The brief fits on one line: keep a Wi-Fi router, a laptop and one desk lamp running through a power cut of up to eight hours, from a single 12 V battery and a small inverter. No generator, no rooftop panels. This is that project worked through on paper, from the outlets back to the battery, with every number taken from a calculator rather than from a rule of thumb.
The rule of thumb is where it starts to go wrong, so it is worth writing down first. Amp-hours times volts divided by watts: 100 Ah × 12 V ÷ 67 W is 17 h 55 min. More than twice the target. Buy any 100 Ah battery and stop reading? Not quite.
The project: 67 watts, eight hours, one battery
The loads are assumptions for a typical small office, not measurements, and each one can be swapped for a figure from a plug-in watt meter. They add up to the number everything else hangs on.
| Device | Power | Note |
|---|---|---|
| Wi-Fi router and modem | 12 W | Runs all the time, so it sets the floor |
| Laptop on its charger | 45 W | An average for office work, below the 65 W on the charger label |
| LED desk lamp | 10 W | The only light kept on |
| Total at the outlets | 67 W | What the inverter has to deliver |
Eight hours of 67 W is 536 Wh delivered at the outlets. The rest of the project is about how much battery it takes to put 536 watt-hours out of a 120 V socket.
From the outlet back to the battery
The inverter is a transformer with a bill attached
A 12 V battery cannot feed a laptop charger directly, so an inverter steps it up to 120 V. Inside a classic low-frequency inverter that job is done by a transformer with a turns ratio of roughly 1 : 10, and the transformer calculator shows what that ratio means for current. A 300 VA inverter, a common small size, may deliver 2.5 A on its 120 V side. On the 12 V side an ideal transformer would draw 25 A. With 90% efficiency, an assumption to replace with the figure from the datasheet, the input is 333.3 VA and the current 27.778 A, with 33.33 W lost as heat at full load.
The project will never run the inverter at full load, though. At the planned 67 VA the same calculation gives 558.3 mA on the 120 V side and 6.204 A on the 12 V side, from 74.44 VA of input. So 7.44 W disappears in conversion, all day long. Not far off the lamp's own 10 W, spent on nothing visible.
The same arithmetic scales all the way up to building transformers, where a 75 kVA unit from 480 V to 208 V allows 208.18 A per line on the secondary. Ratings in kVA turn into amps with one division, and the ready-made pages at the end of this article list nine of them.
What the battery actually has to supply
With the inverter loss included, the battery supplies 74.44 W. On a 12.8 V lithium iron phosphate battery that is 5.816 A, while the load alone would suggest 5.234 A. The battery life calculator shows both figures side by side once an efficiency is entered, which is the number to check the cable and the fuse against.
Here the project reaches its first real decision: which battery.
The chemistry decides half the answer
A lead-acid battery, flooded or AGM, is usually run to about half its rating if it is expected to last many cycles. A LiFePO4 battery can use most or all of its rating. The same "100 Ah" on the label therefore means two different amounts of usable energy, and the table below puts the four options through the calculator with the 90% inverter.
| Option | Rated energy | Usable energy | Runtime at 67 W |
|---|---|---|---|
| Ah × V ÷ W, nothing else | 1,200 Wh | 1,200 Wh | 17 h 55 min |
| 100 Ah AGM, 12 V, run to 50%, 90% inverter | 1,200 Wh | 540 Wh | 8 h 4 min |
| 100 Ah LiFePO4, 12.8 V, 90% inverter | 1,280 Wh | 1,152 Wh | 17 h 12 min |
| 50 Ah LiFePO4, 12.8 V, 90% inverter | 640 Wh | 576 Wh | 8 h 36 min |
The rule of thumb said almost 18 hours. A 100 Ah AGM battery, treated the way lead-acid wants to be treated, gives 8 h 4 min, which clears the eight-hour brief by four minutes. The lithium battery of the same amp-hour rating gives 17 h 12 min, and a lithium pack of half the size, 50 Ah, still gives 8 h 36 min.
With the depth-of-discharge box left empty, the result names both answers at once, the full rating for lithium and half of it for lead-acid. For a plain 100 Ah, 12 V battery and a steady 200 W load that is 6 h or 3 h, and the ready-made pages below go from 50 W to 1,000 W.
The few milliseconds nobody plans for
A manual changeover, or a standby UPS switching to its inverter, leaves a gap. During it the router lives on whatever is stored in its own power supply. Assume 1,000 µF on its 12 V input, an ordinary electrolytic value. The capacitor energy calculator puts 72 mJ in it. The router draws 12 W, which at 12 V behaves like 12 Ω, so the time constant is 12 ms and the voltage falls to 10 V after 2.1879 ms, to 9 V after 3.4522 ms.
How low the router can go before it reboots depends on its regulator, which is in no manual. The lesson does not depend on it: an input capacitor buys a few milliseconds, so the changeover has to be faster than that or the router restarts and the video call drops anyway.
Could a supercapacitor replace the battery?
Only for very short jobs. One minute of the router's 12 W is 720 J. Holding that at 12 V takes a 10 F capacitor bank, and even then only if every last joule could be used, which it cannot, because the voltage sinks as the energy goes. 720 J is 200 mWh. The 100 Ah lithium battery holds 1,280 Wh, which is 6,400 times more. Supercapacitors are for seconds and very many cycles; eight hours is battery territory.
The balance sheet: first guess against calculation
Every row is a number the project would have used without a calculator, next to the one it ended up with.
| Item | First guess | Calculated | Difference |
|---|---|---|---|
| Runtime of a 100 Ah AGM battery | 17 h 55 min | 8 h 4 min | 9 h 51 min shorter |
| Current from a 12.8 V battery at 67 W | 5.234 A | 5.816 A | 0.582 A more |
| Inverter input at its full 300 VA | 25 A | 27.778 A | 2.778 A more |
| Battery that covers 8 hours | 100 Ah of anything | 50 Ah LiFePO4 or 100 Ah AGM | half the amp-hours if lithium |
| Router ride-through on 1,000 µF | "plenty" | 2.1879 ms to 10 V | milliseconds, not seconds |
The first row is the one that would have hurt. A plan built on 17 h 55 min either buys a battery twice too small for lead-acid or ruins a lead-acid battery by draining it flat night after night.
Lessons from the build
Count watt-hours at the battery, not amp-hours at the label
Amp-hours only become energy with a voltage attached, and the 12 V and 12.8 V batteries in this project differ by 80 Wh at the same 100 Ah. Convert first, then subtract losses, then divide.
Ten percent for the inverter adds up
7.44 W of loss at a 67 W load is three quarters of what the lamp itself uses. Running the router straight from 12 V, where its power brick allows it, removes it from the inverter entirely, and on its own the router would run 4.4 days on the lithium battery.
The chemistry changes the answer by a factor of two
No runtime is complete without saying which battery it assumes. The 50 Ah lithium pack and the 100 Ah AGM battery land within 32 minutes of each other for this load.
Capacitors are for the gap, batteries for the outage
Milliseconds from an input capacitor, a minute from a 10 F bank, hours only from a battery. Mixing up those scales is how a backup that looks fine on paper still drops the connection.
Starting over
Planned again, the order would be the reverse of how people usually shop: list the loads and their watts, pick the inverter and write down its efficiency, decide the chemistry, and only then look at amp-hours. For this brief that ends at either a 100 Ah AGM battery with no margin at all or a 50 Ah LiFePO4 battery with 36 minutes to spare, and the lithium option is the one that still works when the laptop draws more than its assumed 45 W.
Tools discussed in this article
- Battery Life Calculator: runtime from mAh, Ah or Wh and a load in mA, A or W, with depth of discharge, inverter efficiency, the current drawn from the battery and a C-rate warning
- Transformer Calculator: turns ratio, secondary or primary voltage and turns, and full-load amps from a VA or kVA rating, single-phase or three-phase
- Capacitor Energy Calculator: energy, capacitance or voltage from the other two, in joules and watt-hours, with the RC discharge time to a target voltage
Ready-made calculations
Full-load amps of a three-phase transformer from 480 V to 208 V: 15 kVA, 41.636 A, 30 kVA, 83.272 A, 45 kVA, 124.91 A, 75 kVA, 208.18 A, 112.5 kVA, 312.27 A, 150 kVA, 416.36 A, 225 kVA, 624.54 A, 300 kVA, 832.72 A, 500 kVA, 1,387.9 A.
How long a 12 V, 100 Ah battery lasts, lithium and lead-acid run to 50%: 50 W, 24 h or 12 h, 100 W, 12 h or 6 h, 150 W, 8 h or 4 h, 200 W, 6 h or 3 h, 300 W, 4 h or 2 h, 400 W, 3 h or 1 h 30 min, 500 W, 2 h 24 min or 1 h 12 min, 1,000 W, 1 h 12 min or 36 min.
More physics tools
The rest of the physics shelf, in the order it was built:
- Momentum Calculator, Force Calculator and Kinetic Energy Calculator
- Potential Energy Calculator, Work Calculator and Mechanical Power Calculator
- Torque Calculator, Kinetic Friction Calculator and Pendulum Period Calculator
- Thermal Expansion Calculator, Specific Heat Calculator and Thermal Conductivity Calculator
- Hooke's Law Calculator, Simple Harmonic Motion Calculator and Buoyancy Calculator
- Acceleration Formula Calculator, Density Calculator and Temperature Converter
- Thermal Efficiency Calculator, Gravitational Force Calculator and Snell's Law Calculator
- Refractive Index Calculator, Thin Lens Calculator and Magnifying Glass Calculator
- Wavelength to Color Calculator, Wave Frequency Calculator and Photon Energy Calculator
- Speed of Sound Calculator, Echo Distance Calculator and Ohm's Law Calculator
- Electrical Power Calculator, Amps Calculator and Parallel Plate Capacitor Calculator