86 BELL SYSTEM TECHNICAL JOURNAL 



used in commercial circuits. The natural limits to amplification which 

 will be discussed in this review are therefore of very practical interest. 



Thermal Agitation ^' 2- 3. 4, 5, e 

 The free charge of any conductor is in random motion in equilibrium 

 with the thermal motion of the molecules of the conductor and this 

 flow of charge creates a random voltage across the terminals of the 

 conductor. This voltage usually is observed in a system composed 

 of an amplifier with an input circuit and an output device. Its mean- 

 square value across the output device is given by the expression 



V,o' = 4kT r RGHf, (1) 



1. 



where the symbols have the following meanings : 



k is the Boltzmann gas constant and is equal to 1.37 X 10~-' watt- 

 second per degree, 



T represents absolute temperature, degrees Kelvin, 



R represents the resistive component in ohms of the input impedance 

 as measured across the input terminals of the amplifier, 



G represents the voltage gain of the amplifier, and is equal to the ratio 

 of voltage across the output device to voltage across the input 

 terminals of the amplifier, 



/ represents frequency in cycles per second, 



R and G are in general functions of frequency. 



In the simple case where the amplifier has a constant gain over a 

 frequency range F and no gain outside of this range, and where R is 

 also constant over the same frequency range and is at the normal 

 temperature of 300 degrees, the mean-square noise voltage across the 

 input terminals of the amplifier is 



Vt^ = 1.64 X 10-2oi?7^. (2) 



This is the voltage that would be produced by a generator supplying 

 to the resistance R the power 



W^^= 1.64 X lO-20/r. (3) 



R 



The power W, sometimes expressed as 1.64 X 10"^" watts per cycle, 

 is independent of R and may be regarded as the apparent input power 

 of the thermal agitation. It depends only on the frequency range of 

 the amplifier, since the temperature cannot be varied conveniently or 



1 For all numbered references see bibliography at end of paper. 



