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Friis formula

Mathematics

Friis formula is very important when receiving signals at higher frequencies. The formula describes the total noise-factor of an amplifier-system.

Friis formula;

NFtot = NF1 + (NF2-1)/G1 + (NF3-1)/(G1*G2) + ...

NFx and Gx are not in dB, these values must be in linear ratio. We hereby directly see that the first factor, NF1, totally control the NFtot if NF1 is large. We can also see that having a large amount of gain in this first stage will give us a lower total noise-factor. The rule is therefore, low noise and much gain in the first stage! This is not the only truth since the amplifier must also be capable of handling the incoming signal. Therefore one must sometimes lower the gain if the amplifier should be able to cope with the incoming signal.

Type of components

All components in a receiver-system have a noisefactor. The noisefactor canít be negative! Active components, as amplifiers, are usually specified in datasheets. All passive components, as filters and attenuators, has the same noisefactor as attenuation, ie loss is the same as noisefactor.

Example

Let say we have a coaxial-cable with 2dB loss, 2dB equals about 1,58 times, a pre-amplifier with 1dB noise-factor and 20dB gain,1dB equals about 1,26 times and 20dB is 100 times.
In case one the pre-amplifier is mounted before the cable;

NFtot = 1,26 + (1,58-1)/100 = 1,2658 = 1,02dB

In case two the cable is mounted before the pre-amplifier, ie if we put the pre-amplifier indoors at the receiver;

NFtot = 1,58 + (1,26-1)/(1/1,58) = 1,97 = 3dB

Conclusion

One can see that the noisefactor of the first amplifier-stage and all attenuation before the first amplifier can be added. This means that it is very important to have a low loss cable from the antenna to the pre-amplifier.
Noisefactors are sometimes expressed as noisetemperature.


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