CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 201 



between the two loops, the currents being mi and ni respectively, where 



Ans. L' = 0.46[(ro' + n J ) log (c/d) +log (6/c) +2mn log 



0.05(m a +7i 2 ), when 6>c, and L'=0.46[(m J + n J ) log (6 /a) + 

 2mnlog (rf/c)]+0.05(m 1 4-n 2 ), when 6<c. 



Prob. 24. A single-phase line consists of three conductors, the 

 total current flowing through conductor 1, and returning through con- 

 ductors 2 and 3 in parallel. If the current in one return conductor is 

 mi, and in the other ni, where m+n = l, prove that the inductance of 

 the line per kilometer of its length is, in millihenrys, 



U = 0.46 [log (6,,/a,) +m log (6 M /a,) + n log (b,,/a s ) +m log (b,,/fe ls ) 



+m log (b lt /b n ) +n log (& u /b)] +0.05(1 +m'+n 8 ), when 6 12 >6 l3 >6 tt . 



In the particular case when 6 n = 6 M = ?) 1 j, ai = Oj = Os, and m = n = $, the 

 inductance is reduced by 25 per cent as compared to that of a single loop. 1 



62. The Inductance of a Three-phase Line with Symmetrical 

 and Semi-symmetrical Spacing. The magnetic field which sur- 

 rounds a single-phase line varies in its intensity from instant to 

 instant, as the current changes, but the direction of the magnetic 

 intensity and of the flux density at each point remains the same. 

 In other words, the flux is a pulsating one. The field created by 

 three-phase currents in a transmission line varies at each point in 

 both its magnitude and direction. At the end of each cycle, the 

 field assumes its original magnitude and direction. If the spacing 

 of the wires is symmetrical, the field at the end of each third of a 

 cycle has the same magnitude and position with respect the next 

 wire. The field may therefore be said to be revolving in space. 



This revolving flux, like that in an induction motor, induces 

 e.m.fs. in the three phases. The problem is to determine these 

 counter-e.m.fs. in the transmission line, knowing the size of the 

 wires, the spacing, and the load. In transmission line calculations, 

 especially in determining the voltage drop and regulation, it is 

 convenient to consider each wire separately, and to determine the 

 voltage drop in phase and in quadrature with the current. Thus, 

 having expressed the e.m.fs. induced by the revolving flux in terms 

 of the constants of the line, each wire is then considered as if it 

 were brought outside the inductive action of the two other wires. 



We shall consider first the case of an equidistant spacing of the 

 three wires, because in most practical calculations of voltage drop 



*The splitting of conductors discussed in problems 21 to 24 has been 

 proposed transmission lit i tlirir induct- 



ance and at the same time increase their electrostatic permittance (capacity). 



