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On the first half of the cycle, the capacitor attached to the transformer's secondary is charged and the output capacitor receives no charging. On the second half, the both capacitors are charged simultaneously. So the transformer's secondary experiences a current conduction on both halves of the cycle and is thus not subject to the magnetizing of its core.
The full-wave voltage doubler circuit, while it does experience conduction through both halves of the cycle, works differently from the half-wave version. Basically, it charges one capacitor during one half of the cycle and then in the second half of the cycle, it charges the second capacitor. Because these capacitors are in series with each other, the voltages and frequency of the noise double.
The full-wave voltage doubler circuit reduces to two half-wave rectifier circuits placed in series. Now, if two half-wave voltage doublers are placed in series, the result is a full-wave voltage doubler. This will require three more capacitors and four more diodes. All capacitors must be rated for at least the full voltage doubled rail voltage and 150% of this value would be safer. So to the diodes must rated for at least the full doubled voltage. (Do not forget that this circuit only works with a center-tapped transformer.)
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Half-wave rectifier circuit
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But in the simple half-wave voltage rectified power supply shown above, the current will flow through the secondary winding in one direction on one half of the AC cycle and then it not flow on the other half of the cycle: up, off, up, off. These repeated single direction conduction cycles are similar in effect to hooking up a DC voltage source across the transformer's secondary, which would definitely work to magnetize the transformer's core.
The principle of operation for a half-wave voltage doubler, however, is different from that of the simple half-wave rectified power supply, being much closer to the full-wave rectifier circuit. They share the "half-wave" name because of noise frequency and half cycle charging of the capacitor to which the load attaches, but not because of the conduction cycle through the secondary. The half-wave voltage doubler's transformer conducts through both halves of the AC cycle just like the full-wave bridge rectifier circuit.
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What is wrong with magnetizing the core?
Transformers are able to pull off a fairly amazing feat: they are able, in a small package, to convert one AC voltage into another. This feat works by using the magnetic field that results from having a current flow through an inductance (the primary) engulf another inductance (the secondary) so that an AC voltage will develop in the second inductance (secondary). While this process does not necessarily require a core, it does greatly benefit from using a core. The core concentrates the magnetic field so it more effectively couples to the secondary and it greatly increases the inductance of both the primary and the secondary. But when the core becomes saturated (magnetized), it cannot do either very well.
A toroidal transformer is particularly susceptible to core saturation, as its core is very tightly made. On the other hand, the usual stack frame transformer's core cannot as easily be made tight and this sloppiness creates an untended air-gap in the transformer, which will serve to reduce the tendency towards saturation by the core. As the toroidal transformer lacks this small air-gap, it will not withstand much unidirectional current flow before saturating.
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