Page 341 - Power Electronics Handbook

Output voltage control 33 I
The square reference wave may be replaced by a sine wave, the output
being very similar to Figure 13.43 except that the modulation depth vanes
linearly along the cycle. This waveform will also contain odd and even sine
terms, and since X, = 0 the r.m.s. voltage of the nth harmonic can be
derived from equation (13.22).

Vrms(n) = J2 (E [cos nX(2M + 1) - cos nX(2M + 2)]
VB nn
+
-
cos
- E [cos n~(2~) n~(2~ x 100 (13.22)
I)]}
The solutions for equation (13.22) for four values of fdfS are given in
Tables 13.20-13.23. It is seen from these that the harmonic content of the
waveform is very similar to unidirectional switching, as in Tables
13.9-13.12. The harmonic with the largest amplitude is that which occurs
close to the chopping frequency fT, both odd and even harmonics being
considered. As an example, when operating at fdfs = 6 the sixth harmonic
is very large. For zero modulation depth the fundamental is zero and the
sixth harmonic has a value 90% of the d.c. supply, since the output is a
square wave at this frequency. As the modulation depth increases the
fundamental also increases in value and the sixth harmonic reduces,
whereas adjacent even harmonics, i.e. the fourth and the eighth, increase
in value. When operating with fT/fs = 20 the harmonic content up to the
15th is very similar to unidirectional methods, except that there are now
odd and even terms. However, the total harmonic content, which for
bi-directional switching is 100% of the d.c. supply irrespective of the
modulation depth, is much higher. As in previous methods using ‘lower
harmonic reduction’ it is clear that the inverter frequency should be several
orders larger than the output frequency for effective harmonic reduction.
This is therefore a disadvantage of the system as it can lead to lower
efficiencies.

TaMe 13.20 Hermonk content of a modul.ted M-bireetlorrPI wave aitbfi/Js = 4
~~ ~
Harmonic R.M.S. voltage as percentage of d.c. supply
number
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.98
1 8.34 16.0 21.4 29.4 34.9 44.4 49.8 57.2 63.4 10.9
2 0.09 0.61 1.03 6.58 7.28 11.1 12.0 14.9 19.4 22.2
3 2.52 1.23 2.16 0.11 0.64 0.01 0.67 1.22 1.79 1.80
4 89.2 87.2 84.9 80.6 77.0 69.4 64.4 57.0 50.7 42.4
5 2.50 1.25 2.18 3.04 4.38 6.81 8.17 11.2 12.7 15.2
6 0.28 1.79 3.00 2.79 5.18 9.21 12.4 15.8 20.7 25.5
7 8.24 15.3 19.8 22.3 24.7 25.1 25.2 22.6 17.9 13.6
8 0.93 0.87 2.01 1.73 3.34 5.29 7.50 11.6 10.2 13.4
9 8.17 14.8 18.7 24.7 26.2 26.7 24.7 21.1 16.1 8.92
10 0.47 2.88 4.70 10.7 11.9 13.6 13.5 11.0 9.10 5.19
11 2.39 1.41 2.29 0.39 0.57 0.94 2.11 1.44 4.55 5.86
12 27.5 22.0 16.2 7.43 1.14 7.07 10.5 11.8 8.42 4.28
13 2.34 1.47 2.33 6.24 7.08 8.65 7.44 3.31 3.97 1.47
14 0.65 3.82 5.99 10.6 13.6 18.1 18.4 18.9 15.8 11.0
15 7.87 12.9 14.3 8.93 6.41 0.89 3.78 5.53 4.63 1.00