# Theory of Linear and nonlinear circuits - Engberg J.

ISBN 0-47-94825

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calculated as shown for the following component values: G\ = G? = 10 mS, Д3 =

200 ft, Gn,2 = 20 mS, Я„,з = 300 ft and Tm = 2 T0.

R =

I

G1 + G-

+ Д3 = 250 П Л Rn = ~ R = 2 R = 500 П

*0

-йп.1+2 = Rn. ~ Rn.3 — 200 f! л 2 — ^n.i~2 {G1 f C2)2 = SO m3

Gn. 1

G’-м = С71Д+2 ~ = 60 mS or ТтЛ

-Г,, - R T„

2.3.2 Series and parallel connections of one-ports characterized by extended noise temperatures

Omitting any deterministic voltage sources t.he series connection of ! one-ports characterized by their impedances and extended noise temperatures is shown in Figure 2.4. As the imaginary parts of t.he impedances do not. «cncratc noise only the real parts - the resistances - are shown in Figure 2.4. Inserting Equation (2.4)

16

2. Noise in one-ports

R\ >^em,l Л2 , 7jm,2 ^3 ,^mi3

Figure 2.4: Series connection of noisy one-ports characterized by the extended noise temperatures.

into Equation (2.13) gives

I I

( I e |2) = J^ikT^iltiAf = AkMY.Tem.iRi [V2]

i=\ i=l

From this equation and Equation (2.11) it follows that the extended noise temperature for the series connected one-ports is

T

-L PI

J2Li тет,, R<

[K]

(2.17

E/=i й;

Аз TemiI and Ri have the same sign Equation (2.17) is valid for both active and passive one-ports. The only requirement is that J2{=i Ri / 0.

Similarly, for I parallel connected one-ports one gets

ZLl Tern., Gj E/=lG,

[K]

(2.18)

Example 2.2 A tunnel diode circuit is regarded as a one-port with the following data: C$ = —2 mS and ТегП1з = — ~Tq. It is loaded by an admittance with Gl = 10 mS and TKm,L = Tq. The extended noise temperature of the parallel circuit is determined from Equation (2.18):

T,

em,S

Gs

TCm,L Gl _ —7 Tp • (-2) + Tp ■ 10

Gs + Gi

-2 + 10

3 Ta

Example 2.3 In order to isolate an unknown external antenna from an internal ferrite antenna in a receiver the network shown in the figure is used. The nominal antenna impedance Za = 300 fi and its noise temperature Xj ~ 3 000 K. In the network Rt = 300 Ji, R-2 = 100 fi and Дз = 62.5 fi and they all generate thermal noise at the ambient temperature of 17 °C.

1—!

R L

c.-_ с___

eceiver спе агисина auu hciwuin uii uc

regarded as a one-port with a given impedance, noise

temperature and signal voltage. The solution is found by computing the parallel connection of Za and R\ (Equation (2.16)- Rj , Tnj), then the series connection of Ri

2.3. Calculation with noise quantities

17

and R2 (Equation (2.15)- Rjj , Тп>ц) and finally the parallel connection of Rij and

Equation (2.16) - Reg T ^ L n

Ri = 150 Q л Tn.I II 1 Т Sl "Т 0 G/ - 1645 К

Rn - 250 n л 'Гп.п - г , „я. nJ Кц j_ Т ^ + 0 RT7 _ = 1103 К

= 50 fl л т 1 я.е-7 •т G-I - + Т0 = 1 V-7“-4 = 452.6 к

The network divides the antenna voltage by 10 (and the power by 16.67) and the antenna noise temperature (and noise power) is divided by 6.6 so the signal to noise ratio is only slightly degraded (~ 4 dB). At long- and medium-wave bands, where the antenna noise temperature is above 300 000 K, the degradation of the signal to noise ratio is hardly noticeable (« 0.09 dB), so a lossy network can be used to isolate a receiver from an unknown antenna impedance if the antenna noise temperature is relatively high compared to ambient temperature.

2.4 References

[1] Nyquist. H.: “Thermal agitation of electric charge in conductors”, Physical Review, voi. 32, pp. 110 - 113, July 1928.

[2] Friis. H. Т.: “Noise figures of radio receivers”. Proc. IRE, vol. 32, pp. 419 - 422, July 1944.

[3] "IRE Standards on Electron Tubes: Definitions of Terms, 1957, 57 IRE 7.S2”, Proc. IRE, vol. 45, pp. 983 - 1010, July 1957.

[4] “IRE Standards on Electron Tubes: Definitions of Terms, 1962, 62 IRE 7-S2”, Proc. IRE, vol. 50, pp. 434 - 435, March 1963.

[5j Haus, H- A. k, Adler, R. B.: ‘‘Circuit theory of linear noisy networks”, Technology

Press and Wiley, New York, 1959.

[6] Engberg, Л. & Larsen, Т.: '"Extended definitions for noise temperatures of linear noisy

one- and two-ports”, IEE Proc. Pari H, vol. 138, pp. S6 - 90, February 1991.

3

Noise characteristics of multi-ports

The multi-ports considered in this chapter can be single response two-ports, multi-response two-ports or multi-ports with one or more responses at each port,. In single response two-ports only one single input frequency gives an output at the corresponding output frequency and - of less importance in noise theory - this input frequency leads to no other outputs at other output frequencies. A multi-response port can be considered as many ports as there are responses at the desired output frequency at the output port. This means that multi-response ports are treated as multi-ports with as many ports as the sum of ports times responses requires. It is important to note that only one output port is considered and for spot frequency analysis only a single output frequency is of interest. If more than one output port is of interest each output port is treated separately one by one.

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