Combining Resistors
We can combine resistors in series, and in parallel.
Resistors in series add values:
R_{total} =
R_{1} + R_{2} + ... + R_{n}
Resistors in parallel work a little differently. The general
forumula for computing resistance in parallel is:
1/R_{total} = 1/R_{1} + 1/R_{2} + ... +
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} = (R_{1}R_{2})/
(R_{1}+R_{2})
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}.
