# The Peukert Effect

### Motivation

If you're designing an EV (Electric Vehicle), choosing an EV to buy, or just driving an EV, it is worth your while, in big dollars and cents, to understand the Peukert Effect. The following description requires high school math to work through but just paying attention to the conclusions can have big payoffs.

### Basics

The "amp-hour" (AH) rating of your EV's battery pack is one of the principle determiners of EV range. The manufacturer's amp-hour rating for a battery might read "186 AH", suggesting that you can draw 1 amp for 186 hours, 46.5 amps for 4 hours, or 186 amps for 1 hour. For a common lead acid battery, the reality for the highest current example is that you'll only get the 186 amps for roughly 38 minutes, 64% of the hour you expected. The battery seems to hold 119 amp-hours, not 186 (see the blue line below).

If you draw half the current of the above example, 93 amps, you might expect it to run for twice the time, 76 minutes. But the battery will deliver for 85 minutes, 2.23 times the 186 amp time. The apparent capacity also goes up 2.23/2, to 132 amp-hours. Note the downward trend in the Apparent Capacity: as the current draw (EV speed) increases, the capacity (range) decreases. You can drive farther if you keep the current (speed) low .

These kind of results should serve as a warning that something non-intuitive is happening here. And, if you should get comfortable that your EV regularly travels X miles at 45mph then, another day you drive the same route at 60 mph, you can be surprised that batteries are exhausted before you reach your destination. While walking the rest of the way, you can ponder Peukert's Equation.

### Peukert's Equation

The table above comes indirectly from the Peukert equation below. His original equation, shown in the Wiki article, been slightly rearranged to improve readability and to use the slightly different symbology given in the Smartgauge articles (see References). 'Cp' is Capacity as Peukert measured it but, throughout this note, 'C' (not 'Cp') will be represent the capacity of the battery. 'C' will be the capacity stated in the modern way, measured over a period of 'R' hours (usually 20).

The exponent is all-important. If it were 1.0, there would be no Peukert Effect to discuss; a 186 AH battery would deliver 186 AH at all currents. The closer the exponent is to 1.0, the more perfect the battery. For deep cycle lead acid batteries, the exponent 'n' is 'typically around 1.1' and, as the batteries are used, that number eventually creeps up to 1.3 or so. A review of  The Battery Table , however, shows that real world batteries have exponents ranging well beyond 1.3 !

All types of batteries, even Lithium Ion (Li), have the Peukert Effect to some degree. For efficient batteries, like Li, their exponents are just nearer to 1.0 than lead acid. Compare the exponents for Li vs LA (again in the same battery table referenced above).

### Smartgauge corrections, explanations

The original Peukert equation has been reverently carried along for over a hundred years but few have really challenged it. More than one text on EVs advised just treating the Peukert effect as a constant 57%, independent of current, exponent, or anything else. Fortunately, Smartgauge noticed the traditional equation didn't work with present day conventions and they published the adjustments. For years, articles had been publishing the unusable form of the equation - and nobody noticed.

The key to their adjustment is to add a value, 'R', in the following to represent the number of hours used to quote the amp-hour rating ('C') for a battery. The following starts with the Smartgauge 'adjustment' and just shuffles it a bit for readability.

### Calculating with Peukert's Equation

With the SmartGauge corrections in place, it's time to figure out what the range penalty is when the current is increased.

The note about 'pre-calculating' above has to do with applications which will convert a number of current measurements into Peukert values. The 'pre-calculate' term, containing only unvarying quantities, can be calculated before the current measuring begins.

The spreadsheet equivalent is shown in excel equations.

### Range effects of doubling the current

This section tests the Peukert Equation with a conceptually simple scenario, that of doubling the current. This doubling could be the result of accelerating more aggressively or going faster. Another interesting instance arises way back in the EV design stages when the battery pack is designed. In principle, a pack built of 12 volt batteries will have twice the voltage and half the amp-hours of a similar pack built of 6 volt batteries (the next section addresses the realities of this assumption).

The following algebraically modifies the Peukert Equation to compare 2x current with the original current. Whereever 'I', the amperage, appeared in the earlier equation, we'll just substitute '2I' and see what the effects are.

### Choosing batteries, the reality show

While the logic of the last section is certainly sound, there are at least two 'real world' problems to address. The minor one is that the logic above posited that you were faced with 2 battery choices, probably of the same weight and thus, theoretically, the same total energy. One battery produced that energy with, say, 12 volts and 100 AH while the competitor produced it with 6 volts and 200 AH. It isn't so easy to see this 2x business while looking through a table of real batteries. And, more importantly and much more subtle is the range of peukert exponents. What if a 6 volt battery has a much better peukert exponent than a considered 12 volt?

The tougher problem, at least for the average car conversion enthusiast, is that the pack voltage is limited to 120 or 144 for the less expensive, "brushed" motors. So the typical "series wound" motor is going to impose a low voltage limit which negates any options for considering high voltage, low amperage configurations. So early in the design of the car or its conversion, a PM (Permanent Magnet) DC motor or an "AC Induction" has to be selected to avoid the range penalties calculated in the last section.

The simultaneous consideration of so many important factors (pack voltage, motor choices, car weight etc) led me to write a computer model to assist making such choices; it is principally focused on the "battery issue". A great calculator (and one I don't have to support) can be found at EV Convert. A sample session, with choices pertinent to my EV is shown in this link.

### Peukert's effect on driving style

In principle it should be obvious that keeping the currents low is very desireable in an EV, especially when the Peukert exponent is large, because of battery type or age. But, there was another surprise lurking in this study. A check of Peukert values over a range of typical currents for my pickup truck showed that all "reasonable speeds" had Peukert effects in the 66% to 71% range. The currents were already too large to enjoy small Peukert effects in the 90% range. Being on the flatter part of the curve, a 125 amp draw (plus or minus) wasn't, in the Peukert sense, going to make much of a difference. The range effects of the higher current draw were really due to the weight and poor aerodynamics of the truck.

The total effect on driving style can be simply summarized:

• try to keep the current low when accelerating away from a stop sign or stop light. This means 'accelerate slowly', assuming conditions allow.
• avoid stop signs. You're guaranteed to have to stop and replace the energy of previous motion. Even if you make some of the lights on an alternate route, you're ahead. A government study claims that 6% of American transportation energy goes into heating the brakes.
• try to find a time on the freeway that gives a continuous (not stop-and-go) flow at an efficient speed.
• avoid hills whenever possible. The higher current to climb the hill is never offset by the free ride down the hill.