MadSci Network: Physics

Re: internal resistance of battery using voltmeter

Date: Sun Nov 2 11:25:50 2003
Posted By: Madhu Siddalingaiah, Physicist, author, consultant
Area of science: Physics
ID: 1062508024.Ph

Hi Sarah,

That's an excellent question!

Measuring internal resistance is a little more complicated.

In general, the internal resistance of a battery is not a fixed value.
It varies over time as the battery loses energy and also varies depending
on the load, or how much current is drawn from the battery. Engineers
refer to this as a non-linear resistance. You might also find that if
really load down a battery, the internal resistance changes dramatically.

In order to fully characterize the internal resistance, we need to take
measurements over time and for different loads. The result is a set of
graphs, each plotting values of resistance as a function of load current.

This can be an involved process, depending on how much data you want.
We can, however, get an estimate of the internal resistance by taking
only two measurements with a voltmeter. First, we need to measure the
"open circuit" voltage of the battery. This is simply the voltage at
the battery terminals when no current is being drawn. Practically,
it's hard to measure voltage without drawing some current, but most
voltmeters have a high enough input resistance that it can be neglected.

So we first measure the open circuit voltage, Vo. Let's say it's
0.9 volts, as in your case. Now we need to load down the battery and
measure the voltage at the battery terminals again, let's call that
Vl. Let's call the load resistance Rl. Given all those values,
the equation for the internal resistance (Ri) of the battery is:

Equ 1: Ri = Rl * ((Vo/Vl) - 1)

You can derive this equation from the equation for a voltage divider:

Equ 2: Vl = Vo * Rl / (Ri + Rl)

This equation can be derived from application of Ohm law and Kirchoffs law.
You can find that information from any text on electric circuit theory.

Now we need to choose a value of Rl that gives us good numbers.
From inspection of equ 2, we can observe that if Rl equals Ri, the load
voltage will be exactly half of the open circuit voltage. This trick
is commonly used if you have a variable resistor within the range of
internal resistance. This is not always possible, so you might have to
use several fixed resistors of different values.

You will probably find that typical alkaline cells
tend to have relatively low internal resistances when they are new. It's
probably in the range of an ohm or so. You will probably need a resistor
that is not more than 10 times that value to make good measurements. If
you can find a 10 ohm resistor that's OK. Of course, you should start
with higher values to see the effects. Start with 100 Ohms and see what
you get. Then try lower values and run the calculations again. You can
enter your values into a spreadsheet and plot the results.

A word of caution: be careful when using low value resistors with good
batteries, the current draw will be high enough the heat up the resistor
(and the battery!).

I hope that answers your question.
Let me know what you find out!

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