Series and Parallel Combinations

Many circuits can be simplified by combining two or more elements into a single equivalent element. The different elements have rules about the ways they combine when they are in series and when they are in parallel. This reference section will first describe what it means for two terminal elements to be connected in series and parallel, and then how these various elements combine.

Series

A two terminal element is in series with another two terminal element if there is a path from a terminal of one of the elements to a terminal of the other element that does not branch off at any point. The two elements cannot have both nodes in common (they would be in parallel then).

R1 and R2 in series, R3 and R4 are not

It is also important to note that order does not matter because of KCL (there must be the same current going through all of the elements in series because current cannot build up in any of the elements. This is useful for simplifying series connections where there are different types of elements.

                        is equivalent to

Parallel

A two terminal element is in parallel with another two terminal element if they have common nodes for both of their terminals.

R1 and R2 are in parallel, R3 is not in parallel with the other two.

The order of parallel elements also does not matter.

            is equivalent to

Combining elements

There are two basic ways that elements can combine, simple addition or addition of reciprocals. The first is an example of elements that combine by simple addition.

R_eq = 1 + 2 + 3 = 6

In the next example, the reciprocals add.

R_eq = 1 / (1/1 + 1/2 + 1/2) = 1/2

Here is a table of common two terminal elements and how they combine in series and parallel.

Note: Unless they have the same values, voltage sources in parallel and current sources in series will break either KCL or KVL and cannot be combined since circuits containing these types of configurations would fail.