AC Voltage Controllers

With phase-angle control at firing angle $\alpha$ on a resistive load:

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

Alternatively, integral-cycle (burst) control switches whole cycles on and off — no harmonics within the cycle, good for heaters with long thermal time constants.

Phase control is cheap but creates harmonics and a poor power factor. Electronic fan regulators use this principle; BLDC fans use an inverter instead and are far more efficient.