Page 302 - Solutions Manual to accompany Electric Machinery Fundamentals

B-2. Derive the relationship for the winding distribution factor k d in Equation B-22.

SOLUTION The above illustration shows the case of 5 slots per phase, but the results are general. If there
are 5 slots per phase, each with voltage E Ai , where the phase angle of each voltage increases by  from
slot to slot, then the total voltage in the phase will be
E  E  E A2  E A3  E A4  E A5  ...  E An
A
A1
The resulting voltage E A can be found from geometrical considerations. These “n” phases, when
drawn end-to-end, form equally-spaced chords on a circle of radius R. If a line is drawn from the center
of a chord to the origin of the circle, it forma a right triangle with the radius at the end of the chord (see
voltage E A 5 above). The hypotenuse of this right triangle is R, its opposite side is E 2 / , and its smaller
angle is / 2 . Therefore,

1
 E 2/ E
sin   R  2 (1)
2 R 
sin
2
The total voltage E A also forms a chord on the circle, and dropping a line from the center of that chord to
the origin forms a right triangle. For this triangle, the hypotenuse is R, the opposite side is E A 2 / , and
the angle is n 2 / . Therefore,
1
n E / 2 E A
sin  A  R  2 (2)
2 R  n
sin
2

Combining (1) and (2) yields

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