Electronic – Why is the negative terminal connected to both transformers in this half-wave rectifier

Maybe these illustrations will help:

Let's assume that the start and end of the primary and secondary windings are such that the 'starts' are at the top of the picture and the 'finishes' are at the bottom.

When primary current flows from the top of the picture to the bottom, the top of the primary winding is at higher voltage than the bottom. This will induce a voltage in the secondary winding with the highest potential at the top of the winding and the lowest at at the bottom (and a potential somewhere in between, at the center-tap).

You can tell quickly that D1 will be reverse-biased, because the voltage at the top is higher than at the center-tap. Current will flow however it can, which will be out of the center-tap, through D2 and into the bottom of the winding.

When primary current flows from the bottom of the picture to the top, the reverse condition holds true in the secondary: D2 will be reverse-biased, and current will flow from the center-tap through the load and through D1 back to the winding.

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.