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Okay where to start? First of all let's identify the op amp. It's the two transistors in the middle with the common emitter resistor. On the left is the + input of the op amp and on the right the - input, that is the positive and negative feedback terminals respectfully. First of all to have voltage regulation we need a reference voltage. That's the 5V zener at the left. The voltage stability of the reference defines the ripple rejection we can expect from the output. There are other factors but for now let's keep it at that. If we vary that voltage we essentially vary the output voltage in a larger degree. I'll get to that in a second. So if we want the output to remain continously variable and stable, we need a negative feedback from the output. That's the 10k/4k voltage divider from the output. What it does is that it steps down the output voltage within the reference voltage boundaries. The op amp then has a voltage gain equal to the difference of the maximum output voltage and the maximum reference voltage (18/5=3.6 times). It's important to add external transistors to handle all the regulation, because loading the op amp directly, even if it's a power OP AMP will cause some error in the output, maybe even some variation with the load. It's also important to keep the output voltage a few volts lower than the input voltage, like it's here, the output is 18V, the input is 20V. This is because a typical rectifier will drop voltage on higher loads and may pass below the Vout requirement, which essentially leaves you with an unregulated output, with the variances and the high ripples and everything. I usually like to add current limiting at the output, but that requires a few extra circuitry to keep things normal, because when the limiter tries to drop the voltage, to compensate for the increase in current, the op amp doesn't know that, and further increases the voltage on the base of the transistors to compensate, but the current limiter usually prevails, and the op amp hits the supply rail. This leaves you once again with an unregulated output.
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