Normalised RMS voltage of a full wave phase controlled power waveform

The recent article Soldering iron – temperature control failure gave a plot of V'rms vs conduction angle for a simple full wave phase controlled AC waveform, and I have been asked to explain the derivation.

The phase controlled switch turns on at some delayed time from the zero crossing of the AC waveform, and conducts until the next zero crossing.

With the simplest circuits, there is a practical limit to the achievable stable range of conduction angle, and a minimum of about 50° to a maximum of about 160° is typical.

The RMS voltage is the square root of the mean of the square of the instantaneous voltage. We can write an expression for the normalised RMS voltage as a function of conduction angle θ.

The normalised RMS voltage is the square root of the mean of the squares: \(V'_{rms}=\sqrt{\frac1\pi\int^{\pi}_{\pi-\theta} sin^2 (x) dx}\\\). If the load is a pure and constant resistance, we can also calculate normalised power: \(P'={V'_{rms}}^2\\\).

Plotting V'rms and P' against θ° we have:

A very common application of these type of controllers was to incandescent lamps, and in that case R is not constant but changes greatly with change in filament temperature, so the P' curve is not applicable.

If you were to measure V with a good true RMS meter, it should agree with the plot. On the other hand older / simpler multimeters which respond to the peak voltage and are calibrated based on an assumed form factor for sinusoidal wave shape with will not read the RMS voltage as shown above.