Step-down transformer AC to DC voltage conversion problem

Either can work correctly if designed properly. If you have a dumb rectifier supply feeding a 7805, then all the rectifier part needs to do is guarantee the minimum input voltage to the 7805 is met.

The problem is that such a power supply only charges up the input cap at the line cycle peaks, then the 7805 will drain it between the peaks. This means the cap needs to be big enough to still supply the minimum 7805 input voltage at the worst case current drain for the maximum time between the peaks.

The advantage of a full wave rectifier is that both the positive and negative peaks are used. This means the cap is charged up twice as often. Since the maximum time since the last peak is less, the cap can be less to support the same maximum current draw. The downside of a full wave rectifier is that it takes 4 diodes instead of 1, and one more diode drop of voltage is lost. Diodes are cheap and small, so most of the time a full wave rectifier makes more sense. Another way to make a full wave rectifier is with a center tapped transformer secondary. The center is connected to ground and there is one diode from each end to the raw positive supply. This full wave rectifies with only one diode drop in the path, but requires a heavier and more expensive transformer.

A advantage of a half wave rectifier is that one side of the AC input can be directly connected to the same ground as the DC output. That doesn't matter when the AC input is a transformer secondary, but it can be a issue if the AC is already ground-referenced.

For Bridge Rectifier selection: Short-list parts that exceed the required maximum voltage, and the required current, by a fair margin, as described below.

For sine wave output from a transformer, the required voltage would be sqrt(2)=1.4142 times the rated transformer output voltage, as transformers are rated for RMS voltage, not peak. Also, transformers are usually, but not always, rated lower than the actual voltage they produce across the secondary with no load: This drops to the rated voltage when the transformer is carrying the rated full load current. Hence, to be on the safe side, around 2.5 times the transformer rated voltage works well for me.

For current calculation as well, 2.5 times the expected load current is healthy - since you would need the bridge to withstand the initial current surge when any reservoir capacitors following the bridge are charging up after power-on.

Now that you have the voltage and current ratings to look for, listing available parts might show you higher rated parts that are cheaper than those just meeting your requirements - so just go with the higher rated parts.

For instance, in local stores near where I live, a BR68 bridge sells for less than half of a BR36, despite the much higher rating. This is due to economies of scale - the BR68 part is just more commonly used here.

Another consideration, though, is physical size / PCB layout: Higher rated bridges tend to increase in size. Also, sometimes SIP pin-put modules are just more convenient on the PCB, compared to square pin-outs, if vertical space is not an issue.

For discrete diode selection: The same calculations apply as for the bridge. The key advantage of going with discrete parts is that heat dissipation is a bit less bothersome, since each diode has its own surrounding space to dissipate heat.

A minor additional benefit is the facility to indulge in somewhat creative PCB layouts when needed, rather than being forced to give up a specific contiguous area on the board.