Single-pulse PWM inverter control and rms

Single-pulse width modulation controls the inverter output voltage by giving one pulse per half cycle whose width $\delta$ is varied; the pulse is placed symmetrically about $\pi/2$ (and about $3\pi/2$ for the negative half).

Operation

 vc   /\        /\        /\     carrier, 2f
 Ar -/--\------/--\------/--\--- reference
    /    \    /    \    /    \
 g1      |====|                  gate T1,T2
 g4                |====|        gate T3,T4
 v0  +Vs  ____
     0 --'    '---.    .---- wt
     -Vs           '--'
         |<-d->|
      0   pi/2  pi 3pi/2  2pi

1. A dc (rectangular) reference signal of amplitude $A_r$ is compared with a triangular carrier of amplitude $A_c$ and frequency $2f$ ($f$ = output frequency).
2. While the reference exceeds the carrier, gate pulses are given: to T1, T2 in the positive half cycle and to T3, T4 in the negative half cycle of a full bridge.
3. The pulse width is $\delta = M\pi$, where the modulation index $M = A_r/A_c$. Varying $A_r$ from 0 to $A_c$ varies $\delta$ from 0 to 180°, and so varies the output voltage.
4. The output frequency is changed by changing the carrier (and so the gate) frequency.

RMS value of output voltage

The pulse extends from $\frac{\pi-\delta}{2}$ to $\frac{\pi+\delta}{2}$ with amplitude $V_s$:

$$\begin{aligned} V_{0,rms} &= \left[\frac{2}{2\pi}\int_{(\pi-\delta)/2}^{(\pi+\delta)/2}V_s^2\,d(\omega t)\right]^{1/2} \\ &= \left[\frac{V_s^2}{\pi}\cdot\delta\right]^{1/2} = V_s\sqrt{\frac{\delta}{\pi}} = V_s\sqrt{M} \end{aligned}$$

Harmonics

$$v_0 = \sum_{n=1,3,5,\dots}\frac{4V_s}{n\pi}\sin\frac{n\delta}{2}\sin n\omega t, \qquad V_{1,rms} = 0.9V_s\sin\frac{\delta}{2}$$

• $\delta = 180^\circ$: $V_{0,rms} = V_s$ (square wave).
• $\delta = 120^\circ$: $V_{0,rms} = 0.816V_s$ and the 3rd harmonic is eliminated ($\sin 180^\circ = 0$).
• $\delta = 90^\circ$: $V_{0,rms} = 0.707V_s$.

Limitation: at low output voltage (narrow pulse) the harmonic content is very high; the lowest harmonic (3rd) is close to the fundamental and hard to filter. Multiple-pulse and sinusoidal PWM overcome this.