200 THE MAGNETIC CIRCUIT [ART. 61 



Prob. 18. Show by moans of tables or curves that the inductance 

 of a transmission line varies much more slowly than (a) the spacing 

 with a given size of wire, (6) the size of wire with a given spacing. 



Prob. 19. When the diameters of the two wires A and B are different, 

 prove that the inductance of the complete loop is the same as if the 

 diameter of each wire was equal to the geometric mean of the actual 

 diameters. 



Prob. 20. Show that the inductance of a single-phase line with a 

 ground return can be calculated from eq. (125) by putting b=2h where 

 h is the elevation of the wire above the ground. Hint: In Fig. 47 the 

 plane 00' may be considered to be the surface of the earth, assumed 

 to be a perfect conductor. If the earth be removed, an "image" con- 

 ductor B must be added in order to provide a return path for the 

 current, such that the field surrounding A would remain the same. 



Prob. 21. Two single-phase lines are placed on two cross-arms of 

 the same tower, one directly above the other, at a vertical distance 

 of c cm apart. What is the total inductance of the combination, when 

 the two lines are connected in parallel and each line carries one-half 

 of the total current? 



Solution : Consider the four wires as forming four fictitious systems, 

 with cylinders at infinity as returns. Let b be the spacing in each 

 loop, and let b be larger than c. Denote the wires in one loop by 1 

 and 2, in the other by 1' and 2', and let d be the diagonal distance 

 between 1 and 2'. Assume all the wires except 1 to be of an infinitesimal 

 cross-section. Then, the linkages of the flux produced by the system 

 1 with the currents in the four wires are 



i'Lt'-O.OStoV+O^CJt) 1 Ln(c/o) +0.2i (Ji)Ln(b/c) 



+0.2(*i) a Ln'(<f/&) millijoules/km. 



Thus, allowing the same amount for the linkages due to the cur- 

 rent in the wire 1', we get that the inductance of the split line, each way, 

 is 



10 (bd/cd) +0.05] millihenrys/km., 



instead of the expression (125) for the single line. The same result 

 is obtained when b is smaller than c. Hence, by splitting a line in 

 two the inductance is considerably reduced, because partial linkages are 

 substituted for some of the complete linkages. If d were equal to c 

 the reduction would be 50 per cent; but since d is always larger than 

 c the gain is less than 50 per cent. However, when the two lines are 

 very far apart the saving is very nearly 50 per cent. 



Prob. 22. A certain single-phase transmission line has been designed 

 to consist of No. 000 B. & S. conductor with a spacing of 180 cm. It 

 is desired to reduce the reactive drop by about 20 per cent, without 

 increasing the weight of copper, by using two lines in parallel, with 

 the same spacing. What is the size of the conductor and the distance 

 between the loops? Ans. No. 1 B. & S. ; about 8 cm. 



Prob. 23. Solve problem 21 when the load is divided unequally 



