TRIAC Angle of Firing vs Power Delivered to Load, how calculate?

Question

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 delivery 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). However, since you all know the power line is not linear, but a sine function, so 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?

Answer 1

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.

Answer 2

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.