no, you cant actually start a car with 8 AAA batteries. Eight AAA batteries do add up to 12 volts, but they still can’t provide enough electrical current to run the starter motor. But that’s not the whole story. Any battery has a limit on the maximum current it can produce.

To explain why, we have a model to represent a real battery.

The pair of vertical long-short lines (labeled V0;) right of center represent the ideal battery. Or, in other words, how much current the real battery would produce if it wasn’t confounded by resistance (represented by the squiggly line labeled Ri). Yes, all of this ideal battery/confounding resistance stuff is going on inside the actual battery.

But why? Why not just have a battery as an ideal battery? Let me start with an example. Suppose I take this AA battery (listed at 1.5 volts) and hook up this copper wire, which I will assume has a resistance of 1 Ohm (the actual value isn’t important).

Only two things exist in this situation—the battery and the wire (a low value resistance). So, it is governed by something called the voltage loop rule: The sum of the voltages around this loop should be zero. That’s because according to Ohm’s Law, the voltage across the resistor depends on the current. From that, I can solve for the current passing through this wire.

With a battery voltage of 1.5 volts and a resistance of 1 Ohm, this means there should be a current of 1.5 Amps. (In case you didn’t know, 1.5 Amps is a fairly huge current). For comparison, vacuum cleaners use something like 10 Amps. That means this would be one powerful little AA battery. Hence the need for a more realistic model, in order to account for internal resistance.

In this model, voltage across the battery drops as the current increases. Here’s the simple circuit using internal resistance:

As the external resistance (the squiggly lines outside the battery labeled *R*) changes, the voltage drop across the actual battery also changes. If I can measure both the battery voltage *and* the current coming out of the battery, then the following should be true:

So, I will change the resistance and measure the battery voltage as well as the current. With a plot of voltage vs. current, the slope should be the negative of the value for internal resistance (if the model above works). Time to ditch the model and go to real life. Here’s my setup:

Here’s how it works. The voltage and current sensors can record measurements quite quickly. Once data collection starts, I just slide the variable resistor back and forth to get many different resistance values. This data then goes into a V-I plot. But what about the switch? That is there so that I can record the “open circuit” voltage of the battery and not the voltage while under a load.

I am finding the internal resistance of a small, AAA battery, so it will drain quickly. This will give me an idea of how the internal resistance changes as the battery dies. And now for the data. Here is a plot of V vs. I for different battery levels (but the same battery).

So, what about the internal resistance? It seems to be fairly consistent except for the fully charged battery which has an internal resistance of 0.634 Ohms. The other measurements are between 0.25 and 0.4 Ohms with no clear trend. Initially, I thought that the battery’s internal resistance would increase as the battery got drained. But maybe this isn’t the way it works. However, it is quite clear that the open-circuit voltage does decrease with battery use (you can’t tell the order, but this voltage did drop over time).

I suspect the change in internal resistance depends on battery temperature. If the battery gets hotter, then it will have a higher internal resistance. For a fresh battery, there is a higher battery current leading to a hotter (I suspect) temperature. But still, it seems that a constant value for internal resistance might work well enough.

### Homework

I can’t answer all these questions without forcing you to read a way longer post. Also, you’ll learn more if I let you do them as homework:

- Based on the constant internal resistance model, the highest battery current would be in the case of a zero Ohm load. What would the highest output current be for this AAA battery?
- Does internal resistance depend on battery temperature? Set up an experiment to find out.
- Is the open-circuit battery voltage proportional to battery life? I admit, this will take a bit more setup to figure out. You need to first drain a battery and measure voltage and current so that you can calculate the total stored energy. Then you will need to go back in time and measure the open-circuit voltage at different energy levels.
- Do different brand AAA batteries have different internal resistances? (the probably do)
- What about a AA battery or a C-cell battery? What kind of internal resistance do these have?
- What is the internal resistance and maximum current for a penny battery (a battery made by stacking pennies)?