CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 



205 



phase loop. Solution: Let I lf /,, and /, (Fig. 48), represent the vectors 

 of the three currents at an unbalanced load. The current / t in .1 is 

 replaced by /, and /$, and the system is split into two single-phase 

 loops, A-B and A-C. The fluxes due to these systems and linked with 

 A are denoted by n and 0, 3 . They are in phase with the corresponding 

 currents, and are proportional to the magnitudes of these currents, 

 because of the equal spacing. Hence, the triangles of the currents and 

 of the fluxes are similar, and the resultant flux d> v linking with A is 

 in phase with /,. If U - fcp/,, and A = $(?/ 3 , where <P is the equivalent 

 permeance of each single-phase loop, then the result shows that <Pi = $<P/|. 

 If the wires B and C coincided the equivalent permeance (P would be 

 the same, and hence the proposition is proved. The voltage drop, E lt 

 due to the flux t is shown by a vector leading 7 t by 90 degrees. 



63. The Equivalent Reactance and Resistance of a Three- 

 phase Line with an Unequal Spacing of the Wires. In the case 

 of an unequal spacing of the wires eqs. (126) and (127) still hold 

 true, because they do not depend upon the spacing; but eq. (128) 

 becomes 



(130) 



-i. 



where L' 12 is the value of the inductance per unit length, calcu- 

 lated by eq. (125) for the spacing between A and B, and L'i 3 is 

 the value of the inductance per unit 

 length, for the spacing A-C. Substi- 

 tuting the valueof 0^ from eq. (130) 

 into eq. (127) we get 



This shows that with an unequal 

 spacing the effect of the mutual 

 induction of the phases cannot be 

 replaced by an equivalent inductance 

 in each phase, because, generally 

 speaking, the currents i 2 and i* 3 cannot 

 be eliminated from this equation by 

 means of eq. (126). 



Let us apply now eq. (131) to 

 the case of sinusoidal currents and 



voltages. Let the current in the wire ft be i' 2 V2/ 2 sin 2nft, 

 where 7 2 is the effective value <>f the current ; then the first term 

 on the right-hand side of eq. (131) beo \ 



FIG. 48. The currents and 

 fluxes in a three-pi a>r liu<> 

 with a symmetrical spacing. 



