This tells me the resulting RMS voltage when chopping an sine wave (starting alpha milliseconds after zero-crossing). The equation seems to be correct, and I’ve used it to create a table of values for 256 steps from zero mS to 8.3 mS.
But, since power is proportional to voltage squared, I want to start with the power table (calculated to be linear), and calculate the delay.
This means I need to solve for alpha, and I can’t figure out how to do it. I get to the point where I have alpha + sin(2alpha) / 2 = constant, and can’t remember how to isolate alpha.
I’m no math-whiz, but either the equation is correct or it’s incorrect, I don’t think there’s any middle ground here. First problem I have is you’re trying to take the sine of a time value, I’m pretty sure you need to divide by frequency to convert that alpha value from seconds of time into an angle value. Second problem I have is you’re trying to add a value in seconds to a dimensionless ratio.
You clear the sine function by taking the arcsine of both sides of the equation … maybe I’m missing something here but this step looks terrible mischievous.
Power = Current x Voltage = Voltage squared divided by Resistance
I once derived almost that equation. It took two days locked away from the world with a CRC handbook.
You might not be able to isolate the term, not uncommon with non-linear equations. In that case you will need to take an iterative approach. Newton’s method often works, but if you are not trying to optimize, a binary search might be simpler.
For the power you want, start stepping the switching angle 10 degrees at a time. When you finally get less than the desired power, reduce the angle by 5 degrees. If it is still less, then go 2.5 degrees less (because you know it would be too high if you went another 5 degrees, because you already tried that). go 1.125 degrees on the next step, choosing direction based on if power is too high or too low. Each step gets you halfway closer to the exact answer, unless you get lucky and hit it exactly. 10 iterations after you narrow it to a 10 degree window will get you within .01 degree.
If you program this into a programmable calculator, you can have the answer for each desired power level in a few seconds.
It is entirely likely that you can’t isolate alpha at all, and have to resort to a numerical solve. Even fairly simple equations can have this property (e.g. try solving for x in x = cos(x)).
OK, I figured out how to do what I wanted.
Even though I couldn’t isolate alpha, I realized that the power function was basically sin^2, so I created a timing table that was the square root of the arcsin of the angle, and that resulted in a linear curve (more-or-less - it’s a little bent at the high end, probably due to floating-point errors, but it’s close enough).