AC voltage controller rms and ELC use

A single-phase AC voltage controller varies the rms value of the AC voltage applied to a load, at the supply frequency, by delaying the firing of two anti-parallel thyristors (phase angle control).

Circuit and operation

        T1 --->|---
   +---|          |----+
   |    ---|<---  |    |
   |       T2          |
 ( ~ ) vs              R   vo
   |                   |
   +-------------------+

1. $0 < \omega t < \alpha$: T1 not yet fired, $v_o = 0$.
2. At $\alpha$: T1 fired, $v_o = v_s$ until $\pi$, where the current falls to zero and T1 turns off (natural commutation).
3. $\pi < \omega t < \pi + \alpha$: $v_o = 0$.
4. At $\pi + \alpha$: T2 fired, $v_o = v_s$ until $2\pi$.

 vo     .
        | \
 -------|--\--------.------
 0      a   pi      |    / 2pi
                    |  /
                    |/
                   pi+a

RMS output voltage

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

As $\alpha$ goes from 0 to $\pi$, $V_o$ goes from $V_s$ to 0, and load power $V_o^2/R$ from full to zero.

Use in the ELC of a micro hydro (MHP) scheme

Micro hydro plants usually run with a fixed water input and no mechanical governor. The electronic load controller (ELC) keeps the total electrical load constant:

$$P_{gen} = P_{consumer} + P_{dump} = \text{constant}$$

 Gen ---+-------> Consumers
        |
        +--[ AC voltage controller ]--> Dump heater
        |            ^ alpha
        +-> sense f/V -> controller

• The controller senses generator frequency (or voltage).
• Consumer load decreases → frequency tends to rise → $\alpha$ is reduced → more power to the dump (ballast) heater.
• Consumer load increases → $\alpha$ is increased → less power to the dump load.
• So speed, frequency and voltage stay nearly constant without a governor.
• The dump load is usually water or air heaters, and the heat can be used.

Drawbacks of phase control here are harmonics and poorer power factor at mid-range $\alpha$; binary-weighted dump loads or integral cycle control reduce this.