# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Note 4: The denominator of Equation (3.14) includes only gains from input responses where signals are applied.

30 3. Noise characteristics of multi-ports

where Nf = |Г|‘ fc Temi, I is the number of input responses with signals and I + J is the total number of input responses, it follows that

т-^I+J rp Г1 N'

Teop = KT" + ^ ^ [K] (3.15)

^ ЧУ —— - 1 y s_,» , V ■ O'

1 2

In most cases term 2 in Equation (3.15) is very small compared to term 1 so

Теар ~ КГее + ^‘ CT'‘- [K] (3.16)

Ei = l GsT,i

is a very good approximation. For single response two-ports Equation 3.16 reduces to

Teop ~ Tee. + Tem [K] (^-17)

In order to keep track of the signs a three-port is considered. The two signal input ports are loaded with an active and a passive generator immittance and the possible sign combinations are shown in Table 3.1. If instead only port 1 is applied with a signal the sign combinations shown in Table 3.2 occur.

Rl N'l Rsi Tem. I R-S2 Tem,2 G*t,i G*T, 2 £L7’-‘g-.- Teop Eq. ЗЛ6 Teop Eq. ЗЛ4

+ - + - “Г + + + + +

+ - + - + - - - - -

- - + + - + - - - -

- - + + - - + + + +

Table 3.1: Sign combinations regarding Ttop for a three-port with two signal responses.

N'l tesi Г.т.1 Ksi T^,2 GeT,l GcT,2 EL, G,r„ ■дт„ &eT, 1 T, op Eq. 3.L6 T, op Eq. 3.14

+ - ~r - ~r -f- - - -

T - + “ + - ~ ~ - -

- - -Г + - -f - - - -

- - + + - - - - - -

Table 3.2: Sitrn combinations regarding / ter a tbree-port with on*’ signal In general

Teop > 0 when —r~~------ > 0

^i = L

3.2. Definitions of noise quantities

31

Teop < 0 when ^1\- < 0

Ы= 1 GeT,i

where I is the number of inputs with signals applied and Ri is the real part of the load impedance.

In most cases where Геор is used both the source and load will be passive and most often a single response transducer is considered.

It should be noted that Teop is used in the common “figure of merit” G/T, where G is the antenna gain and T — Teop.

Example 3.5 The importance of only considering the lossless part of the load circuit is illustrated by calculation of the extended operating noise temperature of an attenuator. Consider the one in Example 2.3 and load it with Zl = 75 + j 25 fi with a load noise temperature Ti — 580 K. Computing the exchangeable power gain of the attenuator one gets Gc = 60.0 x 10-3 and the method used in Example 3.2 determines

F.j. = \/Gs = 16.67

=> Гее = (Fe — 1)Го = 45 43 К

This determines term 1 in Equation (3.15) to be Tee + Ta = 7543 K. In order to

compute term 2 let the load circuit be substituted by an equivalent noise free impedance

Zl in series with a noise voltage generator whose voltage is determined by

(je|2> = 4fcTr Re[ZL] Д/

The power delivered to the noise free part of the load is

„ = Rg/ ^ e* \ _ WMZl]

Z0ut + Zl (Zout + ZLyf \Z0ut + zL\2

In order to relate this noise power to a part of the operating noise temperature it is divided by kGej^f where Ge_r for Zout = 50 fi is 55.4 x 10-3:

ArL2 4 (RefZf,')2 Гг, ______

= I4oUU К

kG,T^f \z,ut 4- Zb\2 G,T

=► Г,ор = 22043 К

if the iRE definition is used the power density fiow to the load is the same as above represented by noise temperatures, but the power density flow from the load to the output of the attenuator wii! then be subtracted, i his part of the power density fiow is determined by the output terminal voltage U, the current into the load circuit I and

/ jr|2\ _ .j f. r. ptfZi} д f:

Xl2,rs = Ыи I’) = Re ( "'0,n —----------—-- Л

^out 1" (Zout “I" &l) /

- (|e[2> Re[Zout]

32

3. Noise characteristics of multi-ports

Referring this power to an input tempcratuic; gives

Nli<re _ — 4 Tl ReZcut ReZb]

kG'T&.f GeT\Zout + Zl\2

= - 9667 К

FsopiRE = — 2124 К

- a somewhat misleading figure. Further information on this is given in [7].

Example 3.6 Let a 12 GHz satellite-TV receiver require a G/T = 14 dB. If a 90 cm reflector antenna has a gain of 38 dB and the noise temperature Tema = 90 K, which noise factor is then the maximum for the receiver?

T = Teop = (38 - 14) dB К = 24 dB К ~ 251К

=> Te, x Tcop - Tcma = 161К

T

=> Fe = 1 + —^ = 1.56 ~ 1.92 dB

io

If the actual receiver has a noise figure of 4 dB then the antenna diameter D, where the antenna gain is proportional to D2, is determined by

F, = 2.51 (~ 4 dB)

=> Гее = (Fc - I) T0 = 438 К

=> Teop и 528 К - 27.2 dB

G = (14 + 27.2) dB = 41.2 dB

-Д- = 10^ = 2.10

^90 cm

^ D - 0.9У2Л0 = 1.31m

3.3 Average noise quantities and the noise bandwidth

The noise definitions in Section 3.2 above яге ail defined at a single fa spot) output frequency and therefore often called spot noise quantities. They are functions of the frequency and most useful to describe the noisp behaviour of the circuit. Sometimes it is practical to characterize the noise properties in a given frequency band with one number instead of the more useful frequency function and therefore the average noise quantities are introduced. If the gain function of the circuit is not very flat there may be some confusion as to how the reference amplification and the frequency

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