Electronic – A stable negative supply for an op-amp (using 555)

Whole tribes are fighting religious wars over the microcontroller vs. 555 debate. I'm on the microcontroller side.

So the first question is: does it have to be a 555, and what are the arguments for it? (that's two questions.) There may be good reasons for it, but I'd like to know them. "My wife wants me to do it this way" is not a good reason (unless maybe when I see her). Without a good reason I'd have a look at the microcontroller.

The main advantages of the microcontroller are simplicity and accuracy. Simplicity as in you only need a microcontroller and a potmeter to set the duty cycle. Two parts for the complete circuit, or one if your analog control voltage comes from elsewhere in the circuit; then you can drop the potmeter. Accuracy as in 10 bits ADC translates in 10 bit PWM, that's 0.1 % precision at a 1 % accurate clock. Make that a 20 ppm clock if you want to add a crystal. Accuracy is also reproducibility.

OTOH the 555 can work at 15 V if necessary, which the microcontroller can't, and doesn't need programming. The latter is again one of those religious things, but in practice writing a program for this can be done as quickly as designing and calculating the 555 circuit: setup two identical PWM channels but one inverted, the other non-inverted, and in a loop read the ADC and copy the value to both PWM registers.
So the microcontroller outputs complementary signals without external components.

The 555 circuit relies on a capacitor and if you want that at 1 % tolerance you'll pay a few times more than what the microcontroller costs. And even then the frequency will vary with the duty cycle.

All in all there are few 555 applications which a microcontroller can't do better.

The op-amp can't do anything useful in this configuration. You need to give it more supply voltage than the desired output voltage, even though it's "rail to rail".

If you ask it to output 8V or even 11V with a 12V supply, it may be able to do that, provided the load is not too much for the op-amp to supply. In this case, that might be 10mA or so.

I don't know exactly why you're only getting 4V at the output, but I suspect it's because the load is something like a couple hundred ohms or equivalent, so the output current is limiting at 20-30mA.

Even if you put a lighter load on there, my first comment applies- this circuit can not do anything useful without some changes (more supply voltage compared to desired output voltage).