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I want to control the amount of current going through a Triac by adjusting the firing angle accordingly. The Circuit is shown below:

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Let's say i want IRMS to be 0.28A then the controller need to change α to be 90° according the formula:

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My problem is I can't derive the Transfer function in order to calculate the needed parameters. Here is what i got so far: $$u(t)in = Vmax.sin(ωt)$$ can also be described as: $$u(t)in = i(t)R + L\frac{di}{dt}$$ Laplace Transformation gives: $$Vmax \frac{ω}{s²+ω²} = iR + L s i$$ Then the Transfer function of the RL load is: $$G(s) = \frac{i}{Vmax} = \frac{\frac{ω}{s²+ω²}}{R+Ls}$$ I am stuck here. There is no Correlation with α and the function is too complicated. The controller i am planning to use is a PT1 Controller with the following transfer function: $$G(s) = \frac{K}{1+Ts}$$ At the end this will be programmed inside a Raspi Pico W. Can anyone shed some light on this?

EDIT:

using the values provided by @Transisor comment I managed to get the following equation: $$Vrms = Vpeak(-0,2552\alpha + 0,832726)$$ doing a laplace transformation gives: $$ -0.2552Vpeak \cdot \alpha + \frac {0.832726Vpeak}{s} = I(s) \cdot R + L \cdot sI(s)$$ I have neglected $$\frac {0.832726Vpeak}{s}$$ to make the system a LTI system and got: $$G_s(S) = \frac{I}{\alpha} = \frac{-0,2552 \hat{V}/R}{1 + \frac{L}{R}s}$$ does this make sense? can I keep going from here?

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I would second the idea of using a lookup-table. You typically only need an approximate relationship. After all, in many cases you can get good enough control using zero voltage switching (which has other advantages). The controller will take care of any differences provided the slope is not so different that it affects tuning too much.

Now, if you are attempting to control temperature, you might be better to use the power relationship than the voltage relationship, assuming R1 is constant and maybe that the mains voltage is constant (which won't exactly be true either).

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I think you are going to need to calculate the inverse of the integral: \$ \int_\alpha^\pi v\ dt \$ where \$\alpha\$ is the turn-on point. I'd be inclined to use a lookup table - but you may need to know what the peak voltage of the supply is as variations is supply voltage will affect the turn-on point.

For a given phase angle, \$ \alpha \$ the RMS output voltage will be $$ V_{rms} = V_{pk} \sqrt{\frac {2\pi - 2\alpha + sin\ 2\alpha}{4\pi}} $$ (Check this. It's been a while ...)

You should be able to derive the formula above from the root of the mean of the squares (RMS). $$ V_{rms} = V_{pk}\sqrt{\frac 1 \pi \int_\alpha^\pi sin^2x\ dx}$$

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Figure 1. VRMS as a function of Vpk and TRIAC trigger angle.

For your application you can generate as many points as required for the lookup table or run the calculation.

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  • \$\begingroup\$ which lookup-table should i use? \$\endgroup\$ Commented Oct 28, 2024 at 8:46
  • \$\begingroup\$ See the update. \$\endgroup\$ Commented Oct 28, 2024 at 13:06

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