and Inductance of a Concentric Main. 



551 



VIII. Numerical example. 



In connexion with the transmission of electric power at 

 low frequencies, the question of the magnitude of the skin 

 effect in three-core cables has been recently discussed by 

 engineers. As the question is one of considerable difficulty 

 owing to the very complex nature of the magnetic field 

 inside a three-core cable (see Russell, ' Alternating Currents/ 

 vol. i. p. 321), it is important to know what are the corre- 

 sponding losses in a concentric main. 



Let us suppose that the inner core is solid and that mc 

 is not greater than 2, so that we may use formula (101). 

 Let the radius of the inner main be one centimetre. For a 

 very high pressure cable a suitable value* of b would be 

 2*4 cms. Hence, since the section of the outer conductor 

 is made equal to that of the inner, c is 2'Q cms. Substituting 

 these numbers in the formula, we find that 



= ^ 1+ l92) + C( 1 + -T92-> 



where I is the length of the main in centimetres. 



With the frequencies used in practice m is] not very 

 different from 1. We see that the skin effect increases the 

 resistance of the inner conductor by about the half of one 

 per cent., and the increase in the effective resistance of the 

 outer conductor is less than the hundredth part of this. 

 For a low voltage cable we might have c = 5/3 and £ = 4/3. 

 In this case 



E p /, m 4 \ p/\ 0-059m 4 \ 



7^( 1+ iW<( 1+ -TI)2-> 



and hence the increase in the loss of the outer conductor is 

 only about the twentieth part of the corresponding quantity 

 for the inner. If the return conductor were a tight-fitting 

 tube so that b = l and c— \/2, the formula becomes 



r = H 1+ l92) + £( 1+ 192-} 



Even in this case the increase in the loss of the outer is less 

 than the fifth part of the increase in the loss of the inner, 

 although the losses for very high frequencies would-be 

 practically the same in each conductor. 



* Russell, < Electric Cables and Networks/ p. 203. 



