Single-phase AC voltage controller with R load

A single-phase AC voltage controller changes the rms voltage applied to a load by delaying the firing of two anti-parallel thyristors (or a TRIAC) by an angle $\alpha$ in each half cycle (phase control). The output frequency is the same as the input frequency.

Circuit

        T1 --->|---
   +---|          |----+
   |    ---|<---  |    |
   |       T2          |
 ( ~ ) vs = Vm sin wt   R   vo
   |                   |
   +-------------------+

T1 handles the positive half cycle and T2 the negative half cycle.

Operation

1. $0 < \omega t < \alpha$: T1 is forward biased but not fired; no current; $v_o = 0$.
2. $\omega t = \alpha$: T1 is fired; it conducts and $v_o = v_s$.
3. $\pi$: supply and load current fall to zero together (resistive load), so T1 turns off naturally (line commutation).
4. $\pi < \omega t < \pi + \alpha$: T2 is forward biased but not yet fired; $v_o = 0$.
5. $\omega t = \pi + \alpha$: T2 is fired; $v_o = v_s$ (negative) until $2\pi$.

The current has the same shape as the voltage: $i_o = v_o/R$.

Waveforms

 vs    _               _
     /   \           /
 ---/-----\---------/----
            \     /
              \_/
 vo      .
        | \
 -------|--\--------.------
 0      a   pi      |    / 2pi
                    |  /
                    |/
                   pi+a
 gates: T1 at a, T2 at pi+a

The output is an AC waveform with chopped beginnings of each half cycle.

RMS output voltage

With $v_s = V_m \sin\omega t$ and by symmetry of the two halves:

$$\begin{aligned} V_{o} &= \left[\frac{1}{\pi}\int_{\alpha}^{\pi} V_m^2 \sin^2\omega t\; d(\omega t)\right]^{1/2} \\ &= \left[\frac{V_m^2}{2\pi}\int_{\alpha}^{\pi} (1-\cos 2\omega t)\; d(\omega t)\right]^{1/2} \\ &= \left[\frac{V_m^2}{2\pi}\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)\right]^{1/2} \end{aligned}$$

Since $V_s = V_m/\sqrt{2}$:

$$V_o = V_s\sqrt{\frac{1}{\pi}\left(\pi - \alpha + \frac{\sin 2\alpha}{2}\right)}$$

• $\alpha = 0$: $V_o = V_s$ (full voltage).
• $\alpha = \pi$: $V_o = 0$.

So the rms output is controlled smoothly from $V_s$ to 0 as $\alpha$ goes from 0 to $\pi$. The input power factor is $V_o/V_s$, which falls as $\alpha$ increases.

Applications: lamp dimmers, heater control, speed control of fans and small induction motors, soft starters.