# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Example 4.2 An amplifier with the noise parameters R.n = 20 П, Gn = 6.4 mS, and K, = 2 + j L4 mS has a source impedance of 50 Q. By adding a lossless admittance

50

4. Noise parameters

(a susccptancc) m parallel with the input of the amplifier the noise factor of the amplifier can be improved. In order to determine by how many decibels, the noise factor without this parallel admittance is computed as

e oid = 1 + \ Gn -f- Rn |Fs + J'т |2 )

(jj 4

= 1 + (64 + 0'020 [(20 + 2)2 + (0 + 14)l)

= 2.000

A lossless admittance in parallel with the input of the amplifier changes the source admittance from Ys = 20 mS to Yj = Gs + j BA where BA is the value of the susceptance. If Вд is chosen such that В4 + By = 0 then the noise factor is at the local minimum. This means that B\ = - B^ = -14 mS and the new value of the noise factor is

iw™ = 1 + ^ (6.4 + 0.020 [(20 + 2)2 + (-14 + 14)2]) = 1.804

From this it follows that the improvement in the noise figure is

AF [dB] = 10 iog —— = 0.448 dB

1 J 0 1.804

If instead of a lossless admittance a complete noise match is performed the Femin =

1.800 could be obtained. It is, however, a much more complicated solution and the extra 0.010 dB, which theory gives, would almost certainly be lost in the real matching network's internal losses.

4.2 Noise power waves

As Y and Z naramet.ers in many applications have been replaced by S parameters it is natural to look for a set of noise parameters which are also based 011 the power waves formulation and could work conveniently together with the S parameters.

[n order to use the Rothe and Dahlke equivalent from Figure 4.1 - redrawn in Figure 4.7 - in the power wave representation, the T parameters are chosen. The representation is in accordance with Kurokawa [10]. Without noise it follows that

' I _ I Т'-л

P/ I I

. J I * *1

The signal power waves are defined as

! b2

f

4.2. Noise power waves

51

Figure 4.7: The Rothe and Dahlke equivalent.

B> = YLzJjlj (4-2S)

2 \/[Re[Z|]| 14 ■>

V2 ■+ Z-> It

A2 - 2vmm a29)

3, = (4.30)

where Z\ and Zi are the complex reference impedances at. ports 1 and 2, which may be complex and aiso active. The only restrictions are that Re[Zi] ф 0 and Re[Z2] ф 0.

Including the noise sources in Figure 4.7 gives

V{ = V - en (4.31)

/J = /i - in (4.32)

Equations (4.26) - (4.32) can be expressed as

Vi + Z\ I\ _ ,r V2 - Z\ 12 ( j, Vj + Z-j I2 j ^

2^ТЩЩ - п2^ЩЩ - ' 2v/[R?[zTTj 1

Vi - z;h _ T_ v? - z;h ^ r v2 + z2r: _ z\\n n

2>/|5ra 121 2 v'TRilZ^I ' 22 2 v'i’RetZ2)> T 2

Here the left-hand side can be defined as power waves Aj and Вь The right-hand side consists of a noise free part and a part due to the TLOISG SGUrCGS. TiUib, iv*’0 noise power waves are defined as

Cn + Z\ ;n

2 v/TRei^jj

( 1-35)

2 -,/jRe[ZiSl *" '

As seen from Equations (4.35) and (4.36), the noise power waves are introduced on

the input side of the two-port. (It should be noted that Meys [61 uses the same b„

but has changed sign on <i„. Aiso Meys has the real part of Z\ positive.)

52

4. Noise parameters

Equation (4.26) can now be written as

Ai

Bi

Ги Tl2 T21 T22

Bi

Gn

bn

(4.37)

It is often practical to express Equation (4.37) by using S parameters. Isolating B\ and В2 it is found that

TI rp rp rp Гр rp

21 л , 1u -L 22 - -М2 21 . i-21

-Ol = ^-^1 + ----------~------------^2 - — On

-Ml ^11 -*11

D 1 * Г12 1

2 = -4l - =— A2 - — an

J 11 111 -<11

which in terms of S parameters can be expressed as

(4.38)

(4.39)

Вi — 5n Ai -f- 5j2^2 — fln 4- bn

B2 = S2\ Ai -f- S22 A2 — 1S21 an

(4.40)

(4.41)

Ai 1

-1

5’n

1

S21 B2

St

Вi S12 a2

Figure 4.8: S parameter representation of a two-port with noise power waves.

Equations (4.40) and (4.41) are illustrated in the signal Row graph of Figure 4.8. It should be noted that the noise power wave an in Figure 4.8 should be treated as an internal noise soume together v>ith the S parameter representation.

4.2.1 The extended effective noise temperature

From Appendix В the relationship between the source power wave B$, and the exchangeable power of the source Pe.c, which is defined in Equation (3.1), is found to be

( |Д?|2) = Pi pc,s(l ~ |Г5|2) (4.42)

4.2. Noise power waves

53

where p, = Re [_Z,]/|Re [Z,]j and i refer to the port number, (|1?Л'|2) indicates the ensemble average of and

Г3 = |Г,|«- = (4.43)

As shown in Figure 3.1, Definition 3.4 for a linear two-port states that Tej. is the noise temperature of the source with a noise free equivalent of the two-port which gives the same exchangeable output noise power as the (noisy) two-port with a noise free source. Figure 4.9 shows the two cases. There the reference impedance of the source is ZrS = and also the reference impedance of the load, This

is so that an easy connection of source, load and two-port can be achieved as shown in Figure 4.9 (b).

ап 1 Ai I _I| S41 В 2

\

■Г s sj S?2

к J

JZ?! Я; a7

<!>

(a): Noise free source (Bs = 0) Bs 1 1 -4l 1 $21 Bj2

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