Combining Resistors

We can combine resistors in series, and in parallel.

Resistors in series add values:

    R_total = R₁ + R₂ + ... + R_n

Resistors in parallel work a little differently. The general forumula for computing resistance in parallel is:

1/R_total = 1/R₁ + 1/R₂ + ... + 1/R_n

But this is formula a pain in the neck to work with. A slightly simpler transformaton, for two resistors, is:

    R_total = (R₁R₂)/ (R₁+R₂)

Even this, though, is not very usable. A handy rule of thumb for resistors in parallel is:

2 equal R's in parallel total R/2.
3 equal R's in parallel total R/3, etc.

As an example of how elegant this rule of thumb is, consider this arrangment of resistors:

To analyze it, take the two 10k's in parallel first -- they combine to make a 5k. Now you've got two 5k's in parallel, for a total of 2.5k ohms. Simple!

Here's another example, which makes the rule of thumb seem even more clever:

    =         =    

Instead of reaching for your calculator, think of the 5k as two 10k's in parallel. Now you've got three 10k's in parallel, for a total of 3.3k.

We have one other useful trick: spotting the dominating resistor. Remember that resistor tolerances are usually about 10%, so anything that changes our total resistance by less than 10% can be safely ignored.

In practice, this means we can ignore the effect of R_small in this case:

And we can ignore the effect of R_big in this case:

Assuming that R_big is more than ten times the value of R_small.

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