In this method of pulse-width modulation, the harmonic content can be reduced using several pulses in each half-cycle of output voltage. By comparing a reference signal with a triangular carrier wave, the gating signals are generated for turning. on and turning-off of a thyristor, as shown in Fig.1(a). The carrier frequency, $f_{c}$ , determines the number of pulses per half-cycle, $m,$ whereas the frequency of reference signal sets the output frequency, $f_{0}$.

The modulation index controls the output voltage. This type of modulation is also known as symmetrical pulse width modulation (SPWM). The number of pulses $N_{p}$ per half-cycle is found from the expression

$$N_{p}=\frac{f_{c}}{2 f_{0}}=\frac{m_{f}}{2}-----(1)$$

Where $m_{f}=\frac{f_{c}}{f_{0}}$ is the frequency modulation ratio. The variation of modulation index (M) from 0 to 1 varies the pulse width from 0 to $\pi / N_{p}$ and the output voltage from 0 to $E_{\mathrm{d} c}$. For SPWM, the output voltage for single-phase bridge inverters is shown in Fig.1(b).

If P is the width of each pulse, the RMS output voltage can be obtained from the following expression,

$$E_{L(\mathrm{rms})}=\left[\frac{2 N_{p}}{2 \pi} \int_{\left(\pi / N_{p}-P\right) / 2}^{\left(\pi / N_{p}+P\right) / 2} E_{\mathrm{dc}}^{2} \mathrm{d}(\omega L)\right]^{1 / 2}=E_{\mathrm{dc}} \sqrt{\frac{N_{p} \cdot P}{\pi}}-----(2)$$

The general expression for various harmonics in the output voltage is obtained by deriving an expression for a general pair of pulses, such the positive pulse of duration $P$ starts at $\omega t=\alpha$ and the negative one of the same width starts at $\omega t=\pi+\alpha .$ This is shown in Fig.1(b). The effects of all pulses can be combined together to obtain the effective output voltage. Thus for this pair of pulses, $$A_{n m}=\frac{2 E_{\mathrm{dc}}}{\pi} \int_{\alpha_{m-p / 2}}^{\alpha_{\operatorname{m+p} / 2}} \sin n \omega t \cdot \mathrm{d}(\omega t)$$

$$=\frac{2 E_{\mathrm{dc}}}{2 \pi}\left[\cos n(\alpha_m+p / 2)-\cos n\left(\alpha_{m}-p / 2\right)\right]$$

$$=\left(\frac{4 E_{\mathrm{dc}}}{n \pi}\right) \sin n \frac{p}{2} \cdot \sin n \alpha_{m}-----(3)$$

and $$\qquad B_{n m}=\left(\frac{2 E_{\mathrm{dc}}}{\pi}\right)_{\alpha_{m-p / 2}}^{\alpha_{m+p / 2}} \cos n \omega t \cdot \mathrm{d}(\omega t)$$

$$=\left(\frac{2 E_{\mathrm{dc}}}{2 \pi}\right)\left[\sin \left(\alpha_{m}+p / 2\right)-\sin \left(\alpha_{m}-p / 2\right)\right]-----(4)$$

If there are K pulses situated at $\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots, \alpha_{m}, \ldots, \alpha_{k}$ , then we have from Eq. $(3 \ and \ 4)$

$$A_{n}=\frac{4 E_{\mathrm{dc}}}{n \pi} \sin n \frac{p}{2} \sum_{m=1}^{k} \sin n \alpha_{m} \ and \ B_{n}=\frac{4 E_{\mathrm{dc}}}{n \pi} \cos n \frac{p}{2} \sum_{m=1}^{k} \sin n \alpha_{m}$$

In Fig.2 for $K=3$ and $K=10$ , amplitude of first, third, fifth and seventh harmonics as a ratio of the maximum value of fundamental (i.e. $E_{L \mathrm{nm}} / E_{L 1 \mathrm{m}} )$ are plotted against the pulse width expressed as a ratio of distance between two adjacent pulses, i.e. $(P/\pi/k)$. As the number of pulses per half-cycle (i.e.k) is increased, the considerable reduction in lower order harmonics is achieved as show in figure 2.

With this method,since voltage control is achieved with a simultaneous reduction of lower control harmonics,this scheme is comparatively advantageous over single-pulse modulation.However,due to larger number of pulses per half cycle, frequent turning-on and turning off of thyristors is required which increases the switching losses.Also,for this scheme inverter-grade thyristors are required which are costly.