I'm not going to touch on supercapacitors. If you can live with electrolytic capacitors, there's a trick: their low voltage energy density is underwhelming. And, to extract most energy from them, you can't just discharge them by directly connecting them to the load that expects a small range of supply voltages. You'd be only using a fraction of the energy stored in them that way. You need a switching power converter to discharge as deeply as possible until diminishing returns set in.
According to this answer, you'd want to use capacitors rated for 400-450V, since per unit volume they give you most energy stored. You'll want to charge them up to 95% of the rated operating voltage, and discharge them down to 50-100V. The lower discharge voltage depends on how good a switching converter you can put together to efficiently convert the higher voltage into the low voltage output.
Thankfully, this is a solved problem: any high-efficiency 12V-output wide input range switching power supply does a good job at discharging capacitors down from a couple hundred volts, while putting out 12V at high currents. Supplies with PFC run their DC-link capacitor quite close to 400V, so you're in ideal energy density territory. How convenient!
The Need for PFC
Without a PFC, the rectifier voltage would be a bit too low to be practical when supplied in Japan/US/Canada, i.e. from 100-120VAC. You'd have to add a voltage doubler and feed the entire supply from doubled rectifier voltage, and then you end up with a 120/240VAC switch typical of old PCs. Those are not practical and not expected in modern equipment - just leverage the PFC.
Most supply designs transition from a dumb rectifier to a PFC somewhere in the 150-250W power range. Since your load is 120W, and you'd probably want to recharge at 1/3C peak, a rating of 160W would be sufficient. With some allowance for derating a 200W supply with PFC would be an OK choice. The excess capacity is used to recharge the capacitor bank while the load consumes full power.
A Simple Design for a Capacitor Bank
simulate this circuit – Schematic created using CircuitLab
The design proposed above is not ideal by any means, but it has some positive characteristics in terms of its handling of failure modes and least interference with the parent supply's operation.
During normal operation, with C2 fully charged and AC present, the capacitor bank is disconnected from the DC link. D2 is open and a small self-discharge current is drawn through D1.
During startup, C2's charging ballast R1 is disconnected until the output converter starts up successfully. An additional delay is provided by RLY1's inherent turn-on delay. This ensures that the PFC startup conditions are conservative: as-if the capacitor bank wasn't present.
Should the converter fail internally, the output voltage is lost, and the capacitor bank is quickly disconnected. The converter, in a failure mode, doesn't have to cope with an unanticipatedly high DC-link capacitance.
When the AC power is lost, the capacitor bank provides backup power to the converter with an inherent delay needed by C1 to discharge by the drop of diode D2.
During backup power operation, C1 and C2 are in parallel.
As soon as the usable backup energy capacity is spent, the converter's output turns off, and the bank C2 is switched into discharge mode. No discharge current flows during normal operation, saving energy.
The C2 charge indicators X1 and X2 - presumably LEDs - are configured for minimum power loss: X1 runs from 12V and doesn't waste much power in the series resistor. X2 runs from the DC-link voltage only during C2 discharge; its series dropper dissipates power that had to be dissipated anyway.
The capacitor charging circuit is simple: a series resistor R1 to limit charge current through D1 into the capacitor bank C2. If the power-up events are rare, the energy loss on R1 is not substantial and doesn't have undue impact on the energy efficiency of the device. If dictated by the requirements, a switcher-based constant current source could replace R1.
The bank C2 doesn't need to charge super fast: a charge lasting 2-3 minutes would be reasonable given the application need of 20-30s backup time, but this can be adjusted as needed. Don't forget about the extra PFC capacity needed to charge C2, and derate the 12V supply accordingly.
In the quiescent state, when the 12V output voltage is absent, the discharge relay RLY1 connects the C2 in parallel with the safety discharge resistor R2. It should be selected to discharge the bank C2 down to a safe voltage of 48VDC in a "reasonable" time - say 10 minutes. The C2-R2 time constant would then be in the ballpark of 1-2 minutes.
Fuses F1 and F2 are important. F1 protects the capacitor bank from destructive discharge (a.k.a. a big bang) should the DC link become shorted. F2 protects RLY1 and the associated wiring, since it wouldn't be rated for the full output current of 10-15A.
The unsafe voltage on C2 is indicated in an energy-saving fashion by two indicators: X1 and X2.
X1 is active when the DC output is present. Obviously, C2 is at an unsafe voltage then, or either F1 or F2 has opened - a condition to be aware of in either case.
If the DC output is lost for any reason - be it due to discharge of C2 below converter's turn-off voltage, or due to converter failure, RLY1 switches C2 into discharge mode, and X2 takes over.
If either X1 or X2 is lit, C1||C2 is presumed at an unsafe voltage.
X2 and R1 provide redundant discharge paths for the DC-link capacitors, albeit at a vastly different time constants.
Such capacitor banks are arc flash hazards, and without proper precautions, it's too easy to get seriously injured. Have a redundant charge state indicator so that you have a last-resort visual indication that things are charged and dangerous. Please familiarize yourself with arc flash hazards in capacitor/battery bank systems!
The capacitor bank will potentially stay charged for years if the discharge load fails. Treat it as dangerous until you verify it's down to a safe voltage with an external trusted instrument.
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