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# Battery discharge time depending on load

This article contains online calculators which can calculate discharge times for specified discharge current using battery capacity, the capacity rating (i.e. 20 hour rating, 100 hour rating etc) and Peukert's exponent.

Everything below was created after spending several hours searching and reading the internet. I'm not electrician, so please pardon me for any mistakes.

Battery capacity is a measure (typically in Amp-hr) of the charge stored by the battery.
You may think that calculating how long a battery will last at a given rate of discharge is as simple as amp-hours: for given capacity C and discharge current I time will be
$t=\frac{C}{I}$,

However, battery capacity decreases as the rate of discharge increases.

This effect had been known for many years but it was Peukert who first devised a formula that showed numerically how discharging at higher rates actually removes more power from the battery than a simple calculation would show it to do.
Thus the effect is now known as Peukert's effect, the formula for calculating it is known as Peukert's equation, and the important number, unique to each battery type, that is put into the equation in order to perform the calculation, is known as Peukert's exponent.

Here is the Peukert's equation

$t=\frac{C_p}{I^n}$,
where
n - Peukert's exponent
Cp - Peukert's capacity
I - discharge current

The Peukert's exponent shows how well the battery holds up under high rates of discharge - most range from 1.1 to 1.3, and the closer to 1, the better. The Peukert's exponent is determined empirically, by running the battery at different discharge currents. Peukert's exponent changes as the battery ages.

Many batteries do not have Peukert's exponent available in specification. But sometimes they have tables giving different run times at different discharge rates, or a graph of discharge rates against run times. Peukert's exponent can be calculated from these graphs or tables, or by running two discharge tests at two different discharge rates. Calculator below helps to do it

#### Peukert's exponent

Digits after the decimal point: 2
Peukert's exponent

Now, what is Peukert's capacity?
Peukert Capacity is the capacity of the battery measured at 1 amp discharge rate. Batteries are rarely specified with Peukert capacity. Battery manufacturers rate capacity of their batteries at very low rates of discharge, as they last longer and get higher readings that way. This is known as "hour" rate, for example 100Ahrs at 10 hours. If not specified, manufacturers commonly rate batteries at 20 hour discharge or 0.05C.
0.05C is so-called C-rate, used to measure charge and discharge current. A discharge of 1C draws a current equal to the rated capacity. For example, a battery rated at 1000mAh provides 1000mA for one hour if discharged at 1C rate. The same battery discharged at 0.5C provides 500mA for two hours.

Knowing the hour rate of your battery, its specified capacity and Peukert's exponent you can calculate Peukert capacity using the following formula
$C_p=R(\frac{C}{R})^n$
where,
C - the specified capacity of the battery (at the specified hour rating)
n - Peukert's exponent
R - the hour rating (ie 20 for 20 hours, or 10 for 10 hours etc)

Finally, knowing the Peukert capacity and Peukert exponent you can calculate discharge time for given discharge current. Calculator below does this.
But note that it shows discharge time for different depth of discharge. Why should you care about this? In many types of batteries, the battery cannot be fully discharged without causing serious, and often irreparable damage to the battery. Manufacturer usually specifies the Depth of Discharge (DOD) of a battery which determines the fraction of power that can be withdrawn from the battery. For example, most car batteries has DOD of 20%, so only 20% of capacity can be withdrawn.

#### Battery discharge time depending on load

Digits after the decimal point: 3
Peukert's capacity, Ahrs

Rated discharge current, A

Another aspect of Peukert's effect is that discharging at lower rates will increase the run time. Rating capacity of the same battery at 0.01C yields more amphours than rating capacity at 0.05C, and you should care about hour rate used in battery specification.
You may think that very low discharge currents will increase available amp hours beyond the capacity of the battery. This is quite correct, however, during a long runtime self discharge effect of the battery comes into play. Due to self discharge total amphours at very low discharge rates will be less than calculated using Peukert's formula.

Final calculator below shows available runtime for different discharge currents.

#### Battery discharge time depending on load

Digits after the decimal point: 3
Rated discharge current, A

Peukert's capacity, Ahrs

Discharge times and capacities for the range of discharge currents

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