Hi,
Where does that ground connect to, was that in the original drawing?
Circuits like these have connections to either the line or the neutral, with ground just for protection not for the circuit itself.
Nowhere - it is there for the simulation to work at all. Without it you either get an error or garbage. Can be connected anywhere and it basically says '0V is here' so all other can reference it.
Of course that is what I suspected, but I like to be sure.
As others have mentioned, this circuit is highly questionable. If it ever works at all, it looks like it charges a cap which turns the triac on then the cap discharges so on the very next half cycle the triac turns back on again. As the cap probably does not discharge all the way, eventually it stays high enough to keep the triac on. That looks like the way it is supposed to work, if it does in fact work at all.
One of the problems is that we don't know which half cycle the triac will turn off and which half cycle it will turn back on again. This could lead to a non symmetrical AC waveform which means there will be a net DC for a time until the triac turns on fully. For example, if it turned on with the first half cycle then turned off after that half cycle was over, it would end up applying a net DC to the load. If this behavior kept repeating, it would look almost like a half wave rectified sine getting to the load, at least until the triac turned on and stayed on.
There is a chance though that you could mitigate this by making sure the time constant while it stays on and when it stays off is long enough such that it allows several cycles to get through to the load. That way the net DC should be small. The problem then is that a slow start circuit would not be as slow start as we would like it to be because it would have to let maybe 10 full cycles get to the load as soon as it turned on. This means it could be hard to create a slow start circuit without accurate timing such as from a microcontroller. That's of course unless you plan to run this with resistive loads only or other loads that can tolerate that unusual behavior.
The way this is usually done is a small value resistor is placed in series with the load, then after a short time the resistor is shorted out so the load get the full power of the line. A relay is probably best to short out the resistor here. If you decide you need multiple steps, you could use one larger value resistor and one smaller resistor, and short out the larger value resistor first then short out the smaller value resistor.
Of course this has to be tested also on the load you intend to use with it, and on any other load if that load is different.
If you use a microcontroller you can time it very precisely by syncing to the line voltage. It's always a good idea to keep the positive and negative half cycles equal in duration though to reduce the DC offset that could emerge. That simply means that if you trigger the positive half cycle turn on at say 1ms after the first zero crossing, then after the start of the negative half cycle turn it back on after 1ms also.
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