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Both designs will not regulate at low voltage output. It is hard to get this type of error amplifier to work below 2.5 volts.The PSU I want to build will have a PWM controller with variable output voltage (0-20V , 0-20A)
You're right about them not regulating down to 0V; the first could get down to around 3-4V (due to TL431 voltage of 2.5V, current >=1mA & ignoring series LED), and the second down to 3V due to the reference of 3V.Both designs will not regulate at low voltage output. It is hard to get this type of error amplifier to work below 2.5 volts.
VR1 is ok; adjustable range is 3-53VDesign B; I do not understand how VR1 sets the output voltage. There might be an error around VR1.