### How do you calculate battery capacity in amp-hours (Ah) at different “C” rates?

**How do you calculate battery capacity in amp-hours (Ah) at different “C” rates?**

For instance, I know that a battery with a capacity of 200 Ah at C/20 has a different capacity at C/100.

When you size a PV system with a battery bank of, say, 800 Ah, how do you tell what C-rate the battery bank is using?

C-rates are also called “hourly rates,” and are based on the length of time of discharge. A C/20 rate means that battery capacity is calculated based on completely discharging it over the course of 20 hours. So, if you have a 1,000 amp-hour battery bank, charging or discharging at 50 amps would be a C/20 rate (1,000 Ah ÷ 50 A = 20 hrs.).

The informal
solar industry standard for comparing deep-cycle battery capacity is at the C/20 rate, because it approximates the 24-hour discharge period of most off-grid systems. Many battery manufacturers’ data sheets also provide capacity information for C/5 and C/100 rates, which are useful in other industries.

It is possible to calculate battery capacity at any given C-rate, if you know Peukert’s exponent for the battery. Peukert was a German scientist who derived the formula for the relationship between battery capacity and discharge rate. Battery manufacturers do not typically provide Peukert data on their spec sheets, but it may be available by contacting them.

**It = C × [C ÷ (I × H)] k – 1; where**

H = rated discharge time in hours;

C = rated capacity at that discharge rate;

I = actual discharge current in amps;

k = Peukert exponent

It = effective capacity at the discharge rate “I”

H = rated discharge time in hours;

C = rated capacity at that discharge rate;

I = actual discharge current in amps;

k = Peukert exponent

It = effective capacity at the discharge rate “I”

Most flooded lead-acid batteries have a Peukert exponent between 1.2 and 1.4, while most absorbed glass mat (AGM) batteries are between 1.05 and 1.2—but keep in mind that these figures will increase as the batteries age. Peukert’s exponent for a given battery can also be calculated if the manufacturer provides you with the capacity ratings at two different discharge rates. That math is complicated, but there’s a helpful spreadsheet (and detailed information on applying Peukert’s law) at
www.smartgauge.co.uk/peukert_depth.html

A typical
renewable energy system will be charging or discharging at different C-rates throughout the course of any given 24 hours. With a battery monitoring system that logs data to a computer spreadsheet, it’s possible to calculate the C-rate at any given instant. And though it seems strange, your usable battery bank capacity is continuously changing.

Fortunately for the typical home RE system owner, discharge C-rates are, on average, quite low. Consider a 48 V battery bank of 1,000 Ah in a system that’s designed to provide 10 kWh of backup energy per day. Those 10 kWh equal 208 Ah. This divided by 24 hours equals 8.7 A. That’s a daily average rate of C/115 (1,000 ÷ 8.7), far slower than the C/20 rate used for comparison when selecting batteries. It’s true that large loads that are used during those 24 hours will increase Peukert effects and reduce usable battery capacity, but this example is also figured with no solar input. On sunny days, the C-rate of discharge will be even slower.

Folks working with electric vehicles must pay closer attention to Peukert’s law. A typical EV’s battery bank has much lower capacity than one in a typical solar home, since both battery weight and bulk must be minimized in vehicles. Also,
EV motors use battery energy at very high rates—under some conditions, the battery might be fully discharged in less than an hour.

For solar energy applications, simply using proper system sizing guidelines such as online spreadsheets or consultation with your local RE dealer will keep the batteries’ C-rates reasonable, usually much better than C/20. Peukert’s exponent will only raise its ugly head if you drastically undersize your battery bank for your loads, or in specialized applications like electric vehicles.