Page 335 - Power Electronics Handbook

Output voltage control 325

wave equivalent is shown in Figure 13.42, the Fourier coefficient for this
being given by equation (1 3.16) the values of B and w being chosen to satisfy
equations (13.7) and (13.8), as before.
(13.16)

It is, of course, possible to combine the two-phase shifted waveforms of
Figure 13.42 together, as was done in Figure 13.41. This would give a
modified result for the nth harmonic content, as in equation (13.17), where
20 is the phase shift between the two waveforms. Since the square of the
voltage ignores signs, the r.m.s. of the total waveform is given by equation
(13.18).

2J2
- - - 4sinnBsinnol cosnD X 100 (13.17)
[l
=
Vnns(n)
VB nx
(13.18)

In equation (13.17) there are now three variables, two of which, say o
and B, can be used to eliminate any two harmonics, whilst the third, in this
case D, is used to control the magnitude of the fundamental voltage.
Therefore if B and o are chosen to satisfy equations (13.19) and (13.20)
then harmonics PI and Pz will be absent from the output over the whole
range of variation of D.
1 - 4 sin (PlZ3) sin (Plo) 0 (13.19)
=
1 - 4sin (P2B) sin (P20) = 0 (13.20)

Equations (13.17) and (13.18) have been used to obtain a harmonic
analysis of waveforms where the third and fifth, the third and seventh, and
the fifth and seventh harmonics have been eliminated. The results are
given in Tables 13.13-13.15.
The system shown in Figure 13.42 is a method of 'selected harmonic
reduction' so comparison will be made with the quasi-square control and
two-pulse unidirectional waveforms, i.e. with Tables 13.1-13.4. From
these it is clear that the maximum fundamental output voltage is less than
that obtainable from the quasi-square wave, and the lower the harmonics
eliminated, the less the maximum voltage. This is not the case for
unidirectional switching, where the reverse is true, but it must be
remembered that the limitation was then fixed by the necessity of not
allowing adjacent pulses to merge. This is not required for bi-directional
switching since in effect positive and negative pulses are already merged in
the primary wave, and control is achieved by phase shifting two such
identical waveforms up to the full half cycle. For example, the quasi-square
wave gives a peak fundamental which is 90% of the d.c. value. For
unidirectional switching this reduces to 75.1%, 62.2% and 33.3% when
eliminating the third, fifth and seventh harmonics, respectively. For
bi-directional switching comparable figures are 75.5%, 77.5% and 84.0%