160 BELL SYSTEM TECHNICAL JOURNAL 



voltage or current components near any frequency should be added directly 

 with due regard for phase. 



Johnson noise is related to many physically similar phenomena such as 

 Brown ian motion and the random fluctuations in position observed in the 

 coils of very sensitive galvanometers. 



The simplest derivation of (1), (2) or (3) is that given by Nyquist'^ in a 

 companion paper to Johnson's. Consider a long lossless transmission line 

 of length L terminated at each end in resistances equal to its characteristic 

 impedance. Imagine line and terminations in thermal equilibrium at a 

 temperature T, as shown in Fig. 2, If electrical energy flows from the 

 resistance at 1 to that at 2, then equal energy must then flow from 2 to 1, 

 as any net gain or loss of energy would violate the second law of thermo- 

 dynamics. 



Now, suppose that we suddenly close the switches at 1 and 2, short circuit- 

 ing the ends of the line. The line now becomes a resonator, having resonant 



Fig. 2 — Two resistances terminating a transmission line act as generators of thermal 

 noise power traveling along the line. 



frequencies such that the line is n half wavelengths long. The resonant 

 frequencies will be 



f=n{c/2L). (4) 



Here n is an integer and c is the velocity of light. The frequencies are 

 separated by frequency intervals 



A/ = {c/2L). (5) 



The energy which originally flowed right to left and left to right between th^ 

 resistances is now reflected at the ends. It may be expressed as the thermal 

 energy associated with the resonant modes of the line. According to 

 statistical mechanics, there is an energy kT associated with each resonant 

 mode. The energy per unit bandwidth is obtained by dividing this by the 

 frequency interval between modes, given by (5) and is 



w^ kTlLf^ kT/{c/2L). (6) 



Since it takes a wave a time L/c to pass completely through the line, this 

 energy w represents the energy per unit bandwidth which flowed into the 



