Page 427 - Handbook of Electrical Engineering

416 HANDBOOK OF ELECTRICAL ENGINEERING

The average value of the waveform is zero because it is symmetrical about the Y-axis, and so
the coefficient a o for the average value is zero. The sinusoidal function in the coefficient a n varies
with the commutation angle u and approaches a limiting value when u is small,

un
2sin
2 1
As u → 0, →
un 2 n
◦
Which applies to a rectangular waveform. When u is 60 the sinusoidal function has an
absolute value of,

un
2sin
3

◦ 2 =
un 2 πn 2
u = 60 ,

0.9549
=
n 2
Therefore the magnitude of all the harmonics decrease as u increases, which is a reasonable
expectation since the waveform more closely resembles a sine wave.
√
The magnitude of the sum of the four cosine terms in (15.18) is 2 3 for all values of k in
(15.19), otherwise the magnitude is zero.
Table 15.2 shows the magnitudes of b n after scaling them by 1/b 1 ,i.e.creating b 1 = 1.0
as reference.

15.3.3.1 Worked example
Consider a 250 kW DC motor fed by a rectifier system. The line voltage is 415 volts at 50 Hz. The
rectifier is fed by a 400 kVA transformer which has an unusually high impedance of 0.0 + 24.5%.
Assume the motor rated efficiency is 0.9 per unit. Assume the motor terminal voltage is 262.3 volts
and its total current is 425 amps.
415
Phase voltage of the supply E = √ = 239.6 volts.
3
√
3 6
Open-circuit DC voltage of the rectifier V do = (239.6) = 560.45 volts.
π
The supply current

2I d 3
I ac = = 0.7797 I d
π 2
= 0.7797 × 425 = 331.37 amps

400,000
The transformer rated current = √ = 556.48 amps
3 × 415
239.6
1 pu impedance = = 0.4306 ohms/phase
556.48