"And indeed, this is pretty unusual. I'm using Zyxel ADSL modem which has all the rectification circuits inside."
So, tell the doctor, you don't need the AC really, do you? First thing it meets in the modem is a rectifier, followed by a smoothing capacitor. And then the AC is gone. The modem works on a DC power supply.
The 9V AC has a peak value of 12.7V, the bridge rectifier drops about 2V, so the smoothing capacitor sees 10.7V minus some ripple. If you supply the modem with a 12.7V DC supply the result is exactly the same: 2V drop across the rectifier and 10.7V on the capacitor, sans the ripple then. The absence of ripple means that it's pretty certain it will also work at 12V in.
So you don't need a regulated 9V AC, but a regulated 12V DC. You can use a linear regulator or a switcher if you want more efficiency. Things to look out for:
- Smoothing capacitor capacitance should be sized for 180V on your transformer, but the working voltage for the 230V
- Input voltage of the regulator should be OK at both 180V in and 230V in
There may be a minor issue with the bridge rectifier in the modem. If you supply AC to it diodes conduct current on alternating half cycles, with a DC input only two diodes will conduct all the time, so their power will be double. If the diodes get really hot you may replace them by a more heavy version.
Fake name mentions this rectifier:
and indeed if that's used the DC solution doesn't work. Actually I'm not really worried about this. This design has some disadvantages. First, only half of the input power is available for V+. In a modem you'll need most, if not all, power for V+. Second, this is only a half wave rectifier, which either needs very large smoothing capacitors (expensive and space-consuming), or gives a large ripple, which means an inefficient power regulation. I doubt this will be used.