Peukert's Equation

Mr Peukert first devised a formula that showed numerically how discharging at higher rates actually removes more power (see below before thinking this is wrong) from the battery than a simple calculation would show it to do. For instance discharging at 10 amps does not remove twice as much power as discharging at 5 amps. It removes slightly more. Therefore a 100 amp hour battery (at the 20hr rating) could provide 5 amps for 20 hours, but it could not provide 10 amps for 10 hours. The available time would actually be slightly less.

Mr Peukert wrote down a formula for describing how much less time would be available. Please note that in the first paragraph I say "Mr Peukert first devised a formula for....". This is because he is generally regarded as being the man who first discovered the phenomenon. This is incorrect. The effect had been known for many years beforehand and was first noted by a certain Mr Schroder several years before Peukert devised his formula. Mr Peukert simply quantified it in a way that had never been done before. However 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. Note that Peukert's exponent changes as the battery ages.

Please note that there are two ways of looking at this effect. We could say that discharging at higher currents reduces the total available power that can be got out of a battery. So a 100 amp hour battery might become say an 80 amp hour battery at higher discharge rates. This is technically the correct way of looking at it.

However it is easier to assume that the total available power in the battery remains identical whatever the discharge rate. But that discharging at higher rates removes more amp hours. This is the method of explanation used throughout this website and on the Peukert calculator spreadsheet.

Note that whichever method is used, the figures and effect remain identical in both cases. It's just that we consider the second method to be easier to understand and "get your head round".

Peukert's equation can be found all over place. On the internet, in battery data sheets and documents, in battery sales literature, in battery monitoring equipment manuals etc. It is usually written as Iⁿ T = C

Where:

I = the discharge current in amps
T = the time in hours
C = the capacity of the battery in amp hours
n = Peukert's exponent for that particular battery type

The idea is that the time (T) that a certain battery can run a certain load for can be calculated by rearranging the equation to read T = C/Iⁿ

Please note that this equation, seen all over the place, is wrong. Actually, I'd better rephrase that. The equation is not wrong. But the way people attempt to apply it to the battery capacity is wrong.

This equation cannot be used on batteries that are specified at (say) the 20 hour rate, or the 10 hour rate or any other "hour" rate. It will not work. For an explanation of why and what equation you need to use read the rest of this article.

Alternatively go here to find a suitable solution without understanding why.

Even a cursory attempt at using it will show that it simply cannot be correct.

So let's try using this equation and see what we get.

The first problem we come across is that the battery capacity does not state any type of rating. Is this the 100 hour rate? the 50 hour rate, the 20 hour rate? or some other rate?

Most people assume it to be the 20 hour rate so we shall do the same here.

Take a battery rated as being 100 Ahr (at the 20 hour rate - the most usual specification) with a Peukert's exponent of 1.3 (a typical figure for a deep cycle wet cell).

The rating on this battery means it can provide 100 amp hours in total at the 20 hour discharge rate. That is what the rating means. This battery, when new, can provide 5 amps for 20 hours.

However, if we plug these numbers into the usual Peukert's equation (the one that we see all over the place) we get:-

T = C/Iⁿ
T = 100/5^1.3
T = 100/8.1
T = 12.3 hours - yet we know, from the specification, that it can provide this current for 20 hours!

Just plugging the battery's actual known capacity onto the equation gives us the wrong result.

Ok, let's do a quick check on this. Let's do exactly the same calculation but this time we will use 2 of the same battery i.e. 200 amp hours, and the load will be exactly twice as much i.e. 10 amps instead of 5 amps. Common sense (and experience and calculations) tells us that the run time will be exactly the same as a single battery at 5 amps load.

T = C/Iⁿ
T = 200/10^1.3
T = 200/19.9
T = 10.0 hours - But we all know that it should be the same as the above example

Let's just double check on this to make sure we haven't missed something.

The first result above suggests that this battery can actually only provide 5 amps for 12.3 hours. That makes it a 5 X 12.3 amp hour battery at this discharge rate. That means this equation tells us it can provide a total of 61.5 amp hours when discharged at 5 amps.

So let's plug these new numbers into the equation and see where it gets us:-

T = C/Iⁿ
T = 61.5/5^1.3
T = 61.5/8.1
T = 7.6 hours - So it has now decided something completely different.

Everytime we try to use this equation it makes the battery smaller!

Clearly there is something very wrong with the equation.

Well there is, and there isn't. There is nothing wrong with Peukert's equation. It's simply that this is not how it should be used. Peukert's equation, as it is, has to be used on batteries specified at the "Peukert Capacity". That is, the capacity of the battery when discharged at 1 Amp. Batteries are very rarely rated this way.

Whilst Peukert's equation is correct, it is not written in a way to enable it to be simply applied to a certain battery in the way they are usually rated. In order to do this we need to modify the equation so that it takes into consideration the way the battery capacity is quoted. The modified equation is:-

T = C/(I/(C/R))ⁿ X (R/C)

Where:

I = the discharge current
T = the time
C = capacity of the battery
n = Peukert's exponent for that particular battery type
R = the battery hour rating, i.e. 100 hour rating, 20 hour rating, 10 hour rating etc.

What we have done here is modify the equation to operate effectively given the battery capacity and hour rating.

This formula works. However, you must ensure that the correct hour rating is inserted. If the battery capacity is quoted at a different rate then this equation will give very misleading results. 99% of batteries are rated at the 20 hour discharge rate.

If we now try the same experiment with the corrected version of the equation:-

T = C/(I/(C/R))ⁿ X (R/C)

We get:-

T = 100/(5/(100/20))ⁿ X (20/100)
T = 100/(5/5)ⁿ X (0.2)
T = 100/1 X (0.2)
T = 100 X 0.2
T = 20 hours, which we know to be correct.

We also know that doubling this discharge current should result in a bit less than half the time available. We know that much to be certain, due to Peukert's effect. Doubling the discharge current to 10 amps should result in a bit less than 10 hours available run time.

So let's now try that calculation to convince us that our version of the equation does indeed work correctly.

T = C/(I/(C/R))ⁿ X (R/C)
T = 100/(10/(100/20))ⁿ X (20/100)
T = 100/(10/5)ⁿ X 0.2
T = 100/(2)ⁿ X 0.2
T = 100/2.46 X 0.2
T = 40.6 X 0.2 = 8.1 hours

Now that's more like it.

It is a mystery to us how magazine articles have been written using the incorrect application of the equation. Also, the internet is full of websites showing their author's expertise on all matter's Peukert, yet quoting this incorrect usage of the equation. (It is interesting to note that since this article [which is referenced and linked to from all over the internet] was first put up on the web, in 2003, that the number of websites incorrectly quoting Peukert's Equation has fallen dramatically. Many of them have now been changed having remained incorrect for, in some cases, over 10 years. A few remain using the formula incorrectly. Did it really take SmartGauge Electronics to teach the rest of the world how to use this equation? It certainly would seem so).

The calculations are self contradictory when trying to use it and simply do not work.

The only rational explanation we can think of is that people see the formula, assume it to be correct, never actually try to use it, and therefore never realise that it's actually wrong. This even seems to be the case with magazine articles, battery monitor owners manuals and several internet sites. It seems they haven't actually checked their results against reality and just assume the calculated figures are correct. But they do so enjoy showing their expertise and quoting the equation, even though they clearly don't understand it. No other explanation seems feasable.

One last point about the use of Peukert's equation. You may occasionally see the equation written as T = Ca/Iⁿ or in some other order but with an extra figure in there somewhere (the "a" in this example). This extra figure is usually specified as being an empirically derived factor, usually with no explanation as to what it is for. We will call this equation the fudged equation.

It is actually an attempt to modify the formula so that it works given a certain battery capacity and hour rating. However the people who use this "fudged equation" state that the figure has to be arrived at by trial and error by trying a calculation, and adjusting this figure until the 20 hour calculation comes out correct. It's not very elegant involving a lot of guesswork. Especially as the required figure can actually be calculated from the given data!

The corrected version of the equation that we used above was kept in that fashion because it is easy to work with for the examples above. However, anyone with even a basic understanding of simple sums will instantly spot that the equation can be rewritten either as T = C(C/R)^n-1/Iⁿ or as T=R(C/R)ⁿ/Iⁿ (they are mathematically the same). I consider this to be slightly more elegant but slightly more complicated to work with for manual calculations. It's exactly the same as the one worked through in the above examples but rearranged.

And those of you who are still awake will have spotted that the (C/R)^n-1 in the first of these two equations replaces the "a" in the fudged equation. As I said, it seems odd to add an empirically derived figure when it can be calculated from the given data.

Finally, you may also, on certain websites or in certain articles, see it written as T = C/(Ia)ⁿ. Where "a" is, again, an empirically derived figure. This is a similar attempt to the fudged equation mentioned above, and again, the empirically derived figure is guessed at until the 20 hour calculation comes out correct. However in this case, the emprically derived figure is in the wrong place in the equation and the other results will be highly inaccurate.

Now, you've just seen how much less run time than 10 hours is actually available when a 100 amp hour battery is discharged at 10 amps. About 17% less than a quick "amps X time" calculation would show. At higher discharge rates the effect becomes very large indeed.

An often neglected aspect of Peukert's effect is that discharging at lower rates will increase the run time quite substantially. For instance, in our example of a 100 amp hour battery (at the 20 hour rate), with a Peukert's exponent of 1.3, discharging the battery at 5 amps gives us 20 hours run time (so 100 amp hours are actually available). Discharging at 2 amps gives us 66 hours run time. But wait, that's 2 amps for 66 hours, that means the battery has provided 132 amp hours. This is correct. At lower discharge rates, Peukert's effect means the battery has a higher capacity. This is why it is so important to check the rating on battery specifications. Rating this same battery at the 100 hour discharge rate (instead of the more usual 20 hour rate) would result in a higher amp hours "number" to stamp on the side of the battery, thus making the battery look bigger than it really is. The true capacity is exactly the same.

Discharging this same battery at 0.5 amps would give a total run time of just under 400 hours. That means a total of 200 amp hours were provided by the battery.

Important note 1

As described above, as the current approaches very low levels the total available amp hours seems to increase beyond the capacity of the battery. This is quite correct and the effect will be seen graphically later in this article. However, balanced against this is the self discharge of the battery which goes some way to cancelling this effect at very low discharge currents. The final effect is that, at very low discharge rates, the apparent total amp hours available from the battery is never quite as high as a calculation based purely on Peukert's effect would indicate (this is obvious if you think about it as it would mean the available energy approaches infinity at extrremely low discharge rates which is obviously ludricous). Some people incorrectly come to the conclusion that Peukert's Equation does not operate correctly at very low discharge rates. This is not the case. Peukert's Equation does work correctly at all discharge rates. It just seems like it doesn't unless the battery internal self drain is taken into consideration.

By way of example, Discharging the 100 amp hour battery at 0.5 amps (as shown above) results in a run time of just under 400 hours. That is 16 days and during that time a typical deep cycle wet cell battery could well have self discharged by around 15 to 20 amp hours or so thus making it look like Peukert's Equation did not operate correctly..

VERY Important note 2

Peukert's equation is clearly an empirical equation. ie one devised to fit what he actually saw as opposed to one derived from fundamentals. Mathematically, the equation doesn't even make sense dimensionally!

Peukert was a bit vague on whether the available energy as calculated by his formula was removed in a single discharge run or whether it was removed in stages allowing the battery some time to recover between discharges. He was also a bit vague on what the measured state of the battery was following the discharge run and following a recovery time. It appears that because of this uncertainty many people assume that the discharge was done in one single run until the terminal voltage fell to his predetermined discharge floor.

As a result of this (and also perhaps a result of never actually having carried out any relevant tests) they reach the (incorrect) conclusion that after a discharge run the battery will recover and the remaining energy will become available. As a further conclusion of these initial (incorrect) conclusions they state that Peukert's equation only matters with regard to heavy discharges and if the battery returns to a low discharge level the full capacity of the battery will still be available.

Some have gone further and stated that Peukert's effect is purely a result of limited electrolyte dispersal and diffusion slowing down the chemical reactions (this is really battery recovery and/or surface discharge which is covered elsewhere on this website).

This is incorrect. Battery tests simply do not support this conslusion.

Peukert's effect is partially a result of slow electrolyte dispersal/diffusion and partially a result of an internal resistance that varies as a function of the discharge rate. This is what makes it so difficult to calculate. If the effect was due purely and simply to one of these effects then it would be much simpler to compute the final results.

If the effect was due only to slow electrolyte dispersal then counting amp hours would work much better than it actually does. The reality is, it isn't very good.

If the effect was due only to the internal resistance changing as a function of the load then the time remaining function on amp hours counters would work much better than it does.

Believe me, if the effect was due entirely to one or the other then, as a battery monitor designer, my life would be a lot simpler! The fact is, the effect isn't due simply to one or the other, I wish it was, but it isn't.

The reality is that both these effects play their part and to say only one or the other is responsible is incorrect and leads to incorrect conclusions about exactly how lead acid batteries react under heavy discharges. It might also explain why some people get even worse results (if that was possible!) than they might otherwise expect from amp hours counters if they removed this incorrect conclusion from their thoughts.

Finally

Now, whilst these figures are interesting, and quite illuminating to the uninitiated, actually calculating them is incredibly boring. To sit there with a calculator, running through the equation with different figures is tedious to say the least. So........

Being the lovely people that we are.......

We have written you a simple Peukert calculator in Microsoft Excel format and put it on this website. This will enable you to play to your heart's desire. This calculator uses the exact same equation shown above but rearranged to a more elegant format. You can download it by right clicking here and selecting "save target as" or use the link on the left hand side.

This calculator will allow you to enter the battery capacity, the capacity rating (i.e. 20 hour rating, 100 hour rating etc) and Peukert's exponent for the battery type. It will then calculate a range of discharge currents from very low up to a discharge equivalent to the battery capacity. It then displays what is termed the "peukert corrected amps" (which is the equivalent discharge rate such a load will remove from that particular battery) for each discharge current and the available run time, again for each discharge current (note that the time is shown in hours as a decimal not in hours and minutes). Finally it shows the total amp hours available from the battery at each discharge rate.

There is also a window to allow the user to enter any discharge current and it will calculate all the same values for that particular current.

Finally, there is a graph on the page which shows the discharge current along the bottom, and the total available amp hours up the left hand side.

Typical Peukert exponents vary widely between different manufacturers but an average figure for a true deep cycle battery is about 1.3. For AGMs about 1.10 and for hybrids about 1.15. Have a play. You will be surprised at just how much difference a heavy discharge rate makes to the available run time. And perhaps also surprised at just how many amp hours are available from a battery when the discharge rate is very low.

There is a much more detailed explanation of Peukert's Equation here along with a mathematical explanation and proof of how any why our modifed equation works correctly for a given battery capacity specification. This equation is then developed further to produce one that will allow Peukert's exponent to be calculated from a battery data sheet when this information is not available. The maths is somewhat involved.

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Page last updated 02/04/2008.
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