KENNELLY. — EQUIVALENT CIRCUITS OF COMPOSITE LINES. 33 



published tables are available. Identically the same formulas are, how- 

 ever, applicable to single-frequency alternating-current cases, by ex- 

 panding their interpretation from real to complex numbers ; or from 

 one space-dimension into two, using impedances for resistances and 

 plane-vectors for potentials and currents. Unfortunately, however, 

 we have no tables of hyperbolic sines, cosines, and tangents available, 

 as yet, for complex arguments except for the particular case of semi- 

 imaginaries, 2 or plane-vectors of 45° ; so that in working out the alter- 

 nating-current cases, as, for example, in telephony, the engineer is de- 

 layed by having to assume the duties of a computer, and to work out 

 his own hyperbolic sines, cosines, and tangents. However, even thus 

 handicapped, it is claimed that the formulas here presented will not 

 be too lengthy for the engineer to use in important cases. If hyper- 

 bolic tables of complex arguments were worked out and published, the 

 formulas could, with their help, be applied almost as quickly and con- 

 veniently to alternating-current cases as they can be applied at present 



e=L*. 



A* 



B 



-„„..^.Y-Y.Y.Y.Y.V -Y Y. Y..Y .Y..Y.Y.Y.Y.V. Y.,^..^. 



G G' 



Figure 1. Uniform line with distributed resistance and leakance. 



to direct-current cases. If, however, attempts are made to obtain 

 alternating-current results of like precision without the use of hyper- 

 bolic functions, there seems to be no hope of helping the engineer. 

 Only specially trained mathematicians could handle the long and com- 

 plex resulting formulas. 



Preliminary Review of Single-Line Formulas. 



In order to pass to composite lines, we may first briefly review the 

 laws of equivalent circuits for single lines. The fundamental formulas 

 will be given for direct-current (D. C.) and for alternating-current 

 (A. C.) circuits, in parallel columns. 



Let AB, Figure 1, be a uniform single line operated to ground, or 

 zero -potential, return circuit. 

 L = the length of the line in kilometers (or miles). 



2 See Table appended to "The Alternating-Current Theory of Transmission- 

 Speed over Submarine Cables," referred to in the Bibliography. 

 vol. xlv. — 3 



