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# Theory of Linear and nonlinear circuits - Engberg J.

Engberg J. Theory of Linear and nonlinear circuits - Wiley & sons , 1995. - 154 p.
ISBN 0-47-94825
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In Figure 3.2 the /-f* J inputs can be loaded, by active as well as passive immittances, but signals, which are supposed to be imcorrelated. are applied to only I of them. Suppose the extended noise temperature Tcc is <nvpn. thpn the exchangeable output noise density from the transducer itself N, can be calculated by
l+J
N[ = kTee (3.9)
3.2. Definitions of noise quantities
27
I
Г+ 1 T + J
Figure 3.2: A multi-port transducer with I responses with signals and ,/ responses without signals.
where Ge.i is the exchangeable power gain from port response i to the output port. If the extended multi-response noise factor is chosen, definition 3.5 leads to
fc Tn
kT0 ELiG^
;з.10)
From Equations (3.9) and (3.10) the relations between Tee and the multi-response Fe can be derived. Denoting the spot response factor
Г-*1+J (~i
Ei=l Ge,,
the relations are
Fe = И 1 +
Гее = {Ц ~ l) To [K]
(3.11)
(3.12)
(3.13)
Example 3.3 Consider a three-port transducer with extended noise temperature Xe. =
5 To and the exchangeable power gain from ports 1 and 2 to the output G.,\ = 15 and Ge;i = —12 respectively. From Equation (3.9) and Definition 3.5 the single response extended noise factors are determined:
N'e = kT„(G,,i + G,,i) - 60.0 x 10" л
A;T0 (0, i т G,,i) + .V!
\Y Hz
к To yGe,i i- Ge,-2 ~kT0Ge,2
-1.5
The same results are derived from Equation (3.12).
28
3. Noise characteristics of multi-ports
When calculating the multi-response extended noise factor, Liie spot response factor TZ — 1 and Definition 3.5 gives
= кТ0(С'Л + G't) + К = fin к To (G’e.l + Се.г)
which is the same as Fc derived by Equation (3.12).
Example 3.4 Consider a mixer with exchangeable power gain for the normal response and for the image response Gei2. From definition 3.5
p — ^Tq(Gc ,1 + Gt'i) + N’c
e,SSB ~ к To Ge,i
kTo(Geji -f Ge,2) -b iVj
e ,i_y o .□ — ' ~
’■’DSB kT0(G'A + G'l3)
Fe,SSB = -C,1J fe,DSB = К Fe,DSB
Ge,l
If GCi 1 = Gc, 2 = Ge => FCissb = 2Fej}SB
3.2.3 The operating noise temperature
The operating noise temperature Top - sometimes called the system noise temperature - is used to characterize a system under operating conditions and includes all noise contributions which add to the output noise power delivered to the load. Consider the system shown in Figure 3.3 where three sources contribute to the noise power density delivered to the load. One contribution is the noise from the source which could be an antenna with known noise temperature. For the time being only passive sources and loads are considered. The available source noise power amplified by the transducer gain determines the contribution from the source to the noise power density delivered to the load. The next noise source is from the two-port considered (e.g. a receiver). This part is given by the Tee of the two-port and contributes by definition 3.4 as the source at the noise temperature of Tce. The final part is from the load itself. The load generates noise; some of this noise is absorbed by the output port of the two-port (and changes T,,, but often only slightly) and the remaining part is reflected at the output port and then contributes to the noise delivered to the load. This third part is usually very small compared to the other two. While these are amplified by the normally very big transducer gain of the system, the load-generated noise is reduced as it is divided between the load and the output immittance of the transducer.
The IRE definition [5] has been changed in two respects. First it has been extended to cover multi-ports with active as well as passive sources, and secondly
3.2. Definitions о/'noise quantities
29
К 1 r:e Uj
Figure 3.3: Noise contributions to the load.
the noise power density considered at the output is the power delivered to the noise free part of the load circuit. This is because the load noise often flows in the opposite direction to the noise delivered to the load and thus reduce the operating noise temperature [7]. The noise generated in the load and reflected at the output port is included as in the IRE definition.
Definition 3.6 The extended operating noise temperature Teop is defined as
1eap = — [ " [K] (^-14)
^ Z^t=l GfiTyi
where N'L is the total noise power density delivered to the equivalent noise free immittance of the load circuit, к = 1.3807 x 10~23-J K-1 is Boltzmann’s constant, I is the number of signal responses from all input ports where signals are applied and Gtx,i is the extended transducer gain from port response i to the output port.
Note 1: There is only one output port and at that port only one frequency is considered.
Note 2: Noise generated in the load and reflected at the output of the transducer back to the noise free equivalent of the load is part of the numerator in Equation (3.14).
Note 3: All ports other than the output port are considered input ports and they should be loaded with any passive or active immittances except short or open circuits.
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