Will the area under a voltage vs. time graph of a battery discharge curve (with a constant current load) give the amp-hour capacity rating of the battery?
Electronic – Area under voltage vs time discharge curve
batteriesbattery-operateddischarge
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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?
Using a constant current load you'll get a nice curve if you plot the voltage over time.
The C rating is given by the manufacturer as an indication of the safe maximum constant current draw from the battery.
Some searching turned up that it is standard to make the discharge curve over 20 hours. So this would be like 0,05C. I speculate that this will avoid some thermal effects, Because your battery would probably get hot from a 1C discharge.
If it is a AA battery, set the current to about 100mA. AAA - 50mA if it is another kind of battery, guess or measure how much more or less volume it has compared to the AA battery, and factor that into the current. It is just a way of guessing capacity. So if the battery looks 4 times larger than a AA, set the current to 400mA.
You can do this guessing because battery capacity is proportional to volume. After you've done this, you have measured the capacity of your battery, and you can make more precise measurements. Good luck
Best Answer
No, in your case it is not the area that tells you the capacity, but the length of time before the voltage is so low that the battery is considered discharged. For example, if the battery started out a 4.0 V, then decreased over 1.9 hours to 2.7 V when you considered it empty, the capacity is 1.9 hours times the discharge current. If the discharge current was 1.3 A, for example, then the capacity was (1.9 h)(1.3 A) = 2.5 Ah.
The area under the graph is proportional to the total energy the battery delivered. Actually its merely the time integral of volts. But you said that the current was constant, so volts is proportional to power, and the time integral of power is energy.