CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 



207 



and a fictitious resistance 



ri' = (/ 2 //i)*i2'sin 



sn 



(137) 



Both x\ and r\ may be either positive or negative, depending 

 upon the constants of the circuit and of the load. The resistance 

 ri' does not involve any loss of 

 power, converted into heat; it 

 merely shows that energy is trans- 

 ferred inductively from phase A 

 into B or C, at a rate /i^i', due to 

 a lack of symmetry in the resultant 

 field. 



Prob. 27. Show that with a 

 balanced three-phase load 



r,'066(*i,'-i 



(138) 



Prob. 28. When the three wires 

 are in the same plane, the spacings 

 being equal, and the three-phase load 

 balanced, show that the equivalent reactance of each outside wire 



Fio. 49. The currents and fluxes 

 in a three-phase line with an 

 unsymmetrical spacing. 



x ' -3m' + 0.435/X 10- 3 ohm/km., 



(139) 



where Xm is the reactance of the middle wire per kilometer, in ohms. 

 The equivalent resistance of the middle wire is zero, and that of the 

 two outside wires is 



r '-0.753/X10- 3 ohm/km., (140) 



where the sign plus refers to the wire in which the current leads that in 

 the nii Idle wire. 



Prob. 29. ( 'ompare the vector diagram in Fig. 49 with that in Fig. 

 48, and shown that with an unsymmetrical spacing the induced voltage 

 E t is not in quadrature with the corresponding current / so that the 

 action of the other two wires cannot be replaced by an equivalent 

 inductance alone, but only by an inductance and a resistance. Show 

 graphically that the latter may be either positive or negative. 



