# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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"See Appendix C.

24

3. Noise characteristics oC multi-ports

but in some cases more tiiau one response is considered, the noise factor was the first to be “extended” to active sources [4].

Definition 3.5 The extended noise factor of a multi-port transducer, iv (at a specified input frequency or specified input frequencies which all give an output response at the same output frequency) is defined as the ratio of (1) the total exchangeable noise power density at the output port (and at the corresponding output frequency) 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 exchangeable noise power density at the output port which originates from the signal source (or sources) at the input frequency (or frequencies) and at standard noise temperature.

Note 1: Fe is not defined for a one-port.2

Note 2: For multi-ports with more than one signal response, either an extended noise factor is defined for each signal response, or part (2) in the definition includes noise from those port responses which are used for the input signals.

Note 3: Fc is a function of the source immittance(s).

For two-ports this definition gives the source noise power density N'eS = к To-The exchangeable output noise power density N'e o consists of two parts: the noise power generated by the internal noise sources of the network N'e and the input noise amplified by the exchangeable power gain of the network kToGe. As definition 3.4 leads to N't = kTee Ge one gets

Please note that Equations (3.6) and (3.7) are only valid for a two-port with a single input and a single output frequency. For multi-ports conversion from Tee to Fe and vice versa is more complicated and is discussed below.

Definition 3.5 is also consistent with H. T. Friis’s previous definition [6] when

- instead of the exchangeable power gain - the available power gain is used. Go. = S0/Ss where Sa and 5s are the available signal output and source powers in

the frequency range of interest Af.

к Го Ge + kTceGe к T0 Ge

1 + ~ (3.6)

T.c = {Fc - I )TQ [K]

(3.7)

Ga k'l oAf

2Unfortunately the expression F* — 1 + T-m/To is sometimes used for a one-port noise factor, which is excluded by the definition.

3.2. Definitions of noise quantities

25

F

N'

From the definition it is seen that

Fe > 1 for Rs > 0

Fe < 1 for Rs < 0

The noise factor for Fe > 1 is often expressed in dB by

F. [dSJ = 10 log Ft,

(3.8)

and then mostly called the noise figure. Also Fc - 1 = Tae/T0 is called the extended excess noise factor.

Example 3.1 In the figure below a two-port consists of three resistors with known resistances and noise temperatures. When the source resistance Rs is known it is possible to determine the noise factor of the two-port by use of definition 3.5.

= 50 ft

«r VJ а ь

Ri = loo a r2 = 100 ft

R3 = 200 ft

Tem, 1 = To ТетпЛ — 1.333 Jo Fdm,з = 1.875 Tq

By use of Equations (2.18), (2.17) and (2.5) the square of the current in the short-circuited output in a bandwidth of Д/ = 1 Hz (|г01/|2) is found as follows:

Ga — s “H G\ — 30 mS

To Gs Tem i Gi

T„

R„

Gs + G[ 133.3 ft =

Ra = = To

33.3 ft

Gi, = 7.5 mS

T- T,m aRa + Tm,2 Яъ , oc rj*

Tc’n-b =-----------r7TY2----------------= L25To

G.. = Gj 4- G3 = 12.5 mS

= 1.50 To

(= U:T,m.cG,A/ = 300 x IQ-24 A-

!f { U-jl1'} divided by 1 Gthe total exchangeable noise power dens?ty Dt the out-p port is determined. To find that part which originates from the source let Tem \ Tcm,2 = Tem<3 = 0 К and repeat the above calculations. The results are

26

3. Noise characteristics of multi-ports

J \; ! 2 \ о л r . ■ i n — 2 4 * 2

\ a i\J л

F = (\ioj\2)/(4Gc)

e {1чя|2)/(4Сс)

Example 3.2 Consider a transmission line made of metallic conductors and therefore generating only thermal noise at the physical temperature of 17 °C (= To) and a loss of L = l/Ge. In order to find its noise figure the source must have the noise temperature of To also. As a transmission line generates only thermal noise the transmission line and source together can be regarded as a one-port at standard temperature and thus its output noise power density is кТц. The noise power density from the source is A;To which is “amplified” by Ge = ifL. The ratio of these two noise power densities determines the noise factor:

F = _*Z°_ = L

e kT0/L

Now let the transmission line have the physical temperature Тц. To begin with let

Tt[ = T0, then

Ktq + kTo/L = к T0 => N'Tq = kTo( 1 - 1 / L)

As Nlj- ls proportional to Tti the general expression is

Кт„ = к Та{1 - 1/L) n kTo/L + kTtl(l - 1/L) f , T,i:T „

^ = -------------kWL--------------- = I + T0[L~l)

Consider a heterodyne system with a normal response and an image response. If it is used for a broadcast receiver the wanted signal is only present at the normal response and the denominator in definition 3.5 includes only noise from the source at the input response frequency, but the numerator contains noise from both the normal and the image responses. If the receiver is for radio astronomy, signals are present at both responses and therefore the denominator in definition 3.5 includes noise from both input responses. Alternatively it may include one response and a separate noise factor derived for each response. The distinction between the two uses of the noise factor is given by calling them single and double sideband noise factors respectively.

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