CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 203 



that is, the induced e.m.f. is the same as in a single-phase line carry- 

 ing the current i\. Thus, the inductance of a three-phase line with 

 symmetrical spacing, per wire, is the same as the inductance of a 

 single-phase line, per wire, with the same size of wire and the same 

 spacing. The total e.m.f. induced in each wire is in quadrature 

 with the current in the wire. 



In reaching this conclusion the following facts were made use 

 of : (a) The current in each wire at any instant is equal to the sum 

 of the currents in the two other wires; (6) the fluxes due to sepa- 

 rate m.m.fs. can be superimposed in a medium of constant perme- 

 ability; (c) The inductance of the loop A-B is equal to that of A-C 

 because of the same spacing. No other suppositions in regard to 

 the character of the load or the voltages between the wires were 

 made. Therefore, the conclusion arrived at holds true : 



(a) For balanced as well as unbalanced loads; 



(6) For balanced or unbalanced line voltages; 



(c) For a three-wire two-phase system, three-wire single- 

 phase system, monocyclic system, etc. 



(d) For sinusoidal voltages as well as for those departing from 

 this form. 



Semi-symmetrical Spacing. When two out of the three distances 

 between the wires in a three-phase line are equal to each other, the 

 arrangement is called semi-symmetrical. Two common cases of 

 this kind are : (a) When the wires are placed at the vertices of an 

 isosceles triangle; (6) when they are placed at equal distances in 

 the same plane, for instance on the same cross-arm, or are fastened 

 to suspension insulators, one above the other. In such cases the 

 inductive drop in the symmetrically situated wire is the same as if 

 the wire belonged to a single-phase loop, carrying the same current, 

 and with a spacing equal to the distance of this wire to either of 

 the other two wires. Let, for instance, the distance A-B be equal 

 to B-C, and let the distance A-C be different from the two. The 

 proof given above can be repeated for the wire B t and the same 

 conclusion will be reached because the spacing A-C is not used 

 in the deduction. But, of course, the proof does not hold true for 

 either wire A or C. 



When the three wires are in the same plane, the inductance of 

 each of the outside wires is larger than that of the middle wire. 

 This can be shown as follows: Let the t linv \\ires be in a horizontal 

 plane, and let them be denoted from left to right by A, B t C. Let 



