**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 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)For a lead acid battery, it could be expressed as :

t – Time in hours. Its the time that 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. For a 100 Amp Hour battery, this would be

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

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

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.

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

Cp = I^{k} t - This is an alternate way of expressing Peukert's law.

where:

Cp is the capacity at a 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 commonlyaccepted capacity calculation equation for lead acid batteries. n and k are constants obtained by data of maximum discharge current and minimum current.

Peukert’s equation is verified by lead acid [Peukert, 1897; Bumby, 1985; Chan, 2000; Vervaet, 2002; Song, 1998], which estimated the capacity with Peukert’s equation. [Chan et al., 2000]’s opinion is that Peukert’s equation is only suited for lead acid battery, but no experiment analysis and demonstration are shown. [Doerffel and Sharkh, 2006] analyzed the Li-ion battery, the results show Peukert’s equation is good at estimation available capacity under constant current discharge, but is not suitable in variable current profile.

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 is utilized here. The result indicates 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 concept of charging and discharging rates could be described by an example. If a load of 10 amps requires 20 hours to discharge, one might think that a load of twice that quantity, at 20 amps, would run for half the time, or 10 hours, since that would require the same number of electrons coming out of the battery. But 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 often stated as

C=I^{n}T

Where, I is the Discharge current in Amperes, T is the time, C is the capacity of the battery and n is the Peukert constant. This value is unique for different 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 5A for 20 hrs. From the Peukert's relation, we can calculate that the time for discharge will be 12.3 hrs and not 20 hrs. This is because the Peukert law should be used to calculate a specific Peukert capacity, ie, the capacity of battery when discharged at 1 A.

So the modified Peukert equation is

T= C(C/R)^{n-1 }/t^{n}.

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 20 hrs 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 1A. 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 R1and R2 be the corresponding discharge times and C1 and C2 be its capacity respectively. The Peukert constant can be calculated by the equation,

n=log(R2/R1)/[log(C2/R1) - log(C2/R2)]. The Peukert constant is unique for a battery and varies from 1 to 1.5. A Peukert constant close to 1 indicates a well performing efficient battery with minimum losses and higher number means less efficient battery.