Q:

Making LM317 output voltage adjustable down to 0 V

I want to design a variable voltage regulator that will get from it`s maximum output (lets say 10V) down to 0V (or very close) considering the input as regulated 12V (from a PC power supply).
All the designs that I found use the 317 IC and can not go under 1.25V and I am pretty sure that there must be a way to do so.
I could not find any tutorials that explain in an easy way the way that the 317 behaves (beside the classical configuration) so any additional explanation is welcomed.

A:

To get the LM317 down to zero volts you need to bring the control pin down to -1.25 V.

Figure 1. This dual LM317 circuit consists of a current limiter based around LM317(1) and a voltage regulator based around LM317(2). The voltage regulator section is relevant to this post as it is adjustable down to zero volts. Source: ON-Semi datasheet.

The control pin of LM317(2) is pulled low by Q2 which is wired as a simple constant current sink pulling several milliamps from the adjust pin.
D3 and D4 clamp the top of Q2 at two diode drops (2 x 0.7 V) below zero (-1.4 V).
The 240 \$\Omega\$ resistor and 5k pot can then adjust from zero up to the supply limit.

The problem with this circuit is that you need to generate a negative supply capable of sinking the few milliamps. My answer to Smartest way to use current limit using LM317? (where I explained the current limiting section) may help in this regard.

simulate this circuit – Schematic created using CircuitLab
Figure 2. If a centre-tapped transformer is used the negative rail can be generated quite easily. In this example a half-wave rectified signal is smoothed by C2 which doesn't have to be very large.

A:

A somewhat nasty solution is to place two silicon diodes of suitable current rating in series with the LM317 output (after the voltage-setting resistor divider).  This would reduce the output voltage by about 1.4 volts.  This will degrade the voltage regulation slightly, as the diode voltage drop will vary with load current.
A better solution is to provide a -1.25 volt (or so) low current supply to the bottom of the voltage setting resistors, rather than connecting that point to ground.

A:

A regulator is an amplifier which stabilizes the output at a target voltage.   The LM317 does this by comparing the output to a 1.25V reference, and while you can get higher voltages than 1.25 (by voltage dividing the output) you cannot
get lower (voltage dividers top out at 1:1).   
Instead of an LM317, you can use an operational amplifier that senses
near ground, and for a 20V range, divide output by a fixed ratio (20:2.5)
and amplify difference of that divided output to a second divider on a
2.5V reference voltage (TL431 being a suitable reference source).
When the reference divider is at 1, output stabilizes at 20V; when it
is at 0, the output stabilizes at 0V.
See figure 13 here:
http://www.ti.com/lit/an/snoa589c/snoa589c.pdf

