How to detect if resistors are in parallel or series?

Some circuits are easier to see which resistors are in parallel or series. However, when faced with a more complex circuit, I can't figure out which ones are in series or parallel. I am currently facing the following problem:

I am instructed to find the equivalent resistance; but I do not know how to determine which are in parallel or series. Is there a particular method to do so?

• If all of the current exiting one resistor enters another resistor, the two resistors are series connected. If all of the voltage across one resistor is across another resistor, the two resistors are parallel connected. Nov 4, 2015 at 1:49

The trick is to look at the nodes in the circuit. A node is a junction in the circuit. Two resistor are in parallel if the nodes at both ends of the resistors are the same. If only one node is the same, they are in series. So, R1 and R2 are in parallel and R3 is in series with R1||R2.

• Thanks for the response! However, I am still a bit confused about the node joining R2 and R3. Before R2 meets with R1 to the right, R2 meets with R3. So that does not effect R1 and R2 being in parallel? Nov 5, 2015 at 18:27
• That is just the way they are drawn. You'll notice that the right sides of R1 and R2 and the top of R3 are all connected together with wires. There are no components separating them, so they are all connected to the same point in the circuit. We call that point a node. You can draw the circuit any way you like, component don't always appear to obviously be in series or parallel. Looking at it by nodes always works though. You could imagine taking the vertical wire right of R1 and sliding it left past R3. It doesn't change the circuit but it would look obviously parallel. Nov 6, 2015 at 2:11
• The important fact here, is considerar a node as an area instead of point Mar 21 at 22:29

Basically, you simplify, simplify, simplify. For instance, your $R_1$ and $R_2$ are simply in parallel, so you can replace them with a single resistor. Then, depending on which terminals you're measuring from, the merged resistor and $R_3$ will be in parallel or series.

• You cannot replace $R_1$ and $R_2$ by a single resistor, they are not 'simply in parallel' Nov 4, 2015 at 8:09
• Sure you can, and they are. The drawn lines are a bit confusing, but the left ends of $R_1$ and $R_2$ are connected by a wire, and the right ends of $R_1$ and $R_2$ are connected by a wire. Ergo, they are in parallel. Nov 4, 2015 at 11:32
• Oh, sorry about that, you are right. Nov 4, 2015 at 11:47
• This does not answer the op question Mar 21 at 22:31
• @EduardoSebastian Yes, it does. Mar 22 at 23:18

Sometimes a bit of visualization can help, either as a mental exercise or using real objects. Picture the resistors as, lets say, ping-pong balls and the wires as stretchy string (e.g. bungee cord). Pick any two terminals, pull on them and see what happens; repeat for a different pair, etc.. You find that:

• The bottom terminals are the same node, because no resistor is on the path between them
• R1 and R2 are in parallel because they will end up side-by-side for at least one pair of terminals and never in line with each other.
• R3 is in series with R1/R2 because they will end up in line for at least one pair of terminals and R3 will never end up side by side with either R1 or R2.

The easiest method to determine series vs parallel connectivity is to do the following: 1. If one end of R1 is connected to one end of R2 and the other end of R1 is connected to the other end of R2, then the resistors are in parallel. R1 & R2 are in parallel. 2. If only 1 end of R2 is connected to 1 end of R3 and the other end of each resistor is not connected then R3 is in series with the parallel pair R2&R1.