CHAP. Ill MAGNETIC CIRCUIT WITH IRON 29 



Prob. 11. A ring of forged steel has such dimensions that the average 

 length of the lines of force is 70 cm. The ring has an air-gap of 1 .5 mm., and 

 is provided with an exciting winding concentrated near the air-gap so as 

 to minimize the leakage. What is the flux density at an m.m.f. of 4000 

 ampere-turns? First Solution : Assume various values of B, calculate the 

 corresponding values of the ampere-turns, until the value of B is found, for 

 which the required excitation is 4000 ampere-turns (solution by trials). 

 Second solution : Let the unknown density be B and the corresponding 

 magnetic intensity in the steel be H. The required excitation for the steel 

 is then 7QH, and for the air-gap 0.15 X 0.8 XlOOO-120B ampere-turns. 

 Therefore, 



70// + 1205=4000. 



The values of B and H must satisfy this equation of a straight line,and 

 besides they must be related to each other by the magnetization curve 

 for steel forgings (Fig. 2). Hence, B and H are determined by the intersec- 

 tion of the straight line and the curve. The straight line is determined by 

 two of its points; for instance, when H =40, B - 10; when H =24, B - 19.3. 

 Drawing this line in Fig. 2 we find that the point of intersection corre- 

 sponds to B = 16.3. l Ans. 16.3 kilolines per sq. cm. 



Prob. 12. Solve the preceding problem, assuming the ring to be made 

 of silicon steel laminations: 10 per cent of the space is taken by the 

 insulation between the laminations. 



Ans. Flux density in the laminations is 15.2 kl/sq. cm. 



Prob. 13. In a complex magnetic circuit, an air-gap 3 mm. long and 

 26 sq. cm. in cross-section is shunted by a cast-iron rod 14 cm. long and 

 10 sq. cm. in cross-section. What is the number of ampere-turns neces- 

 sary for producing a total flux of 215 kilolines through the two paths in 

 parallel, and what is the reluctance of the rod per centimeter of its length 

 under these conditions? Ans. 1160 ampere-turns; 0.933 mill i-rd. 



Prob. 14. The magnetic flux in a closed iron core must increase and 

 decrease according to a straight-line law with the time, then reverse and 

 increase and decrease according to the same law in the opposite dim -lion. 

 Show the general shape of the curve of the exciting current, neglecting 

 the effect of hyteresis. 



Prob. 15. Show that if in the preceding problem the flux varies ftOOOfd- 

 ing to the sine law the curve of tho exciting < -urrent is a po. 

 Show how to determine the shape of this curve from a given magnetiEa- 

 of the material. This problem has an application in the ral.-u- 

 iatk>Q Of the exciting CURn1 in a transforms. 



Prob. 16. In the magnetic circuit shown in Kip. l tho useful flux 

 passes through the air-gap between the two steel poles; a part of the Mu\ 



'The student will see from the solution of thi* problem that in the 

 of a series main t it is much easier to find the m.m.f. required for 



a given flux than vice vena. On the other hand, in the ease of two mag- 

 : aths in parallel (such as in prob. 13), it is easier to find the flux for a 

 given m.m f 



