Function to describe alkaline battery voltage under constant current / load

Question

Introduction

Commonly used alkaline batteries (Duracell etc) have a pretty interesting discharge curve. For example, an AA duracell has the following voltage characteristics under constant load,

and constant current:

(http://www.procell.nl/pc1500.pdf)

The battery voltage is also affected by temperature. For example, from some of my recent tests the fluctuations correspond roughly to the time of day (temperature) they were measured. If the batteries were hotter, they measured a higher voltage.

Question

Before I try to do this myself, does anyone know a mathematical function to describe the voltage of an alkaline cell under constant current and/or constant load? An added bonus would be a function that takes temperature as a parameter. Of course a linear approximation often suffices, but I'm after a better fit.

The purpose is to fit the curve (representing the function) to battery voltage readings taken from a device under test. This will enable predictions of battery life, and battery life at different temperatures.

Answer 1

An equation/model that described the effects of time, current, temperature, etc. on battery voltage would be very useful. It would be even better if a microcontroller could use that model to deduce/estimate the internal state of the battery -- in particular, the state of charge (SoC) and the depth of discharge (DoD). Ideally by watching a battery as it is normally being used, but perhaps probing the battery with occasional brief pulses of positive and negative current would be informative.

My understanding is that many people approximate a battery as some internal voltage source in series with the battery internal resistance (or a more complex RC network). Rather than try to find an equation that directly gives the output voltage of the battery given the instantaneous internal battery state and the instantaneous current pulled from it, they assume the internal voltage source stays fixed (for a given kind of battery chemistry) and find some equation that slowly adjusts the internal resistance of the battery -- close to zero when the battery is fully charged, and slowly increasing resistance as the battery discharges. (Other rapid-transient effects are modeled by fixed capacitors and fixed resistors in the RC network).

• Jonathan Johansen. "Mathematical modelling of Primary Alkaline Batteries". gives curves that very closely match your first curve, and a explanation in terms of the internal chemistry. (Can you tell I prefer such "Babylonian" explanations?)
• Mathworks. generic battery model. Uses a fixed internal resistance and a complex equation to describe the internal voltage. This gives a curve that very closely matches your first curve. Alas, to me it looks like the kind of Euclidean equations that give more-or-less the right answers, but don't help me understand what's really going on.
• Min Chen, and Gabriel A. Rincon-Mora. "Accurate Electrical Battery Model Capable of Predicting Runtime and I–V Performance"
• M.R. Jongerden and B.R. Haverkort. "Battery Modeling".
• Wikipedia: Peukert's law. Peukert's law is an equation that estimates the run-time -- from fully charged to fully drained -- from 4 other parameters, including a Peukert exponent.
• Guoliang Wu, Rengui Lu, Chunbo Zhu, and C. C. Chan. "Apply a Piece-wise Peukert’s Equation with Temperature Correction Factor to NiMH Battery State of Charge Estimation". Guoliang Wu et. al. show one way to adjust the Peukert exponent to compensate for temperature. So we're up to 5 values. Alas, my understanding is that both Peukert's law and Guoliang's improvement are purely empirical fits to a bunch of data -- it doesn't explain why the run-time varies in that way. They only gives one point on your graph -- the time when your graph crosses the manufacturer's full discharge voltage -- roughly 0.8 V for alkaline batteries.
• Ralph Hiesey. "Some comments on “Peukert’s” compensation—why we don’t use it".
• Ahmed Fasih. "Modeling and Fault Diagnosis of Automotive Lead-Acid Batteries".
• Mikäel G. Cugneta, Matthieu Dubarrya and Bor Yann Liawa "Peukert's Law of a Lead-Acid Battery Simulated by a Mathematical Model".
• Quan-Chao Zhuang et. al. "Diagnosis of Electrochemical Impedance Spectroscopy in Lithium-Ion Batteries" p. 192 shows a model of a battery composed of a bunch of resistors and capacitors.
• Duracell MN1500 AA datasheet has a nice graph of resistance versus depth of discharge. All Duracell datasheets, in case the link changes.

I hear that one manufacturer uses a state-of-charge model of a battery with 408 different values. Is there a better model?

Answer 2

Off the cuff as someone who started formal training in both electronics and mathematics as a teenager, and ended up with an M.S. in Applied Mathematics (Univ. of MD, 1991) and 20 years of experience as an electronics design engineer - including using Lithium Iodide cells for CMOS memory backup in 1981 (yeah, "way back in the day") - I will tell you flat out that any hope for a "closed form" mathematical equation (into which you can "plug in" the parameters) is right up there with wishing upon a star. It is not about electronic instruments or their use of primary or secondary cells for operating power: it is all about electrochemistry!

So go study Physical (and Electro) Chemistry as that is the actual science, not technology, behind these devices. All common batteries (as we know them) are chemical sources of electrical energy - or did someone forget to mention that simple and obvious fact? (Does this matter truly require a BSEE to fully grasp intellectually?)

Answer 3

The instantaneous voltage equation over time may be interesting.

But what is important to most for a given cut-off voltage, what is the watt-hours of service, for a given load value and type {constant I, R, or P).

Other variables which affect service life include duty cycle per day average service temperature.

You can try to convert the results from here to a formula using temperature to affect the capacity.

Answer 4

You can use MS Excel to generate a polynomial curve fit based on your data, then have Excel display the equation on the chart. You may have to diddle the first term a little to get it to fit exactly. I use this feature quite often, and take the equation displayed and generate another curve with it. This way I can make those little adjustments to the first term of the polynomial until the calculated curve fits my data, then I have a pretty good approximation of the actual data.