CHAP. XIJ INDUCTANCE OF TRANSMISSION LINES 199 



(c) Linkages outside the loop, that is, from x=6-fa to x = 

 infinity, 



(d) Linkages within the wire B, that is, from 



x=ba to x=b+a. 



The linkages (a), (b) and (c) are the same as in a concentric 

 cable (Fig. 46), because the shape of the lines of force and the 

 number of turns with which they are linked are the same. The 

 partial linkages (d) are somewhat difficult to express analytically. 

 When the distance 6 between the wires is large as compared to 

 their diameters, the whole current in B may be assumed to be con- 

 centrated along the axis of the wire B, instead of being spread over 

 the cross-section. With this assumption, the partial linkages (d) 

 are done away with, the linkages (6) are extended to x =6, and the 

 linkages (c) begin from x =b. The expressions for the linkages (a) 

 and (6) are given by eqs. (110) and (109) respectively. The link- 

 ages (c) are equal to zero, because in this region the lines of force 

 produced by A are linked with both A and B, and therefore 

 with i i=0 ampere -turns. Thus, the inductance in question is 



L' =0.46 Iog 10 (6/a) +0.05 (125) 



This gives the inductance of a single-phase line in perms per centi- 

 meter length, or in millihenrys per kilometer length of the wire. 1 

 To obtain the inductance per unit length of the line this expression 

 must be multiplied by two, because the linkages due to the flux 

 produced by the system B are not taken into account in eq. (125). 

 However, for the purposes of the next two articles it is more con- 

 venient to use expression (125), and to consider separately the 

 inductance of each wire, remembering that the two wires of a loop 

 are in series, and that therefore their inductances are added. 2 



Prob. 17. Chock by means of formula (125) some of the values for 

 the inductance and reactance of transmission lines tabulated in the 

 various pocketbooks and handbooks. 



1 It is of interest to note that the exact integration over the partial linkages 

 (d) leads to the same Eq. (125), so that this formula is correct even when 

 the wires are close to each other. See A. Russell, Alternating Current, Vol. 

 1 (1904), pp. 59-flO. 



9 The inductance of two or more parallel cylinders or any cross-section can 

 be expressed through the so-called " geom.-trir mean distance," introduced 

 by Maxwell. For details see Orlich, Kapazitat und InduktiviW (1909), 

 pp. 63-74. 



