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I am running my dsl modem and router on my 12V UPS battery. But there is a 100 watts (12V DC to 220V AC) small chinese power inverter in-between. My this technique was successful as my modem and router never got restarted again as they used to do sometimes even on the UPS.
12VDC Battery >> 100W Power Inverter 220VAC >> Modem & Router
I didnt connected the modem directly to the battery because it never gives constant 12 DC volts, it varies from 10 to 13 volts depending on charging. But you can try your technique at ur own risk, hoping it will not harm ur modem.
Also make sure while charging ur battery through a charger that your charger has capability to stop charging once charging is complete, otherwise u will end up with a exploded battery.
You technique is good but there is wastage of energy during conversion.
Power inverter converts 12V to 220V, then the power adapters convert again from 220 to 9V/12V required by your modem and router.
If you are familiar with electronic circuits, you can make more efficient use of battery power, consider following setup:
12V Battery ----- DC/DC converter ------ modem and router
the DC/DC converters can be found easily in the market for around Rs 300/400. These are usually for use in cars by plugging into the cigarette lighter point and have adjustable output voltages.
Alternatively, you can replace DC/DC converter with a regulator IC circuit to have sustained output voltages. The circuit will cost you less than 100Rs and will give you 12V output even if input voltage is higher or lower. I can’t recall the exact upper/lower limits right now but I am sure it can handle ±2V variations.