# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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3.3. Aver age noise quantities an d the noise bandwidth

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band are defined. These problems are exposed and the equivalent noise bandwidth is defined.

In the IRE definitions of average noise quantities the transducer gain is used. When the definitions are extended to active sources it is necessary to replace it with the extended transducer gain from Equation (3.3), but the really confusing problem is when source immittance or load immittance changes sign in the frequency band of interest. This leads to the possibility of zero exchangeable noise power and gives results which can not be given a physical meaning. If the IRE definitions are going to be extended to active devices it could be more interesting to look at the flow of the noise powers. Then the results may be given an understandable physical meaning, but the equations will be rather complicated. As extended average noise quantities are used rather seldom, it is chosen to keep the mathematics as simple as possible, and thus in cases with both positive and negative noise power flow the values of the noise quantities do not right away give an impression of whether an amplifier has good or bad noise properties.

3.3.1 The average effective noise temperature

The definition of the average extended noise temperature is given as a formula and if all source immittances for all input frequencies have the same sign and the load immittance also has the same sign for all output frequencies a more explanatory equivalent formulation can be given.

Definition 3.7 The average extended effective (input) noise temperature of a multi-port Tse is defined as

-г- - Е'=| Г г,(/)сгт,,(/ - h)df

- r if - ‘ ‘ (3-lb)

where Гее(/) is the extended effective noise temperature of the multi-port as a function of the frequency /, and GeT,Af - ft) is the extended transducer gain from port response i to the output port at an input frequency / — /; which originates a corresponding output frequency /.

If Tee(/) for all f have the same sign and also if GeT.Aj — ft) f°r all / and г have the same sign, Equation (3.18) is seen as a weighted average of T„. In words this can be stated as follows:

For a multi-port with a load immittance which has the same sign for all frequencies and with source immittances which ail (and for all frequencies) have the same sign, the average extended effective (input) noise temperature Tee is the extended noise temperature applied to all input immittances of a noise free equivalent of the multi-port which delivers the same noise power to the load as the noisy multi-port

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3. Noise characteristics of muiti-ports

3.3.2 The average noise factor

For the average extended noise factor the same comments apply as for the average effective noise temperature.

Definition 3.8 The average extended noise factor of a multi-port Fe is defined as

ZUJ^C'TAf - fi)df {" '

where Fc(f) is the extended noise factor of the multi-port as a function of the frequency /, G,jti(f — fi) is the extended transducer gain from port response i to the output port at an input frequency / — /. which originates a corresponding output frequency /, I is the number of signal responses and /+./ is the total number of responses from sources to load.

If Fe(f) has the same sign for all / and GejAf — fi) has the same sign for all / and i, Equation (3.19) can be seen as a weighted average of F.. In words this can be stated as:

For a multi-port with a load immittance which has the same sign for all frequencies and with source immittances which all (and for all frequencies) have the same sign, the average extended noise factor Fe is the ratio of (1) the total noise power delivered to the load when the extended noise temperature of the source (or sources) is/are the standard noise temperature ('290 K) at all frequencies (and input ports), to (2) that part of the noise power delivered to the load which originates from the signal source (or sources) at standard noise temperature.

Defining the response factor

- . ELY STG,TAf-h)df .

K - w—7^7'—71—77777 ('3-20)

E;=1 Jo fi)df

as the ratio of “gain bandwidth product” of all responses to that of the signal

responses, the relation corresponding to Equation (3.12) is proved by

/~t~~ \

+ 1 (3.21)

To

fi)df ('Liii It'T"Af)G-T,i(f ~ h) if + , ) t=l Jo ^-reT,iif ~ fi) ч/ V ‘ ’ ^i--l Jo GeT.if / ~ Ji) ,:7 J

E:=1J ft Fed) G.tAJ - /,-) df EL J0~ G>r.i(/-/;)<*/

From Equation (3.21) Tee is found to be

'K

q. e. d.

n - 1 | To (3.22)

3.3. Average noise quantities and the noise bandwidth

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3.3.3 The average operating noise temperature

The IRE definition [5] uses signal gain which involves the signal bandwidth in the definition of the average operating noise temperature. As many new modulation schemes have been developed in the last decades it seems inconvenient to tie a noise definition to the type of signal. Thus the signal gain is replaced by the extended transducer gain from, the signal input ports to the output port in the definition of the average extended operating noise temperature.

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