# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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Definition 3.9 The average extended operating noise temperature of a multi-port Teop is defined as

eap ' kzUiTG'Uf-Wf 1 â€˜ M3)

where N'i(f) is the total noise power density delivered to the equivalent noise free immittance of the load circuit at the output frequency /, Ðº â€” 1.3807 x 10~23 J K-1 is Boltzmannâ€™s constant, I is the number of signal responses from all input ports where signals are applied an d GeT,i(f - /,) is the extended transducer gain from port response i to the output port at an input frequency / â€” /, which causes the output frequency /.

Note 1: 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 (3.23).

Note 2: The resulting Nâ€™r {/) may consist of parts with opposite signs at a frequency / and also jVÂ£(/) may change sign with varying frequency. This means that when active immittances occur great care should be taken in interpreting the value of Teop.

Note 3: All ports other than the output port are considered input ports and they should be loaded with any passive cr active immittances except short or open circuits and the extended noise temperatures oi these loads are the actual ones.

Note 4: The denominator of Equation (3/23) includes only gains from input respouses where signals are applied.

From

36

3. Noise characteristics of multi-ports

follows

_ , z!:tJfoxTcm(f)G'T,,(f-f,)<if

LLfiÂ°G'T,i(f-fi)df

Ef=i

(3.24)

As stated in note 2 care should be taken in interpreting the value of Tcop when active immittances occur. Even a very noisy transducer may have almost zero Teop as the noise power may flow in opposite directions at different frequencies. On the other hand in all those cases where the noise power parts delivered to the load add up with the same sign the average extended noise temperature is a very descriptive quantity.

3.3.4 The equivalent noise bandwidth

For each response a gain function exists. In many expressions in noise theory this gain function is integrated over all /. It is often convenient to replace the integral with the product of a fixed reference gain GejT (often the maximum gain) aud a bandwidth which is called the noise bandwidth B,\, when the gain function is the (extended) transducer gain. Even if the extended transducer gain can be used the interpretation is most clear for the common transducer gain function which is possitive for all /.

Definition 3.10 The (extended) equivalent noise bandwidth Bn is defined as

^ [Hz] (3.25)

GeTr

where Gejv is the reference value of the extended transducer gain function Gct{}) (at a corresponding reference frequency fT).

Note 1: Usually Ger r is chosen as the maximum value of GeTr{f), but it can be chosen freely, so it is important to know which Gctt (or fT) is used when the noise bandwidth is specified.

As seen from Figure 3.4 it is important to specify which Gerr or /,. is used when the noise bandwidth is used. It should be noted that ail axes in Figure 3.4 are linearly scaled and that the area under the gain function is equal to the area determined bv GcXt -0/V- The ratio of B\r/Ð’-Ñ†Ð² for n Â£C-circuits with identical Q factor, resonance frequency and without any mutual coupling is given in Table 3.3.

Example 3.7 A tuned amplifier consists of an LC-circuit, a transistor and another LC-circuit with the same Q and resonance frequency as the first, and no coupling exists

3.3. Average noise quantities and the noise bandwidth

37

(a) (b)

Figure 3.4: Examples of noise bandwidths: (a) a single LC-circuit, (b) a mistimed double LC-circuit.

n: 1 2 3 4 r /-Â» 0 j 0 00

BvfBw .v / -lsjiio â€¢ 1 kT Ð¥.Ð™, 1.221 1.155 1.13 l.ii | l.io 1.06

Table 3.3: for n uncoupled identical LC7-circuits.

between the two resonant circuits. The amplifier shows a B3dB of S kHz. When the average noise factor of the amplifier is F = 4.0 and the noise temperature of the source is Tems = 580 Ðš (~ 2 Ð“Ð¾) the problem is to find the signal to noise ratio at the load for an input signal power level of i pW.

Top ~ Tsms + (F â€” 1) To =

5l = J?, = Si

Ni kTopBy Ðº 5 T0 1-221 Ð±Ð·Ð°Ð²

It is supposed that the signal gain Gs = Si/S; is equal to the transducer gain Gt ~ Ni!{k TJp By), which usually is the case. If the power spectra of Sl and Ni are different due to a special type of modulation and/or very frequency dependent Tâ€ž= the above calculation may give a wrong result. The IRE definition, which uses G's in the definition of Top, will of course give the correct result, but the problem is that the signal spectrum then must be specified together with the Tnp figure to give a full description of the noise In the system and also when measuring the Top.

The response factor defined in Equation (3.20) can be expressed with use of noise bandwidths for multi-ports which are not loaded with short or open circuits

5 To

5120 ~ 37.1 dB

38

3. Noise characteristics of multi-ports

at any frequency as

Ð¹ = ELVgjr.ifl.v.i

ELl G'.Tr.i BN,i

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