**Introduction to Peukert Law**

Peukert's law, presented by the German scientist Wilhelm Peukert in 1897, expresses the capacity of a battery in terms of the rate at which it is discharged. As the rate of discharge increases, the battery's available capacity decreases.The greater the discharge rate, the lower the delivered capacity.

The formula for Peukert's law could be expressed in different ways.

(i) The equation that tells how long a battery will last under a particular load, it could be expressed as :

t – Time in hours. It is the time the battery will last given a particular rate of discharge (the current).

H – The discharge time in hours that the Amp Hour specification is based on. For example, if you had a 100 Amp Hour battery at a 20 hour discharge rate, H would equal 20.

C – The battery capacity in Amp Hours based on the specified discharge time.

I – This is the current that we’re solving. For example, if we wanted to know how long a battery would last while drawing 7.5 amps, we would enter 7.5 here.

k – the Peukert Exponent. Every battery has its own Peukert exponent. Sometimes the manufacturer will provide it and other times we may have to figure it out.

(ii) For one-ampere discharge rate, Peukert's law is often stated as,

** **

where:

Cp is the capacity at one-ampere discharge rate, which must be expressed in ampere-hours,

I is the actual discharge current (i.e. current drawn from a load) in amperes,

t is the actual time to discharge the battery, which must be expressed in hours.

k is the Peukert constant (dimensionless)

**C**_{I} = KI ^{(1-n) } - is the commonly accepted capacity calculation equation for lead-acid batteries. n and k are constants obtained by data of maximum discharge current and minimum current.

n and k are calculates using the following equations,

Peukert’s law is a valuable tool for estimation. However, it has limitations. Among them are

- The effect that temperature has on batteries is not included in the equation.
- Battery age is not considered. The Peukert exponent increases with battery age.
- If you’re calculating for a low discharge rate, the equation does not account for the fact that each battery has a self-discharge rate.

The concept of charging and discharging rates could be described by an example. If a load of 10 amps requires 20 hours to discharge a battery, one might think that a load of twice that quantity (i.e., 20 amps) would run for half the time (i.e, 10 hours). But using the equation for Peukert’s law, this is found to be much less than 10 hours. That is why, when the capacity is mentioned, it is usually specified for a particular discharge rate in amps.

Peukert’s relation describes how the capacity is related to the total time to discharge the battery using a constant known as Peukert Constant. For a one-ampere discharge rate, Peukert's law is stated as,

where, I - the discharge current in Ampere, T is the discharge time, C is the capacity of the battery and n is the Peukert constant. This value is unique for every battery. The equation is an empirical relation describing the battery discharge capacities. When the unique value of the Peukert constant is equal to one, the discharge capacity will be independent of the discharging rate. When its value is more than one, the discharge capacity is decreased. The law is relevant only when the discharge rate is constant and the temperature increase inside the battery is limited.

Suppose the capacity of a battery is 100 Ah, that discharges at a rate of 5 A for 20 hours. From the Peukert's relation, we can calculate that the time for discharge will be 12.3 hours and not 20 hours. This is because the Peukert law should be used to calculaye a specific Peukert capacity, i.e, the capacity of the battery when discharged at 1 A.

So the modified Peukert equation is

where R is the battery hour rating. The term (C/R)^{n-1} will correspond to the battery hour rating and capacity.

The new equation can be verified for the above example and we will get 20hrs as the answer. The Peukert equation holds good for any capacity or hour rating. Original Peukert equation states that the battery capacity is the total Ah that can be drawn from the battery at the discharge rate of 1 A. but this is not true practically. The term corrects the given capacity specification to match up that 1 A current drawn.

In order to find the Peukert number of a battery, discharge the battery at two different discharge rates. Let T1 and T2 be the corresponding discharge times and C1 and C2 be its capacity respectively. The Peukert constant can be calculated by the equation,

The Peukert constant is unique every battery and varies from 1 to 1.5. A Peukert constant close to one indicates a well performing efficient battery with minimum losses and higher than one means less efficient battery.

**Peukert's Equation With Temperature correction factor to NiMH battery**

In order to verify the practicability of Peukert’s equation at room temperature and low temperature, 1/3C, 1C, 2C, 3C rate discharge experiments done in a study [2], which we have used here. The result of the study indicates that Peukert’s equation is suitable for estimating available battery of NiMH battery in 25 °C. The result also indicates, at low temperatures, Peukert’s equation is practical for NiMH battery at low current and unpractical in high current. A piece-wise Peukert’s equation with Temperature Correction Factor to NiMH Battery State of Charge (SOC) estimation is used in the simulation, which improves the precision of SOC estimantion^{[2]}.

The data and equations for simulation of the Peukert's law including the temperature effect are taken from the study. A 27 Ah, 1.2 V, NiMH battery had been used in the study at room temperature 25^{o}C. Experiments were conducted at constant current discharge rates of 81 A, 54 A, 27 A, and 9 A. Peukert's constant ( n and k) are related to the temperature for different ampere ratings, as shown below,