192 THE MAGNETIC CIRCUIT [ART. 59 



very high frequencies, the skin effect becomes noticeable, that is, 

 the current in the inner conductor is forced outward and that in 

 the outer conductor inward. In the limit, when the frequency 

 is infinite, the currents are concentrated on the opposing surfaces 

 of the conductors, and the partial linkages are equal to zero. 

 Thus, each of the expressions in question must be multiplied by a 

 variable coefficient k which, for a given cable, is a function of 

 the frequency. At ordinary frequencies k = l, and gradually 

 approaches zero as the frequency increases to infinity. 1 



Prob. 1. A concentric cable is to be designed for 750 amperes, the 

 current density to be about 2.2 amp. per gross sq.mm., and the thickness 

 of the insulation between the conductors to be 6 mm. What are the 

 dimensions of the conductors assuming them to be solid, that is, not 

 stranded? The fact that they are in reality stranded is taken care of 

 in the permissible current density. 



Ans. a = 10.5 ; b = 16.5 ; c = 19.6 mm. 



Prob. 2. Plot the curve Ogrs of distribution of the flux density 

 in the cable given in the preceding problem. 



Ans. At x= a, B= 143; at x = b, B= 91 maxwells per sq.cm. 



Prob. 3. What is the total flux in megalines per kilometer of the 

 cable specified in the two preceding problems? Ans. 15.1. 



Prob. 4. Show how to plot the curve of the distribution of flux density 

 in a three-phase concentric cable, at some given instantaneous values 

 of the three currents. 



Prob. 5. What is the inductance of a 25-km. cable in which the 

 diameter of the inner conductor is 12 mm., the thickness of insulation 

 is 3 mm., and the dimension c is such that the current density in the 

 outer conductor is 10 per cent higher than in the inner one ? 



Ans. 25 [0.0810 +0.050 + 0.0122] = 3.58 millihenry. 



Prob. 6. A cable consists of three concentric cylinders of negligible 

 thickness; the radii of the cylinders are r t , r 2 , and r 3 , beginning with 

 the inner one. What is the inductance in millihenrys per kilometer, 

 when a current flows through the inner cylinder and returns equally 

 divided through the two others? 



Ans. 0.46 [log (r a /r t ) +0.25 log (r,/r,).] 



Prob. 7. In the cable given in the preceding problem the total 

 current i flows through the middle cylinder, the part mi returns through 

 the inner cylinder, and the rest, ni, returns through the outer one. What 

 is the total inductance per kilometer of length? 



Ans. 0.46 [w 2 log (r,/r t ) + n 2 log (r f /r a )]. 



1 For the field distribution in and the inductance of non-concentric cables 

 aee Alex. Russell, Alternating Currents, Vol. 1 (1904), Chap. XV; for the 

 reactance of armored cables see J. B. Whitehead, " The Resistance and React- 

 ance of Armored Cables, Trans. Amer. InsL Electr. Engrs., Vol. 28 (1909), 

 p. 737. 



