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What breaker and cable for a 3,000 W inverter?

24 V bank: a 200 A DC breaker and 4/0 AWG (120 mm²) battery cable.
48 V bank: a 100 A DC breaker and 2 AWG (35 mm²) battery cable.
12 V: not practical at 3,000 W.

The same rule the lab's safety check uses, the same in any country: it is DC, so the mains voltage does not matter.

Battery bankMost it drawsBreaker (at the battery +)Battery cable
12 V294.1 A—not practical: over 260 A, use a higher-voltage bank
24 V147.1 A200 A DC breaker4/0 AWG (120 mm²)
48 V73.5 A100 A DC breaker2 AWG (35 mm²)

Why these sizes

  1. An inverter draws the most when the battery is low: 3,000 W ÷ (bank volts × 0.85), the 0.85 covering the low battery voltage and the inverter's own losses.
  2. The breaker is the next standard size at 125 % of that, so a full load does not trip it, and it sits at the battery's + terminal, where a short would start.
  3. The cable must be rated for the breaker. At low voltage the currents are large: a higher-voltage bank halves the current and the cable size.

The cable is fine-stranded battery or welding cable, short and the same length on + and −; check the inverter manual for its own fuse rating and cable size, which come first. Most 3,000 W systems are built at 24 V.

Open it on the benchAn off-grid system with its battery breaker: change the inverter and let the safety check size the breaker.

For planning and learning. Use the appliance's nameplate (amps, MCA and MOCP) when you have it, follow your local electrical code, and have a licensed electrician do work on a building's wiring.