Page 430 - Handbook of Electrical Engineering

HARMONIC VOLTAGES AND CURRENTS 419

two parts are,

◦
Part 1. For the 180 rectangle waveform,
4 4
b n180 = , the fundamental b 1180 =
πn π
◦
Part 2. For the 60 rectangle waveform,
2 2πn 4πn 8πn 10πn
b n60 = cos − cos − cos + cos
πn 6 6 6 6

The value of the fundamental coefficient b 160 is,
1 1 2
b 160 = (4) =
π 2 π
√
The magnitude of the two parts is divided by 3 to obtain the primary line current of the
delta-star transformer. The result is then added to the line current of the star-star transformer. The
total magnitude of the supply line harmonic coefficient b nsum is given by,
1 4 πn 1 2πn
b nsum = √ + cos + √ cos
πn 3 6 3 6
1 4πn 5πn 7πn
− √ cos − cos − cos
3 6 6 6

1 8πn 1 10πn 11πn
−√ cos + √ cos + cos
3 6 3 6 6
and
n=∞

b nsum sin nωt
i sum (ωt) = i max
n=1
The value of the fundamental coefficient b 1sum is,
√ √
1 4 4 3 2 4 3

b 1sum = √ + + √ =
π 3 2 3 π
◦
◦
◦
The fundamental coefficients from the 180 , 120 and 60 waveforms are found to be in the
√
ratio 2: 3:1 respectively. The fundamental coefficient of the supply current is double the magnitude
◦
of the 120 waveform coefficient, which is the desired result.
◦
◦
The 180 waveform contains triplen harmonics for n taking odd values. The 60 waveform
also contains the same triplen harmonics but with opposite signs, which therefore cancel those in the
◦
180 waveform. None of the waveforms contain even harmonics.
The following harmonics are contained in the waveform,
n = 12 k ± 1
Where k = 1, 2, 3,..., ∞. The lowest harmonic present is the eleventh.