Electronic – Power Supply Diode Keeps Burning Out

A typical use of the power supply schematic in the O.P. is for powering one (1) analog circuit, which needs both positive and negative power supply rails. In principle, you can power two (2) completely independent circuits from it (+9 to 0 and 0 to -9). This would be a peculiar scheme, though, because the ground of one circuit is at 9V (plus or minus) w.r.t. ground of the other one. Still, if the circuits are in fact independent, such scheme would work. In practice, this scheme is not used in general purpose desktop power supplies.

A classic desktop power supply (like the one in the YouTube video linked in the O.P.) is of a "dual positive type". Each channel has its own independent secondary winding (or even a separate transformer) and its own rectifier. The channels can float with respect to each-other, and that allows to connect them in series.

Not quite safe.

First, the transformer is fine. However...

12V AC would deliver 16.8V DC minus the diode bridge drop of 1.2V, so 15.6V nominal. This is pushing the limits of the 16V capacitor, but looks OK at first glance.

But if your AC mains is above nominal, or if the transformer is poorly regulated*, then the voltage at no load will rise above the nominal 15.6V by anything up to: 10% for the mains voltage, and 15 or 20% for the transformer's regulation; I'll use a total of 25% - that would give a worst case voltage of 19.5V across it.

So be prepared for the capacitor to have a short, messy, and possibly spectacular life. You might get away with it for a short while, but better to replace it with a 25V one.

• Transformer regulation : to deliver 12V AC at its rated current, a transformer is designed to give 10%, 15% or 20% more when unloaded. Big transformers (> 2amps) may have 10% regulation; I'm assuming 15% for yours, but its datasheet may tell you for sure.