224 BELL SYSTEM TECHNICAL JOURNAL 



Investigating the ciueslion of concentrating tiie line inductance at 

 uniformly spaced interxals, Professor Pupin gave his famous solution 

 in a paper** presented before the Institute in May, 1900. Dr. G. A. 

 Campbell in his paper^ of Mafch, 1903, also gave a mathematical de- 

 velopment of the loading theory along somewhat different lines. 



These early investigations showed that a coil loaded line should have 

 several coils per wave length in order to simulate a uniform line. The 

 more closely the coils are spaced the more exact is the degree of equiv- 

 alence, and when there are ten coils per wave length the equivalence 

 is very close. On the other hand, the cost of the loading increases as 

 the spacing is shortened. Thus, from the standpoint of commercial 

 application, the question "What is the smallest number of coils per 

 wave length that will give satisfactory transmission?" is very im- 

 portant. In the investigation which was made to determine the 

 magnitude of the changes in attenuation, velocity and impedance, 

 as the number of coils per wave length is reduced, abrupt changes in 

 these characteristics were found to occur at the spacmg of two coils 

 per actual wave length. The critical frequency at which this spacing 

 applies in a loaded line became known as the cutoff frequency, since 

 at this frequency and higher frequencies the attenuation loss is so 

 extremely large as to amount practically to a suppression, or cut-off 

 effect. 



At the cut-off freqeuncy the velocity of the coil loaded line is lower 

 than the velocity of the corresponding smooth line approximately 

 in the ratio of 2:7r; consequently, at the cut-off frequency there are 

 approximately tt coils per wave length, in terms of the velocity of the 

 corresponding smooth line. 



The following expression deiines the cut-off' frequency in a coil 

 loaded line having zero distributed inductance: 



ir\/LsC 

 in which 



/c = cut-oft frequency, 



L=coil inductance, 



s =coil spacing, 



C = line capacitance per unit length. 



■• "Wave Transmission over Non-Uniform Cables and Long Distance Air Lines," 

 M. L Pupin, Trans. A. L E. E., Vol. 17, 1900, p. 445. Refer also to Pupin, U. S. 

 Patents Nos. 652, 230 and 652, 231, June 19, 1900. 



5 "On Loaded Lines in Telephone Transmission," G. A. Campbell, Philosophical 

 Magazine, March, 1903. 



