NOISE IN RESISTANCES 159 



An initial warning should be made that quantum effects treated in 

 Nyquist's original paper on Johnson noise, but afterwards much neglected, 

 are entirely disregarded here. 



I. Johnson Noise* 



In 1926, in an investigation of amplifiers with exceedingly high grid 

 resistances, J. B. Johnson discovered that a resistance acts as a noise genera- 

 tor having an open-circuit voltage with a mean square value 



v^ = 4kTRB. (1) 



Here and subsequently, lower case letters v and i will be used in referring to 

 noise voltages and currents. In (1), v- is the mean square value of noise 

 voltage components of frequency lying in a small bandwidth B (sometimeg 



> (matched 

 power into load is: ^ load) 



Fig. 1 — Relations between noise power, noise voltage and noise current can be derived 

 by assuming the noise source to be a voltage in series with a resistance. 



called df or A/), k is Boltzman's constant, and R is resistance. We easily 

 see from Fig. 1 that the maximum noise power which can be made to flow 

 from the resistance into a load (that which will flow into a matched load) is 



P = ^ = ''TB. (2) 



This '^available noise powxr" is a convenient alternative formulation. 



If an impedance has a reactive as well as a resistive component, the open 

 circuit noise is given by (1) where R is the resistive component; if an admit- 

 tance has a conductance G the noise may be represented as an impressed 

 current (that which flows when the admittance is short circuited) of 

 magnitude 



J2 = 4kTGB. (3) 



We see from (1) that if two resistances are connected in series, the total 

 squared noise voltage is the sum of the squares of the noise voltages produced 

 by the resistances separately, and from (3) we see that the noise currents of 

 conductances connected in shunt also add by summing squares. This rule 

 of addition holds for adding the noise of all independent sources. Of course, 

 if noise from the same noise source reaches a point by different paths, the 



