Why are batteries measured in ampere-hours but electricity usage measured in kilowatt-hours?

I was reading about energy usage in batteries and don't quite understand why it is measured in different units than home electrical usage. An ampere-hour does not include a measure of volts. But my understanding though is that a battery has a constant voltage (1.5V, 9V, ...) just as much as home electrical usage (120V, 220V, ...). So I don't see why they have different units by which they are measured.

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fyi. It looks like your question touches the subject of battery gauging. Here's a recent overview article: batteryuniversity.com/learn/article/… –  Nick Alexeev Jan 8 '12 at 3:33

$kW \cdot h$ are a measure of energy, for which grid customers are billed and usually shows up on your invoice in easily understood numbers (0-1000, not 0-1 or very large numbers; ranges which, unfortunately, confuse many people).

$A \cdot h$ are a measure of electrical charge. A battery (or capacitor) can store more or less a certain amount of charge regardless of its operating conditions, whereas its output energy can change. If the voltage curve for a battery in certain operating conditions are known (circuit, temperature, lifetime), then its output energy is also known, but not otherwise, though you can come up with some pretty good estimates.

To convert from $A \cdot h$ to $kW \cdot h$ for a constant voltage source, multiply by that voltage; for a changing voltage and/or current source, integrate over time: $$\frac{1 kW\cdot h}{1000 W\cdot h}\int_{t_1}^{t_2} \! I(t)E(t)dt ~;~~E~[V],~I~[A],~{t_{1,2}}~[h]$$

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Grid customers are billed for both power and energy. A battery's "output energy" can't change (neither created nor destroyed). I think you mean "output power". The energy storage capacity of a secondary (i.e. rechargeable) battery can change over the lifetime of the battery due to changes in its structure (because work has been done on it). –  Josh Jan 21 '12 at 7:18
A battery's output energy and charge does change, @Josh, most notably with respect to changes in temperature and current draw, but the number of coulombs it took to deliver that energy is more constant. I wouldn't mind if you'd explain in chat how one could be billed for power alone. –  tyblu Jan 21 '12 at 7:46
My bad, your usage is correct. I thought it was important to draw the distinction between power and energy here, but they are directly related. Energy output will vary with power output because energy is the integral of power. The whole point of a battery is to store energy and of course its output will diminish as you use it. Power isn't usually billed alone to residential customers (though part of what we pay does go to the infrastructure), but it is billed to large customers as "Demand Charges". –  Josh Jan 24 '12 at 3:30

The way a battery works, the total coulombs it can push around falls out more directly than the total energy it can store. The voltage is not constant. It varies by state of charge for one, and the relationship between the two can be quite different between battery chemistries. All this is to say that A-h is more relevant to battery manufacturers than W-h or Joules.

Joules can of course be relevant to circuit design, so this information is available, just not included in the 2 second sound bite called the Amp-hour rating. Battery datasheets can get quite complex. As with most things, there are a host of tradeoffs and thorough information is more than a single number. If you do have to pick just two numbers to quickly characterize a battery, Volts and Amp-hours are as good as any, and are what the industry has converged on.

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The problem with Ah is that it is dependent on the load, and it in usually not specified at the short info located on the battery, its not even always available from the manufacturer. –  Gunnish Dec 12 '12 at 19:26
@Gunnish: All battery datasheets I've seen give you some idea of what the battery can produce over a variety of conditions, with A-h being very common. Yes, the A-h a battery can deliver varies with the current profile, temperature, and other parameters, but so does the total energy. Everything is a function of lots of other parameters when it comes to batteries, but the total deliverable charge is probably more constant than others, like the total deliverable energy. –  Olin Lathrop Dec 12 '12 at 19:46

A note about battery voltage: Rated battery voltage is "nominal". A fully charged 12 volt lead acid battery actually starts out around ~14.4 volts and drops off as you draw energy from it. The actual battery voltage depends on a number of factors not limited to state of charge, battery age, load profile, chemistry, etc,... For instance, A lithium ion battery of 3.7V (nominal) may start out at 4.15 volts and diminish to ~2.7 volts before requiring recharge.

Watt-Hours (or kW-H) is an indicator of the energy storage capacity of the battery, whereas amp-hours would refer to how many amps minimum you can draw from a battery at full charge for an hour before it was no longer capable of providing that level of flow (perhaps at or above the rated voltage?). They are closely related, but not equivalent. Some batteries are designed more for high current draw devices, whereas others are designed to last a long time for lower current draw devices.

Appended: Now that I look at my cell phone battery, I notice that it has all three ratings printed on it. It is a Lithium-Ion battery whose nominal voltage rating is 3.7V. It's energy capacity is marked as 4.81 Watt-Hours. It's electric charge rating is 1300 milliAmp-Hours. This seems to indicate that Energy = Voltage * Electric Charge (at least in terms of the battery ratings), though I think that this equation is hiding the fact that there is an integration of P=VI going on and that V is more like an average value than a constant, which probably gives a pretty good approximation.

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The voltage of a fully charged battery (that has been left alone for 24H) is pretty close to 13 V rather than 14.4 V. 14.4 V is the voltage the charger uses under a charging phase. –  Gunnish Dec 12 '12 at 19:29

One factor not yet mentioned is that because batteries have a certain amount of internal resistance, drawing more current will cause the voltage to sag. Suppose, hypothetically, that a particular battery that's been discharged a certain amount will supply 12 volts when supplying 10mA, or 10 volts when supplying 100mA. Drawing 10mA from the battery for 10 hours will discharge it about as much as drawing 100mA for an hour, but in the former scenario the battery would have supplied 20% more "useful" energy. Key point: a larger fraction of the energy in a battery will be lost when trying to drain it quickly than when trying to drain it slowly.

Power lines also have a certain level of resistance, and similar factors may apply, but the line voltage reaching a residental customer's meter is generally not appreciably affected by that customer's usage. A power company could supply one amp at 105 volts using 20% less energy (per unit time) than would be required to supply one amp at 126 volts. If customers were billed per amp-hour, the power companies would have an incentive to supply their energy at the lowest possible voltage. Billing per kWh means the customer's billable usage will be proportional to the amount of energy the power company has to generate to supply it. Incidentally, some devices (e.g. induction motors) will often draw less current at higher voltages (while doing the same amount of work), while other devices like incandescent lamps and heaters will draw more current at higher voltages (while producing substantially more light and heat).

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