# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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cp =

cei

c..

Cei

c„

= b(Y„

(6.100)

It should be noted that this equation does not require reciprocity, but in the rest of this section reciprocity is assumed.

Hillbrand and Russer [7] have shown how the noise correlation matrix in ABCD form relates to the noise parameters. Let С a denote this correlation matrix. From [7] or Equation (6.71) it is seen that

С, 4,11 = Rn (6.

£'.4,12 = | (Fem.n - 1) - Rn\SoF — RnY' (6.

С A, 21 = С*АЛ2 = Rn. Y-, (6.

С A,22 = -йп I^SOFI' = Gn + Rn\Yy\2 (6.

Cpd is the correlation noise matrix in admittance form which is obtained from С a by

CFd = V С a V* (6.105)

where V is expressed in terms of elements oi the admittance matrix Y- (Equation

(6.90)) as

Г -V „ I 1

У - i 1 f>. ;№)

L -и ]

The reverse transformation is given by

130

6. Noise of embedded networks

from which

^fi min — 1 + '2(C'a,12 + С.А.ц} SOF)

Rn

Y,

С A,\ 1 \

Сл.22

Cam \ С.4Д1 or Rn, Gn and Yy are expressed in terms of С a as

f- j

п[Сдлг]

Cam

(6.108)

(6.109)

(0.110)

Rn = Cam Gn = Ca,22 — С A,21

Cam C.4,2i

Cam

Y-y =

Caa

(6.111)

(6.112)

(6.113)

In many cases this deembedding procedure, which here is performed with matrix algebra, can be performed as shown in Example 6.3 in section 6.1 and in section 6.2, where sections of transmission line in a similar way can be removed from the encapsulation by adding the negative lengths.

6.4.2 Calculating noise parameters at a new frequency

Now the correlation matrix of the active device C.j is known at one frequency. Since the two-port is linear it is possible to write as a superposition of two terms:

Cd = TNT1 + SNtS1 (6.114)

where T and S are transformation matrices which are functions of the equivalent circuit parameters. Nt represents the thermal noise contributions included with the active device and is known from the equivalent circuit parameters. N represents the noise from the noise generators in the intrinsic transistor and from [11] Equation (6.114), N can be expressed as

N = T-4Cd - S N; S^jT^1 (6.113)

'Sow consider a frequency change. The frequency dependence of N is know:;.1 /V. represents thermal noise and thus the noise generators are independent of frequency, h\;t the iritiiiencp o! frequency dependent elements shapes the noise spectra of the thermal noise generators in their transformation to the reference planes cor responding to the matrix Nt. The transformation matrices, T and 5, should also

4.V is often independent of frequency, e.g. m cflsp of л ?'F!T without flicker

6.4. Calculating noise parameters from deembedded data

131

be calculated at the new frequency and the noise parameters are calculated in accordance with the method outlined below.

Deembedding procedure:

1. At some frequency, where the noise parameters Fcmin, Ysof and Rn are

known, calculate С4 from Equations ((HOI) - 16.104).

2. From measurements or simulation calculate the package admittance Yv and

partition it as shown in Equation (6.83).

3. Calculate the device admittance matrix Yd from its equivalent circuit.

4. Calculate D by Equation (6.91).

5. Calculate V by Equation (6.106).

6. Calculate Cpd by Equation (6.105).

7. Calculate and partition Cp by Equation (6.100).

8. Calculate Cd by Equation (6.99).

9. Calculate T, S and Nt for the particular device.

10. Calculate N by Equation (6.115).

Change to a new frequency and at this frequency:

1. Calculate T, 5 and Nt.

2. Calculate Cd by Equation (6-114) using N from step 10 above.

3. Determine YP from measurement or simulation and partition as in Equation

(6.83).

4. Calculate Yd from the equivalent circuit of the device.

5. Calculate D by Equation (6.91).

6. Calculate Ye by Equation (6.90).

7. Cn.lcula.te and partition CJ by Equation (6.100).

8. Calculate Cpd by Equation (6.981.

9. Calculate V bv Equation (6.106).

10. Calculate С a by Equation (6.107).

11. Calculate the new noise parametprs fronn Equation* ^6 b's'i - i'fi.ilQ).

132

6. Noise of embedded networks

6.5 Transformer coupled feedback

Consider an active two-port with feedback consisting of two transformers - one connected to give voltage series feedback and the other to give current parallel feedback as shown in Figure 6.15. Let the two transformers be identical. This type of circuit is often used but how does its noise two-port look? Neither of the above mentioned methods leads to a result.

Figure 6.15: Amplifier with double feedback.

The method used here is to compute the noise factor of the circuit in Figure

6.15 and also to compute the noise factor of its equivalent circuit with the unknown noise two-port. As these two noise factor expressions are identical the new (primed) noise parameters can be found by setting the two expressions equal for four different values of source admittances with the real part positive. This method has also been used to get the results in section 6.1.

In Figure 6.15 the two transformers are characterized by their Z parameters and the transistor by its У parameters:

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