Electronics/Circuit fundamentals

Series and parallel circuits

After thiswhat you will be able to doReduce series and parallel resistor groups to equivalent resistances, and identify what stays equal and what adds in each connection.

Questionwhat this lesson answersA circuit can keep one path or split into branches, yet charge cannot disappear at a junction. What stays the same in series and parallel, and how can a network become one equivalent resistance?

Not coveredwhat this lesson leaves outWe analyse ideal direct-current networks made from resistors and wires. We do not use Kirchhoff's full loop method, account for wire resistance, or study time-changing circuits.

The words series and parallel describe connections, not components. Two resistors can be connected in series, in parallel, or in a larger network containing both arrangements. Begin with the simplest distinction: does charge have one route or more than one?

One route: series

Place a ${\SI{2}{\Omega}}$ resistor and a ${\SI{4}{\Omega}}$ resistor one after the other in a single closed path. Any charge crossing the first resistor must cross the second. In a steady circuit, the current is the same through both:

The source’s total voltage drop is the sum of the two individual drops. Using ${V = IR}$ for each resistor,

The one resistance that would draw the same current from the same source is therefore

The ${\SI{2}{\Omega}}$ and ${\SI{4}{\Omega}}$ resistors behave, as far as the source is concerned, like one ${\SI{6}{\Omega}}$ resistance. The voltage divides between them in proportion to their resistances: the larger resistor gets the larger drop because the same current crosses both.

Two routes: parallel

Now connect the same two resistors so that both start at one node and both end at another. Each branch has the same endpoints, so the voltage difference is the same across both:

The current can divide at the first node and recombine at the second. Charge conservation gives

Writing that total current as ${I_{\text{total}} = V/R_{\text{eq}}}$ gives

For the ${\SI{2}{\Omega}}$ and ${\SI{4}{\Omega}}$ branches,

The parallel equivalent is smaller than either branch. Adding a route gives charge another way to cross, so the source supplies more total current for the same voltage. That is the opposite of series, where adding a resistor makes the only route more difficult.

The connection is the explanation

It is tempting to memorise “series means current” and “parallel means voltage.” The better rule is to look at the path:

The equal quantities follow from the geometry of the connection. One uninterrupted path cannot lose charge between components, and shared endpoints force the same potential difference. The quantities that add are the ones collected before a path reunites or across the full path.

Real circuits often nest the two arrangements. Reduce one obvious series or parallel group to an equivalent resistance, then inspect what connection remains. If no group can be reduced cleanly, the full loop and junction rules are the next door. For this arc, the important habit is already in place: draw the endpoints, identify the paths, and only then choose the equation.