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There appear to be two kinds of power supply adapters in market.

The first/older kinds were such that they were reasonably heavy and had a small transformer inside the adapter itself. The newer ones in market (those that come with a mobile phone for instance) are surprisingly lightweight; I find myself wondering whether there is even a transformer inside.

How do these new-fangled adapters achieve step-down?

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These are called switched-mode or switching power supplies.

Yes, they still have a transformer inside. That's necessary, otherwise the low-voltage side wouldn't be touch-safe.
The transformer is kept very small. That's possible because it works at a very high frequency; the mains voltage is rectified and a high-frequency switcher feeds the transformer. The higher the frequency the better the energy transfer between primary and secondary are. Remember that transformers only work on variations in current/magnetic field, so a higher frequency makes for more changes.

Power supplies like this are not only recognizable by their light weight, they usually accept a wide range of input voltages as well, like 100V to 240V.

edit
This is a clever method which allows using a small transformer:

enter image description here

Here the high frequency comes from two spikes per 50/60Hz period, where a capacitor discharges into the transformer when a small thyristor switches the voltage near its maximum. Due to the brief transfers (once per 10/8.33ms) this is only suitable for low power applications. The schematic shows a 150mW supply.

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    \$\begingroup\$ actually rectified first, converted to high-frequency AC, stepped down, and then rectified again. \$\endgroup\$ – tcrosley Sep 21 '11 at 10:36
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    \$\begingroup\$ @Everyone - Yes, it would, but you don't have a 5A diode because you can't draw 1kW from the adapter. You may have up to 20W (e.g. 4A @ 5V), but that will translate to less than 100mA @ 240V. \$\endgroup\$ – stevenvh Sep 21 '11 at 11:00
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    \$\begingroup\$ @tcrosley, why do they do it that way instead of rectify and then DC-DC (buck converter)? \$\endgroup\$ – Matt B. Sep 21 '11 at 16:48
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    \$\begingroup\$ @Matt B. You need isolation for safety. Bucking the rectified AC is fine to make a rail that's not touch-safe (i.e. an internal housekeeping supply for a microwave or washing machine) but if the end user can touch it, it has to meet safety requirements (including isolation). \$\endgroup\$ – Adam Lawrence Sep 21 '11 at 17:19
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    \$\begingroup\$ @Everyone: There's an application note that shows in great detail how one could implement these types of switched power supplies: powerint.com/sites/default/files/PDFFiles/der265.pdf (Disclaimer: I do not work for Power Integrations, I just happened to stumble across it). It's consistent with tcrosley's comment: the high voltage end is rectified, converted to HF AC, then rectified and filtered. \$\endgroup\$ – In silico Sep 21 '11 at 17:20

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