# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

**Download**(direct link)

**:**

**38**> 39 40 41 42 43 44 .. 85 >> Next

Cd = D-'{Cpd - PCVP^)D^ i

Under special conditions the matrix D may not possess an inverse, and the deem-bedding procedure fails. This may occur when certain non-reciprocal elements, such as isolators, are considered part of the package.

In order to find the matrix transformation matrices P and D in terms of the Y

parameters of the package and the active device, some quantities are defined:

Yp : four-port admittance matrix of the package.

Yd '■ two-port admittance matrix of the active device.

Ypd : two-port admittance matrix of the packaged device.

Cp : noise correlation matrix of the package.

C,j : noise correlation matrix of the active device,

Cpd '■ noise correlation matrix of the packaged device. np : vector of the noise current sources of the package. jl : vector of the noise current sources of the active device. f. : identity matrix of order k.

Ir. Figure 6.13 the active device is shown with noise current generators and terminal currents and voltages. !:s Figure 5.1 1 the package is shown wita two ports to the outside world and two ports to 'he active device. It is seen tha.t the active device is a two-port, the package a four-port and the packaged device a two-port. The terminal voltages and currents of the package can be divided into two groups one belonging to the external ports (with subscript e) and another belonging to

126

6. Noise of embedded networks

Ко J3

<M Vd4

Figure 6.13: Active two-port with noise current generators.

Figure 6.14: Package with noise current generators.

the internal ports (subscript i). In this way the following vectors are defined:

6.4. Calculating noise parameters from deembedded data

127

In the same manner the package noise source matrices are defined:

■ 1 n^p

n!,

. Ti'2 J 71 pi — np ii

ПА

3 1

The admittance matrix of the package is partitioned into four 2x2 submatrices:

У,„ : У,

УР =

(6.83)

v.. i Yi,

and the signal and noise nodal equations for the package can be expressed in terms of the above vectors and matrices either succinctly as

or in partitioned form as

г = YPv + np

— YeeVe “Г Уё: t-1; 4" Пуе г, = Yiev, 4 YuVi -r npt-

(6.84)

(6.85)

(6.86)

A similar procedure is followed for the active device. With reference to Figure 6.13 the following current a.nd voltage vectors are defined:

4 =

and the nodal equations for the active device can be written as

4 = YiVi + jd (6.S7)

Applying the boundary conditions

■ /й ’ Vdz h

. ^. 1 — . ки : Jd — . j* .

to Equation (6.87), inserting (<5.37) into (6.86) and solving for t.\ the following equation is obtained:

vi = - (Ys 4 y/-';y„Lv -f nr, ! j,) {{;:<?]

Inserting Equation (6.88) into (6.85) gives

i, — YecVe — У.; (Yu 4 Yi)^' (YtoV, 4. np, - J,l) 4 npr.

This formula can also be written as

128

6. Noise of embedded networks

ie = Y„v„ + in (6.89)

where the first term represents the signal component of the current and the second term the noise component. It is evident that Ye represents the admittance matrix of the two-port packaged device. It is given by the expression

Ye = Yee + DYte (6.90)

where the matrix D is defined as

D = -Y,,i{Y;, + Yi)-1 (6.91)

The noise matrix term in denotes the sum of the contributions of the package

and the active device. In matrix form it can be represented as

г„ = P np -r Dji (6.92)

where

P = %'-D] (6.93)

Now the correlation noise matrix of the packaged device

Cpd = (6.94)

where (■••) denotes the ensemble average over processes with identical statistical properties.3 As no correlation exists between the noise sources in the device and the package,

Cpd = P {nvn]p) P] + D {jp jt) D] (6.95)

The definitions

Cp = (npn*) and Cd = Up it) (6.96)

lead to

Cpd = PCpPf + DCdD■ (6.97)

wMrh is Equation (6.81). This equation can be reduced to one involving matikeb of order 2 only, when Cp is partitioned in the same way as YP (Equation (6.83)).

^pd ~ "b О Cie ~r* On O' “f* D [Cii C,l) (6-98)

"Please non? that the statistical noise .averages, hence correlation matrices, are normalized to

2 fc To Д/, where fc = 1.33 x JK'1 is Boitzmanrrs constant, To = 290 К is the standard

noise temperature and Д/ is the noise bandwidth in Hz. The factor 2 (instead of 4) is chosen because both positive and negative frequencies are included, fn any case, this factor cancels out in the final noise parameter expressions.

6.4. Calculating noise parameters from deembedded data

129

The deembedding procedure is to find C,i from (6.98) and thus Cd = D~l (Cpd - Сес)0'-' - Clt! Df_1 - D~l Cei - Сц (6.99)

This equation is equivalent to (6.82) but it includes only matrices of order 2. It now remains to find the expressions for the correlation matrices С.~аr and CD for the packaged device and the package in order to be able to compute the correlation matrix Cd for the device from (6.99).

Twiss [iOj has shown that for a linear and passive network, such as the package, which generates only thermal noise, its noise correlation matrix is

**38**> 39 40 41 42 43 44 .. 85 >> Next