116 proceedings of the american academy. 



Imitative Accuracy of Artificial Lines. 



As the preceding formulas indicate, an artificial line does not corre- 

 spond electrically to the real smooth line having the same linear con- 

 stants (resistance and leakance per km.) as its nominal linear con- 

 stants, but to some other real smooth line having somewhat different 

 linear constants. In other words, an artificial line has an imitation 

 error due to lumpiness. The amount of this error will differ with the 

 degree of lumpiness, and would obviously disappear if the number 

 of line sections were made indefinitely great. In general, the fewer the 

 sections the greater the lumpiness, and the greater the lumpiness error. 

 With any given artificial line, however, the lumpiness error depends 

 upon the particular quantity considered, and is not the same for all 

 quantities. Thus, let a! and r ' be the nominal attenuation-constant 

 and surge-resistance of the uncorrected artificial line, by (1) (2) ; while 

 a and r are the corresponding constants, corrected for lumpiness, 

 according to (4) and (5) . Then the ratio of received ground current 

 over the artificial line to that over the real line of same nominal linear 



/l* ' O] T~l [~1 lit 



constants will be - — — — — . Again, the ratio of sending-end resist- 

 r sinn La 



ances with the far end grounded will be -—. ; — =r— ., a distinctly differ- 



b r ' tanh La" J 



ent ratio ; while in respect to, say, voltage at the free distant end, the 

 ratio will be again different. Consequently there is no single correction 

 factor for the lumpiness of an artificial line, and each particular quan- 

 tity will have to be corrected, according to the preceding formulas. 



Equivalence between Single-Section Artificial Lines and 



Uniform Smooth Lines. 



In Figure 10, let AOB represent a uniform smooth actual line of 

 L kms. in length, with a linear conductor-resistance of r ohms per km. 

 and a linear dielectric conductance of g mhos per km. Its attenuation- 

 constant will then be a = Vgr hyps, per km., and its surge-resistance 



=iA 



r = v - ohms. Its hyperbolic angle will be La, and its semi-hyper- 

 bolic angle \a hyps. Then let a single section of artificial line be con- 

 structed, as in Figure 11, with a total conductor-resistance of / ohms, a 

 leak at the centre of g' mhos, or R' = \/g' ohms. This single section 

 of artificial line will be the complete external equivalent of the actual 

 uniform line in Figure 10 in the steady state, if: 



