# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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The noise Iree circuit matrices are the ; . 2 and chain or ABCD matrices, and the correlation matrices are - in the same order -- "Wen bv

oy =

L \l'2ll i \г2г

16.68)

CZ

\i;l',il) ( u 1 u2 )

( 6.69 i

122

6. Noise o{embedded networks

Figure 6.11: Admittance, impedance and chain representations.

C, =

I

2 Д/

(uu’) {tti*} i‘u') (”')

(6.70)

where Д./ is the bandwidth, the factor 2 occurs because the two-sided Fourier transform has been used and {- - •) denotes the ensemble average over processes with identical statistical properties.2

Often the correlation matrices can be calculated without knowing the noise sources. If the two-port considered consists of only passive elements the thermal noise from it results in a correlation matrix of either of the two forms:

CY

CZ

■2kTRe[Y] 2 к T Re [Z]

where к = 1,38 x 10~23 J K-1 is Boltzmann’s constant and T fKj is the noise temperature of the two-port.

For active two-ports the chain correlation matrix is given by

Г . = 2 (• T

Rv

5(FCTnin - I) - RvYsof

2 (fern in 1) Rn^ SOF

Rn I^sof!2

( b.

where R~ is the equivalent noise resistance, ir.: ~lrl is the minimum noise factor and Ysof the corresponding source admittance.

C’ = rCCT]

(6.72)

2See Appendix A.

6.3. Noise parameters of interconnected two-ports

123

From impedance From chain

To admittance [1 01 1° !. Гп П-2 Yll ^22 0 1 1

To impedance Г Zn zl21 [ Z^i z22 J [1 ol _0 ij I 1 —Z\i {0 -Z-n

To chain [ 0 в 1 n L " 1 -Л ' 0 -C ’ 1 0 1 n 1 . " " J

Table 6.1: Transformation matrices.

where С and C' denote the correlation matrix of the original and resulting representation, respectively. The transformation matrix T is given in Table 6.1 and the dagger (-4') denotes the Hermitian conjugate (of-4).

Interconnections of two two-ports in parallel, in series or in cascade result in a correlation matrix given by

Cy =; Cyi 4- Cy, (parallel) (6.73)

Cz = CZl 4- Сz? (series) (6.74)

CA = C,tl -f- Al CA^ ,4j (cascade) (6.75)

where the subscripts 1 and 2 refer to the two-ports to be connected.

The noise parameters are obtained from

Ysof =

\

С A,22 /lm[C.4,12j\2 , ■ /ЫСлдг

c.4.11 V СЛМ J ' J { С А. П

= 1 4-

c.4,12 4-

kT

Rn — С a,11

The noise factor as a function of tU£.

F - 1

г'Од -

'IkTRe'Zr]

(6.76)

(6.77)

(6.78)

(6.70)

(6.80)

This method can be used instead of section 6.1 as demonstrated in [7]. It is also s^ful when active two-ports are interconnected.

124

6. Noise of embedded networks

In a more recent paper Dobrowolski [8] has used a wave representation and enlarged the network to be considered. His network consists of interconnected passive multiports which introduce only thermal noise and active linear two-ports. It is also

а. requirement that the two-ports are interconnected two and two. Dobrowolski’s results are applicable to computer aided design of noisy microwave circuits.

б.4 Calculating noise parameters from deembedded data*

Pucel et al. [9] have shown a procedure to calculate noise parameters as functions of frequency for field effect transistors, FET’s, and high electron mobility transistors, HEMT’s. This is especially important at frequencies so high that it is very difficult to measure the noise parameters. If the noise parameters at one frequency and the parasitic elements of a transistor (with its encapsulation) are well known, the intrinsic elements and noise generators with their correlation can be computed by deembedding. As for FET’s and HEMT’s these noise parameters are approximately independent of frequency and the behaviour of the intrinsic equivalent circuit is well known, so it is possible to compute the S and noise parameters in a large frequency range. Thus computation replaces measurement which is a great advantage as noise parameter measurements are rather difficult at frequencies above 10 - 20 GHz. The authors claim sufficient accuracy up to at least 40 GHz.

6.4.1 Matrix formulation of the deembedding procedure

Figure 6.12: У parameter and noise correlation matrices for a packaged two port.

The deembedding uses the equivalent circuit shown in Figure 6.12 from which the correlation matrices of the packaged device, CP!i, and the intrinsic device, Cj. are

Some of the text in this section is adapted from Puce] et ai[9] f©1992 IEEE). Reprinted by permission of IEEE.

6.4. Calculating noise parameters from deembedded data

125

related by the linear matrix equation

CPd = PCPPi + DCdDi (e-S1)

where CP denotes the admittance correlation matrix of the packaged four-port and P and D represent package and device transformation matrices expressible

in terms of the admittance parameters of the package and device. The dae^er

denotes the Hermitian (conjugate transpose) of the associated matrix. Equation

(6.81) is used when the intrinsic parameters aie embedded in parasitic elements of the encapsulation.

Deembedding is performed by solving Equation (6.81) for C^:

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