€Ð¾Ð³* transferred to the source) both multiplied by the exchangeable power gain of the

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