Series and Parallel

Two sentences that generate every rule

Series: one path. The same current flows through everything, and the voltage divides.

Parallel: one voltage. Everything sees the same voltage, and the current divides.

The pool asks these directly — in which type of circuit is the current always the same through all components? (series) and in which type is the voltage always the same across all components? (parallel) — and every combination rule follows from them.

Combining resistors

In series, resistances add:

$R_{total} = R_1 + R_2 + \cdots$

Two resistors in series carry the same current, so their voltage drops add, so by Ohm’s law their resistances add. 100 Ω and 220 Ω in series give 320 Ω.

In parallel, conductances add, which comes out as the reciprocal rule:

$\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$

100 Ω and 220 Ω in parallel give 1 / (1/100 + 1/220) = 68.75 Ω.

Two shortcuts worth having

• Equal resistors in parallel: the total is one resistor divided by how many there are. Two 100 Ω in parallel is 50 Ω; four is 25 Ω.
• Two resistors in parallel: product over sum. (100 × 220) / (100 + 220) = 22000 / 320 = 68.75 Ω. Faster than the reciprocal form for the common case.

The check that catches every error

A parallel total is always smaller than the smallest branch. If you compute two resistors in parallel and get something larger than either one, you have inverted something. This check costs a second and catches the most common arithmetic slip in the whole subject.

Where the pattern breaks: capacitors

Resistors and inductors follow the same pattern — series adds, parallel is reciprocal. Capacitors are the reverse:

This is a favourite exam trap and it is not arbitrary. Capacitance grows with plate area, so wiring capacitors side by side (parallel) is effectively building one bigger plate — capacitances add. Stacking them in series is like increasing the plate spacing, which reduces capacitance.

Reading a real circuit

Real circuits are neither purely series nor purely parallel; they are series and parallel sections nested inside each other. The method is always the same:

1. Find the innermost group that is clearly one or the other.
2. Reduce it to a single equivalent value.
3. Redraw with that value in place.
4. Repeat until one resistor remains.

There is no shortcut past this, and there is no need for one — each step is either an addition or a product-over-sum.

Check yourself

1. Three 300 Ω resistors in parallel. Total?
2. A 50 Ω and a 50 Ω resistor in series, that pair in parallel with another 50 Ω. Total?
3. Two 10 µF capacitors in parallel. Total capacitance?