The arithmetic, and where the 1000 comes from
Amp-hours = (daily kWh × days of autonomy) ÷ (system volts × depth of discharge) × 1000
Ten kilowatt-hours a day, two days of autonomy, a 48 volt bank, 80 percent depth of discharge: 20 kWh divided by 38.4 gives 0.5208, times 1000 is 520.8 amp-hours. The thousand is only unit bookkeeping — kilowatt-hours to watt-hours — and it is where most hand calculations go wrong by a factor of a thousand in one direction or the other.
Notice what depth of discharge does to the answer. Keep everything else the same and drop from 80 percent to 50: the same 20 kWh now needs 833 amp-hours. That is not a small adjustment, it is 60 percent more battery for the same job, and it is the single biggest reason lithium banks look expensive per amp-hour and turn out competitive per usable kilowatt-hour.
Depth of discharge is a design choice, not a spec
Every battery has a curve relating how deeply you cycle it to how many cycles it gives before capacity falls to some threshold. Cycle it shallow and it lasts a long time. Cycle it deep and it does not. The figures below are the ranges commonly quoted, and they are orientation only — the specific product's datasheet is the authority, and the spread between two batteries of the same chemistry can be large.
| Chemistry | Commonly designed to | What limits it |
|---|---|---|
| Flooded lead-acid | 30-50% routine | Sulfation and plate wear. Deep cycles cost cycle life quickly. |
| AGM / gel | 30-50% routine | Same wear mechanisms, less tolerant of abuse than good flooded cells. |
| LiFePO4 | 80-100% | Much flatter cycle-life curve. Most of the limit is the BMS cut-off, not the chemistry. |
Two things that are not in the table and matter as much. Cold: lead-acid loses a meaningful share of its capacity near freezing, and most lithium cells must not be charged below freezing at all unless the battery has heating or the management system blocks it. And rate: lead-acid capacity is quoted at a slow discharge, commonly C/20, and pulling it faster gives you materially less than the nameplate. A 100 Ah lead-acid battery emptied in four hours is not a 100 Ah battery. Lithium is far less sensitive to this but not immune.
Series, parallel, and making the arithmetic close
Series adds voltage and keeps amp-hours. Parallel adds amp-hours and keeps voltage. Four 12 volt 100 Ah batteries in series is a 48 volt 100 Ah string, which is 4.8 kWh. Two of those strings in parallel is 48 volts and 200 Ah, which is eight batteries and 9.6 kWh. The check that catches most errors is that series count times battery voltage must equal the bank voltage exactly — not approximately, exactly — and strings times battery amp-hours must reach the target.
A practical caution on parallel strings: they only share current if they are electrically identical, which means the same age, the same model, the same state of health and cable runs of the same length to the busbar. Uneven cabling means one string works harder, ages faster and then works harder still. Several parallel strings of lead-acid is a known way to get a short-lived bank; where the capacity is needed, larger cells at a higher system voltage is the cleaner answer.
Why 48 volts keeps winning
Power is volts times amps, so the same 5 kW draw pulls about 417 amps from a 12 volt bank, 208 from 24 volts and 104 from 48. Conductor cross-section scales with current, so a 12 volt system at any serious power ends up on cable that is genuinely awkward to bend, terminate and pay for, and the voltage drop over even a short run becomes a large fraction of a small supply. Forty-eight volts is where the cabling becomes sane and where most modern inverters and battery modules are designed to live. Twelve volts still makes sense for vans and small cabins, where the loads are small and the runs are short.
Before anyone connects anything
A battery bank is not a beginner wiring job. A modest lithium bank can push thousands of amps into a shorted spanner without blinking, and DC arcs do not self-extinguish the way AC arcs do. Every battery and every string needs correctly rated overcurrent protection close to the source, a disconnect you can reach, and conductors sized for the fault current rather than the running current. Lead-acid adds hydrogen venting to the list, which means the enclosure needs ventilation and no ignition sources. Lithium adds a management system that has to be able to actually disconnect the battery, and an inverter and charger whose charge profiles the battery manufacturer accepts — a lithium bank on a lead-acid charge profile is a warranty claim waiting to happen. Anything that ties a solar system to a building supply or to the grid is permit and inspection territory in most places. Interconnection agreements, rapid shutdown provisions, labelling, disconnect placement and who is allowed to do the work vary by jurisdiction and by utility, and the code edition your Authority Having Jurisdiction has adopted is what governs the installation, not a web page. Use these numbers to plan and to price, then have the design reviewed by someone who knows what your utility and your inspector expect.
Questions people ask
How many batteries do I need to run my house for a day?
Start from the loads you actually intend to back up rather than from the whole house, because they are rarely the same. A typical whole-house daily consumption of 25 to 30 kWh needs an enormous bank; a backup subset of fridge, lights, network gear, a well pump and a furnace blower is often 5 to 8 kWh a day and needs a fraction of it. Put the subset figure in the field above with one day of autonomy. As a rough shape: 10 kWh a day for one day at 48 volts and 80 percent depth of discharge is about 260 amp-hours, which is three 100 Ah 48 V modules or twelve 12 V 100 Ah batteries in a 4S3P arrangement.
Is LiFePO4 actually cheaper than lead-acid in the long run?
Usually, but the comparison has to be per usable kilowatt-hour delivered over the life of the bank, not per nameplate amp-hour at purchase. Lithium typically gives you 80 to 100 percent of nameplate instead of 30 to 50, so a given amount of usable storage takes roughly half the nameplate capacity. It then delivers several times as many cycles at that depth. Against that, the up-front price per nameplate kilowatt-hour is higher, it needs a charger and inverter with compatible profiles, and it has real temperature restrictions on charging. Where lead-acid still competes is a rarely-used backup bank that sits float-charged for years and cycles a handful of times, because there the cycle-life advantage never gets used.
What depth of discharge should I actually design to?
The honest answer is that it comes off the cycle-life curve in the datasheet for the battery you are buying, weighed against how many cycles a year your system will do. A daily-cycled off-grid bank does 365 cycles a year, so a battery rated 800 cycles at 50 percent depth is a two-year battery at that depth and much longer at 30. A backup bank that cycles twenty times a year can be worked far harder without the cycle count ever mattering. The ranges in this calculator are defaults for orientation. Change them to whatever your product and your cycle count justify.
Can I add batteries to the bank later?
Only with real caveats, and this catches people out. New cells paralleled with old ones do not share current evenly: the newer, lower-resistance battery takes more of the load and more of the charge, which ages it toward the old one rather than lifting the old one up. For lead-acid the usual advice is not to mix ages by more than a few months. Lithium modules with their own management systems tolerate it better, and some manufacturers explicitly support later expansion within stated limits, but check that the manufacturer supports it for your exact product before you plan around it. If growth is likely, the cheaper path is to build the bank at the size you will eventually want, or to design the system so a second complete bank can be added alongside the first.