# Question about temperature dependence of Battery capacity

Discussion in 'General Electronics Chat' started by steveb, Nov 20, 2009.

1. ### steveb Thread Starter Senior Member

Jul 3, 2008
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I have a battery question that has proved difficult to answer. Apparently it is a tricky question because consultation of over 30 research papers, a 1000 page battery book and questioning several engineers has not revealed the answer. This, despite the fact that it's basically a very simple question. I have a high regard for the depth of knowledge and deep-thinking ability of the people here, so here goes!

First, here's a statement of well-established fact. It is known that the maximum battery capacity (i.e. AHr rating) increases as temperature increases (within reasonable operation limits, of course). This means that a warm battery is capable of storing more energy/charge, compared to a colder one.

If you have a colder battery that is fully charged to it's maximum capacity, and then heat it, will the battery have more usable energy, or the same energy? In other words, will the battery still be charged to its (now higher) maximum capacity, or will it now be charged at a capacity less than it's new maximum value? In the second case, you would then need to charge the battery further to take advantage of the higher capacity, while in the first case, you would damage the battery if you charge it further.

I have come up with my own unverified answer, based on logic, but I need a real answer and a scientific explanation of why it is true. My own thought is that a fully charged battery stays near it's maximum capacity as temperature changes and the stored energy goes up and down as temperature goes up and down. I say this because otherwise a battery fully charged at room temperature would be damaged when cooled because it would then exceed it's maximum allowed charge. However, I'm in a situation where guessing, even if rooting in logic, is not allowed.

The basic reason for my question is that I'm making an equivalent circuit model for batteries based on what is available in the literature. I've now found several usable models which include temperature dependence, but seem to predict that capacity changes with temperature, but stored energy does not change with temperature. In other words, it contradicts my logical argument above.

2. ### Audioguru New Member

Dec 20, 2007
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On Energizer's website there are detailed Applications Manuals for Alkaline, primary Lithium, Ni-MH and old obsolete Ni-Cad batteries. There are graphs of what temperature does.

3. ### beenthere Retired Moderator

Apr 20, 2004
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I'm not sure this is the case -
A battery has to run a chemical reaction in order to produce new charge. As chemical reactions slow by one half per 10 degC decrease in temperature, that is probably why a cold battery (at least lead-acid) can't sustain the current output cold.

My car's battery did not lose charge overnight when the temperature went to -25 (F). But, if I wanted that big old 383 to fire up in the morning, I had to take the battery inside to keep it warm.

4. ### steveb Thread Starter Senior Member

Jul 3, 2008
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Perhaps I didn't word that quite correctly. What I'm really referring to is "usable capacity". Various references show the usable capacity to decrease at colder temperatures which seems to be consistant with what you are saying. This means that if you cool the battery and then heat it without a load, you don't lose anything. However, if you cool the battery and then use it, the total charge drawn out at full discharge will be less than that at the warmer temperature. This effectively means that less charge is available, and less energy is drawn out. What isn't clear to me is whether this fully discharged cold battery will then have usable energy if you heat it back up again, without charging it.

Various circuit models attempt to model the battery as a capacitor and assume that the effective capacitance is lower at lower temperatures. This works fine if the battery is at constant temperature during the cycle. However, I'm trying to use the models to allow for battery heating during a cycle. In order to set up the correct dynamic equations that include temperature changes, I need to understand this better.

To give more insight into the issue, consider the capacitor equation in a circuit with variable capacitance. (Note that capacitance changes with temperature.)

$i={{dQ}\over{dt}}={{dCV}\over{dt}}= C\;{{dV}\over{dt}}+V\;{{dC}\over{dt}}$

Here you can see that the usual capacitor equation $i=C\;{{dV}\over{dt}}$ is not valid.

I'm not sure which of these, or if either of these are good models to use to capture usable capacity when temperature is changing.

5. ### steveb Thread Starter Senior Member

Jul 3, 2008
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Thanks for that good suggestion. The information there is useful. It shows that useful service of the battery is less at lower temperature. This does not completely resolve my misunderstandings, but surely helps.

6. ### steveb Thread Starter Senior Member

Jul 3, 2008
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Just to provide futher information which may be useful. One example of a research paper with a battery model is the following:

M. Chen and G. A. Rincon-Mora, Accurate Electrical Battery Model Capable of Predicting Runtime and I-V Performatce, IEEE Transactions on Energy Conversion, Vol. 21, No. 2, pp. 504, June 2006.

This is freely available on the internet and there doesn't seem to be a copyright issue with it.

A further clarification of using capacitance to model a battery is needed, since I'm sure a red flag goes up when most people think of doing this. Note that the capacitance is not directly used to model the battery voltage, but is used to represent the state of charge. Then a nonlinear mapping function is used to relate the battery open circuit voltage to the state of charge.

7. ### jpanhalt AAC Fanatic!

Jan 18, 2008
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If I had to take a swing at it, a fully charged cold battery that shows an increase in capacity when warmed would be less than fully charged at the higher temperature.

Check out equilibrium constant in Wikipedia (http://en.wikipedia.org/wiki/Equillibrium_constant) and scan down to temperature dependence.

BTW, I have an autographed picture of van't Hoff. I used to keep it in my bathroom for inspiration.

John

Last edited: Nov 20, 2009
8. ### Wendy Moderator

Mar 24, 2008
20,735
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My feeling is the cold slows chemical reactions, including the absorbers. This has the same effect as less capacity, but there are fundamental differences. One being when the battery is returned to room temperature it is restored to its former capacity.

Been wrong before, be wrong again, but that was my thought.

9. ### steveb Thread Starter Senior Member

Jul 3, 2008
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That seems reasonable to me and seems to agree with the models available. Basically, the effective capacity (i.e. Ahr rating) is a function of temperature, with little hysteresis or memory effect on temperature changes. I'm sure this is only approximately true and even then only within a reasonable range of temperature changes (i.e. storage range). But, for practical use, it should be good enough.

My uncertainty pertains to how to properly model the changes in state of charge SOC as both temperature and current are changing at the same time. If I change either one separately, the situation is clear to me. Or, at least the circuit models seem to make sense to me. The model I referenced above implies that capacity depends on temperature and the state of charge obeys simple charge conservation at that temperature. In other words, current flowing in directly adds the charges to the SOC, and current flowing out directly subtracts the charges from SOC. In equation form this is:

${{d\;SOC}\over{dt}}={{-i_{batt}\over{Q_c}}$, where $i_{batt}$ is the battery source current and $Q_c$ is the maximum charge capacity (i.e. capacity in Ahr converted into charge in Coulombs).

I'm not even sure that I'm phrasing my question properly, but I can give some examples of questions I can't answer with confidence, once both temperature and current are changing at the same time.

1. If a fully charged (i.e SOC=100%) battery at room temperature is cooled to 0 degrees and then discharged completely to SOC=0%, will it have a SOC=0% when it is heated back to room temperature, or will it have some residual charge available: for example, SOC=20%?

2. If a fully charged (i.e SOC=100%) battery at room temperature is cooled to 0 degrees, will SOC=100%, or will it exceed 100%?

3. If the latter case in 2 above, why is storing batteries in the refrigerator a good idea? Wouldn't that damage the battery? Or perhaps, the definition of maximum capacity allows for the fact that the battery may be stored at low temperature?

4. What is the proper definition of SOC if both temperature and current are allowed to change? If temperature (and hence capacity) are fixed, then SOC is simply the ratio of stored charge to maximum charge capacity. If temperature is changing, is SOC referenced to the temperature dependent capacity, or is it referenced to the initial capacity at the start of the cycle, or maybe it is reference to capacity a nominal temperature?

5. With any accepted definition of SOC, what is the proper dynamic equation for SOC when both temperature and current change? The relation need not be perfect, but it should not grossly contradict the real behaviour.

10. ### studiot AAC Fanatic!

Nov 9, 2007
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Surely when you pump in a given quantity of charge at any temperature you separate the same number of ions?
Then the joker comes when you want to retrieve the charge, as the internal resistance increases with decreasing temperature so you can't get back so much - the terminal voltage is lower.

11. ### Smoke_Maker Active Member

Sep 24, 2007
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My understanding is the battery dose not change in capacity but changes in the way you can get the energy out of the battery. A battery is like a slurpy the colder the battery the slower it comes through the straw and the warmer it is the faster you can get it out.

Then you have the Puekert's factor, the higher the rate of discharge the less you will get out of the battery at that discharge rate.

12. ### thatoneguy AAC Fanatic!

Feb 19, 2009
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As far as the theory: A fully charged battery at 0 degrees will be fully charged when it is brought to room temperature. I've experienced this with Lithium batteries as well as Alkaline batteries. When cold, the internal resistance increases due to lower chemical reaction speeds, limiting the available draw.

There is a point where heating batteries will not increase output, but cause them to self-discharge, usually around 100 degrees Farenheit, but can be noticed at lower temperatures. This applies to not only Alkaline, but Lead-Acid Batteries as well, although car batteries have a much higher temperature tolerance on the high end, usable Amp Hours do not increase a whole lot beyond 100 Degrees Farenheit.

And Now, For the Hijack of the thread, testing the above.

For giggles, I'm doing an experiment with some brand new Energizer 9V batteries.

At 74 Degrees F from an IR Therm, and using an AC Ohmmeter (ESR Meter) for measuring internal resistance. I have found ESR meters are Very Accurate way of measuring battery internal resistance. The measurement is always near what a battery analyzer shows for internal resistance, but the ESR meter doesn't need to do a current draw for the test! You only need to remember the values for "Dead Batteries", roughly 0.15Ω for a AA battery, and 1Ω for a 9V.

#1 measured 0.75Ω in internal resistance, 9.18 Volts open Circuit.

#2 measured 0.76Ω, 9.15 Volts

#3 measured 0.76Ω as well. 9.16 Volts.

The three batteries used in this test are from a blister-pack 4 pack of 9V Energizers, with a printed expiration date of 03-2013.

Onto the measurements:

First Test (general idea while #1 is chilling):
Battery Number 3 was measured for Short Circuit Current. Peak output was 6.1 Amps, dropping to 5.8 after 1/2 second. Average for the 1/2 second drain pulse was 5.9 Amps. The battery temperature increased from 74 degrees to a maximum of 85 degrees from that One Half Second short. Internal resistance increased to 0.98Ω, and open circuit voltage was as low as 8.78V 10-20 seconds after the current dump.

Allowing 30 minutes for #3 to return to room temperature, non-destructive measurements were repeated. The internal resistance increased slightly to 0.79Ω, but the open circuit voltage of the battery was depressed to 9.01V. Roughly 15% of it's estimated capacity is diminished.

Due to the terminal layout of 9V batteries, keep them from bumping into conductive objects and being shorted, even for fractions of a second!

#1 was put in the freezer. At 24 degrees farenheit, Internal resistance was measured at 1.4Ω, open terminal voltage 9.11 V. Short Circuit Current draw for 1/2 second started at 2130mA, and decreased to 800mA within 1/2 second. Terminal voltage after current draw was 8.39V. Temperature increased to 42.5 degrees, compared to the cold reference battery, which was measured at 25 degrees. The battery shows "Dead" on a battery tester before the current dump test, as well as after.

#2 was put in the freezer, and taken out, with no current load. Will check value once it returns to room temperature.

While awaiting the freeze and double checking results of #3, I'll post the differences, or details of what exploded, broke, or got ruined in the "Destructive Tests".

Once I figure out how to keep my battery analyzer from warming up battery number 1, I'll let you know the mAH Attained with a 100mA drain for both batteries.

Last edited: Nov 20, 2009
13. ### steveb Thread Starter Senior Member

Jul 3, 2008
2,433
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No, I'm not 100% sure about this. I'm learning as I go here. This statement is just based on the data I've looked at, and also the claims in the modeling papers I'm using. The data shows the capacity (usable capacity at least) to be temperature dependent. Generally the capacity is much lower at cold temperatures (0 deg C) and then it increases with increasing temp. Then, it seems to reach a peak capacity at a temperature generally above room temp (e.g. 40 deg C), although I've noticed from the Energizer data that NiMH seems to have peak capacity below room temperature, at about 20 deg. C.

14. ### Smoke_Maker Active Member

Sep 24, 2007
126
15
I have been involved in electric vehicle drag racing, we heat AGM batteries up to about 130 degrees for the race, not to increase capacity but to increase amp output to 2000 amps.

A 100 AH battery has 100 AH regardless of temperature but the discharge rate is temperature dependent.

15. ### steveb Thread Starter Senior Member

Jul 3, 2008
2,433
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Thanks, I'll think about this more. One of my uncertainties is how the given models compare with the Puekert's equation. The reference I provided above claimed that the internal resistance accounts for the capacity de-rating with current, but it seems to me that resistance affects terminal voltage and power, but not current and capacity. This is yet another confusion I have.

It looks like I'm getting more uncertain and asking more questions as I go, but I've been through this process before. It just means to keep working. There is quite of bit of information I need to go through this weekend it seems. I appreciate the help from all.

16. ### GetDeviceInfo Senior Member

Jun 7, 2009
1,571
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amp/hours is the discharge rate, as advertised by the manufacturer at some specified temperature.

I have been evaluating batteries for outdoor devices, with environments ranging to -30 C, and am keenly interested in anyones findings at colder temps.

17. ### beenthere Retired Moderator

Apr 20, 2004
15,815
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Just to throw in a teaser from an old book - "Submarine", by Richard Beach. He mentions that the old S class submarines could have the two batteries connected in series or in parallel, but does not mention the circumstances that dictated the two connections.

These lead-acid batteries were just enormous. There are tales of ball lightening coming off contactors, and of the very large conductors tearing off the retaining clips holding them to bulkheads from the repulsion caused by going from forward to back emergency power settings.

18. ### thatoneguy AAC Fanatic!

Feb 19, 2009
6,357
718
Updated post above with step two, frozen.

Internal resistance is doubled at the lower temperature, and although the initial 1/10th second current output barely topped 2 Amps, It dropped quickly down to 800mA and below within the half second. I should have graphed the current on the computer, the room temperature drain is a downward sloped line, whereas the frozen one has a couple bumps in the line, and has an incredible slope.

I store my rechargeable and Lithium batteries in a fridge, along with super glue and epoxy so that they all stay good longer. I will warm up the rechargeables and repeat if somebody is interested, though the results would be similar, save for lower internal resistance on NiCd batteries for the entire temperature range, as well as more consistent at many temperatures.

Alkaline were the only ones I had handy to start the "Extremely Scientific" test above with.

Do not take any of the information given as "Battery Specifications"!!! This is an extremely small sample set, though measured with decent equipment (Fluke DMM/Temp, Cadex Battery Analyzer), it is only a rough idea.

I use a ton of batteries in cameras, flashlights, and scopes, so this is an area of my interests.

Last edited: Nov 20, 2009
19. ### steveb Thread Starter Senior Member

Jul 3, 2008
2,433
469
Definitely I'm interested. There is nothing like real data to verify models. Hopefully at some point I'll come up with a model that I have confidence in. Then, I can compare this against any real data to verify.