Page 84 - Solutions Manual to accompany Electric Machinery Fundamentals

   d  BA

Under a pole face, the flux density B is always parallel to the vector dA, since the flux density is always
perpendicular to the surface of the rotor and stator in the air gap. Therefore,

   B dA

A differential area on the surface of a cylinder is given by the differential length along the cylinder (dl)
times the differential width around the radius of the cylinder ( rd ).


dA   dl rd where r is the radius of the cylinder
Therefore, the flux under the pole face is

   B d
dl
r
Since r is constant and B is constant with respect to l, this equation reduces to
  rl  B d


Now, B  B M  cos    t   B M cos (when we substitute    t   ), so
  rl  B d

 rl     /2 /2 B M cos   d   M rlB sin    /2 /2  rlB M    1    1  


  2rlB
M
3-10. In the early days of ac motor development, machine designers had great difficulty controlling the core
losses (hysteresis and eddy currents) in machines. They had not yet developed steels with low hysteresis,
and were not making laminations as thin as the ones used today. To help control these losses, early ac
motors in the USA were run from a 25 Hz ac power supply, while lighting systems were run from a
separate 60 Hz ac power supply.
(a) Develop a table showing the speed of magnetic field rotation in ac machines of 2, 4, 6, 8, 10, 12,
and 14 poles operating at 25 Hz. What was the fastest rotational speed available to these early
motors?
(b) For a given motor operating at a constant flux density B, how would the core losses of the motor
running at 25 Hz compare to the core losses of the motor running at 60 Hz?
(c) Why did the early engineers provide a separate 60 Hz power system for lighting?

SOLUTION
(a) The equation relating the speed of magnetic field rotation to the number of poles and electrical
frequency is
120 f
n  e
m
P

78