# TRIAC Firing angle vs power delivered to load, how calculate?

I'm building a phase control system for simple power control to a given load (heater for example.)

I need an equation to find the relationship between angle of TRIAC firing vs power delivered to load (in percent.)

Let's say that my AC mains frequency is 60Hz. If I need to deliver 50% of power to the load I need to fire the TRIAC every half half wave of the AC (every 4.16ms after the zero-crossing.) Since, as you all know, the power line is not linear but a sine function, I cannot get a simple rule of three to find the power x time relationship.

What I need is an equation that solves for a given power (eg. 35%) I get the time after zero-crossing that the TRIAC needs to be fired.

Any ideas?

• Add the circuit diagram. Is it a single-phase or a three-phase load? Is the load purely resistive (probably since you said it was a heater) or a resistive-inductive series load? Commented Oct 6, 2021 at 17:38

See formula from Jony130 here: https://www.physicsforums.com/threads/how-to-calculate-rms-voltage-from-triac-phase-angle.572668/

On the other hand, if your load is not a lamp, but a slow resistive thing with huge current draw like heaters, you may better go for a standard zero crossing SSR.

The solution is just like PWM (or phase control), except that it is slower. Assume you have a time period of 0.8 seconds (800ms), and you only turn on/off the load at zero crossing. Zero crossing triac triggering takes care of turning on, and the triac itself will turn off after a zero cross. Now, if you turn on the SSR for one half-cycle (8 ms) and keep it off for 99 following half cycles, you have an 1% power. If you keep it on for 22 half cycles, and off for 78, you have 22% power, and so on.

This however assumes that the load reacts slowly, i.e. this trick will cause serious flickering if you're using a light bulb, or humming in case of some load. On the other hand this trick is quite friendly for slow loads, makes no transients and no unwanted harmoics on the power lines.

• I would add a constraint that after a skipped half-cycle, the next half-cycle should be of the opposite polarity. Otherwise, assuming low DC circuit resistance, you run some risk of building up a DC current in e.g. the pole transformer or any isolation transformer and magnetizing and saturating it. Probably most a concern at duty cycles that would fire only, or primarily, on the same polarity (50%, 25%, etc). Not sure how much of a risk this would be, but at least some edge cases. Commented Dec 9, 2021 at 20:03

I'm not really much of a math guy, but basically you would integrate to find the Average power over the whole period of the waveform and then subtract (the integrated) average power over the fraction of the waveform where it is turned off. The result will be the average power for that firing angle.

Here the "rule" of current .vs. angle ... (SOE, resistive load)

Case of SCR (1 half wave, thyristor) ... Adapt for triac (2 half waves).

Currents, AVG or RMS values :

Power function for triac :

For getting alpha .vs. Power (relative), simply solve this equation. (100 % power = 1).