Suppose we want to know the overall resistance between the terminals in the network above. Although this looks complicated, we can make it much simpler by combining resistors in series and parallel to get an overall resistance. If we're chiefly interested in their aggregate behavior (the overall current through them and/or the voltage across the whole group), that will tell us everything we need to know without having to figure out what's going on with each individual resistor. (Problem adapted from Basic Engineering Circuit Analysis, 10th Ed., by Irwin and Nelms.)
If you recall, the basic rules for combining resistors in series and parallel are:
Resistors in series will produce a larger overall resistance, and resistors in parallel will produce a smaller overall resistance.
We'll tackle this problem step by step, starting at the end of the network furthest away from the terminals.
The two highlighted resistors are in series, so we can add them together to get an equivalent resistance of 10 kΩ.
The next step is a little tricky - all three of these resistors are in parallel. (Don't let the diagonal one fool you - all three of these are connected together at both ends!) If we do the math, we get an equivalent resistance of 1.428 kΩ.
Things are nice and simple again. We've got two resistors here we can combine in series.
And finally, we can combine the last two parallel resistors to find an overall resistance between the terminals of 3.268 kΩ.
A little tedious, but combining resistances like this is helpful if you want to simplify a more complex circuit for analysis, or if you're trying to get an idea of big-picture behavior in a circuit with a number of different parts.
More Than You Ever Wanted to Know About Electrical Engineering, Part 3: Series and Parallel Circuits
We've talked a bit about the relationship between voltage, resistance, and current and Ohm's Law. It's easy to see how that applies with a simple circuit like this:
But what if your circuit looks more like one of these?
Wouldn't it be nice if you could combine those resistors together into a single resistance and treat them like the first circuit?
You can, in fact, do exactly that. Let's take a look at the circuit on the left first.
This configuration, with a number of components one after the other, is called a series circuit. Note that the same current flows through all the resistors here - we can simply add resistances to come up with an equivalent resistance value. The equivalent resistance of a series circuit will always be bigger than the value of any of the individual resistors. One big restriction of flow instead of a series of smaller restrictions.
Using this equivalent resistance, we can find the current flowing through this circuit using Ohm's Law.
Cool. So what about the circuit on the right?
This circuit topology (the circuit splits into a number of branches, each with their own components) is called parallel. Note that the current splits here. Combining these into a single resistance is a little counterintuitive - the inverse of the equivalent resistance for a parallel circuit is the sum of the inverses of the individual resistances. Instead of encountering a series of obstacles one after the other, each branching current will encounter a single smaller obstacle. The overall effect is of less resistance - the equivalent resistance for parallel circuits will always be less than any of the individual resistances.
(For a case with only two resistances in parallel, this simplifies to R1*R2/(R1+R2). Although this is a useful shortcut equation, note that it will not work for more than two resistances in parallel!)
Here's what we get when we plug this equivalent resistance in to the circuit:
In this circuit we get a current of about 90 mA, whereas in the series circuit, we got just 6.2 mA. This is remarkable. None of our components have changed - we've got the same 9 V battery and the same three resistors. But just by rearranging them, we've managed to get more than ten times the current.
