... because in my simulation it doesn't seem to be working.

The control voltage is being set quite low, but the ramp waveform is stuck high. I really need it to follow the control voltage for my PWM generator to work. I'm sure I'm missing something...

I am using an LM324 at the moment for convenience as it has a push-pull output. But later I will use a proper comparator.

10 bonus points for figuring out what this circuit actually does .

Blue trace represents ramp voltage. Green represents control voltage. As you can probably see the ramp voltage exceeds the control voltage

 

The LTSpice NE555 is a macromodel that is optimized for speed. It doesn't act like a real 555, either BJT or CMOS; you can't buy one like it. It's good enough for quickie models just to get an idea of the output frequency and things like that, but otherwise you really need a better model.

I've attached a component model I got out of CircuitMaker Student; it seems to be a lot closer to what a real bjt 555 operates like.

I changed the pin layout so it's easier to use than the one that comes with LTSpice.

Put the .asy file in LTSpiceIV\lib\sym\Misc
Put the .sub file in LTSpiceIV\lib\sub

 

Does the NE555 timer in LTspice model the control voltage?

Yes , it does, unless you hold on to the trigger signal too long that the timer has already timed out.