# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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N

■Fs 5 ц ^22

I )

B\ Si 2 ^2

(b): Noise free two-port (an — 0, bn = 0)

Figure 4.9: Flow graph illustrating the Tee definition.

Calculating the exchangeable output noise power from Figure 4.9 gives

II Cf Г. 12

<v“ , = P-"

e, out — r - -j _ j 9'J2 v '

I1 - S^rL;2

P 2

i - |-?У2

j\ S’o: };h^ — .S o

x ( h-Tc—r—! с——~T~c ■ c—^ i ) (4.44)

у 1 - \S\ii s T bi2b2li S*- L + ^221 l) л- ^ 11 «Э221- 51- L 1 /

where с с г

^12^21 *- 5

.54

4. Noise parameters

|c £hp mitniit rsfl

put reflection «.о^шсьйил oi in.e two-port, and

_ |1 - 5-kril2 ,ln l2v

Pi —;-----ГсГ " \\вг\ )

1 “ l^2'2i

= P 2

11 - ^TlI2 ' ~ P22I'

S-nBs

16)

11 — (5’иГ5 + + ■УггГ/,) + 5 ц ■SmI'sI1

By definition Ar“ou( must be identically equal to Nf! out, from which

Bs — brl (in

and thus

- (|0s|2) = {(rs6« - on)(rsbn - Qn)*>

= ( K|2) + lr5|2( l*n|2> - 2Re[r5(a;b„)] (4.47)

From Equation (4.42), this can be expressed as

Pi^,s(l - |Г5|2) = (|a„!2) + |Г5|г(|6„|2) - 2Re[r5(a;6„)] (4.48)

The exchangeable source noise power determines the extended source noise temperature TM. Thus

,VC,S = kT„Af (4.49)

where Д/ is the noise bandwidth and

(K(2> + |rs!2(N2> - 2Re(rs(a;M] ,irn,

= p* (4-50)

From Equations (4.50) and (4.43) the sign ot T!e as a function of the signs of the

source impedance and the input reference impedance can be found as shown in Table

4.2.

In Equation (4.50), the noise waves are represented as (jo„j2) and (jivj"), which can be regarded as the available power of the ingoing and outgoing power waves at port 1, and as (a‘bn), which represents the cross-correlation between the ingoing and outgoing aoise powfti waves. A set of noise parameters, (T„, To, T7 and .p-,). called T noise parameters, are defined as

(|a„|2) = к TaAf (4.51)

<i6„j2} = к TpAJ (4.52)

(a'nbn) ^ kT-,AfpJ'^ (4.53)

4.2. Noise power waves

Re[2sj RefZij !Г J 11 ^ 1 Гее

> 0 > 0 < 1 > 0

> 0 < 0 > 1 > 0

< n > 0 > 1 < 0

< 0 <0 < 1 < 0

Table 4.2: Magnitude of |Г5-1 and the sign of T?e as functions of the signs of Re[Zsi and Re[^].

From this definition, the extended effective input noise temperature is given by

Ta 4- |Г5|‘Т3 — 2 jr^'l T-y cos(^s r-v)

Tee = Pi-------------------------fTj^T--------------------~ ^,>4)

It should be noted that the definitions. Equations (4.51) - (4.53), are the same as Meys’s [6,7]. They are, however, used in a different wav. as shown in Equations

(4.37) and (4.54) (for the real part of Zj positive). To distinguish them from each oilier, the indices have been chosen differently.

The extended noise factor as a function of the T noise parameters and the зоитсе reflection coefficient is found from inserting (4.54) into (3.12):

^ , , Ta + j - 2 jFsI cos(95 + y>7)

Fe = 1 + Pi -------------Tf-r,-----ГР~7ТТ--------------- (-i-5o)

Toil - |rsP)

From the complex correlation coefficient (a*b,,)/|(in|2) (\bv.\2) it follows that

0 < T-, < \fTaT)i (4.56)

where Г-, = 0 corresponds to no correlation between the ingoing and outgoing noise power waves and Г-, = \JTa Тд corresponds to full correlation.

4.2.2 T noise parameters

Another set of noise parameters - the T noise parameters which are Ta. Tj. and Г-, expfj:f-t) - lias been defined above and the extended effective noise temperature 7,,e expressed by the Г noise parameters in Equation [ 1.5-1). T,„ ran also be written as circles for constant T,c in the source reflection coefficient plane. Equation (4.54) Ccin be written 3,s

f

4. Noise parameters

which, for Tee constant, is an equation for a circle with centre Гст a.nd radius Rj given by

Гсг = ^■■-Гт-Т- e-^ (4.58)

1.3 -Г Pi i«

Я - / 4 _ Га - PlTe.

T V(Г/J + PiTee)2 Tp + PlTce (-o9)

When Гее is constant, then from Equation (3.1'2), i%, = 1 + T,,JT0 is constant.

Therefore circles can also be drawn for Fe constant with centre and radius given

by substituting Tee = (Fc - l)7o from Equation (3.13) into Equations (4.58) and (4.59).

If Rj — 0 then two extrema for Tec (or Fe) are determined. These and their corresponding source reflection coefficients are

Tcem.r. = I [pi (Ta - T,) + У(Га + I»2 - 4Z?] (4.60)

Fem,n = 1 + ~ [pi (T. - Tp) + y/(Ta + T,3y -4H2] (4.61)

for

(4.62)

(4.63)

and

for

Гsot = TsOF

2 T-, e~J ^

Ta + T0 + p, y\ro + Tpy - 4 T2

Ta + Tg — Pi J(Ta + Tg)2 — 4Г2 . V -j ч>-,

T - - ■Lee max — ^ 2 Т.,

[p, (Ta - Te) - y{Ta + 'Tj)2 - 4X?|

F - 1 x e max — L + l^1 '^a _ -^9) _ \Л^“ ~

Tsot' = Г SO F'

2 Tye~->^

+ j/j — Ft + Ipy — 4T7

T3 + T3 -f p, v/(Ta 4- Tj)2 - 4T2

(4.64)

(4.65)

(4.66)

4.67)

It should be noted from Equations (4.60) and (4.64) and further from (4.61) and (4.65) that

T„mm > T„m„, and F.> F~ tnnj; (1.68)

4.2. Noise power waves

57

Figure 4.10: Contours for constant T.e in the source reflection coefficient plane for pi = + 1 with data from Example 4.3.

The circles divide into two parts by a singularity when the centre goes towards infinity for Tg + p\ T,c = 0. These two parts are separated bv a line n. which is determined as

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