Recap of Resistors in Series and Parallel

Recap of Resistors in Series and Parallel

In previous lessons, you learnt about resistors and were introduced to circuits where resistors were connected in series and in parallel. In a series circuit there is one path along which current flows. In a parallel circuit there are multiple paths along which current flows.

When there is more than one resistor in a circuit, we are usually able to calculate the total combined resistance of all the resistors. This is known as the equivalent resistance.

Equivalent series resistance

In a circuit where the resistors are connected in series, the equivalent resistance is just the sum of the resistances of all the resistors.

Definition: Equivalent resistance in a series circuit

For n resistors in series the equivalent resistance is:

\[R_{s} = R_{1} + R_{2} + R_{3} + \ldots + R_{n}\]

Let us apply this to the following circuit.

The resistors are in series, therefore:

\begin{align*} R_{s} & = R_{1} + R_{2} + R_{3} \\ & = \text{3}\text{ Ω} + \text{10}\text{ Ω} + \text{5}\text{ Ω} \\ & =\text{18}\text{ Ω} \end{align*}

Optional Video by Khan Academy on Circuits and Ohm's Law

Equivalent parallel resistance

In a circuit where the resistors are connected in parallel, the equivalent resistance is given by the following definition.

Definition: Equivalent resistance in a parallel circuit

For \(n\) resistors in parallel, the equivalent resistance is:

\[\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \ldots + \frac{1}{R_{n}}\]

Let us apply this formula to the following circuit.

What is the total (equivalent) resistance in the circuit?

\begin{align*} \frac{1}{R_{p}} & = \left( \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} \right) \\ & = \left( \frac{1}{\text{10}\text{ Ω}} + \frac{1}{\text{2}\text{ Ω}} + \frac{1}{\text{1}\text{ Ω}} \right) \\ & = \left( \frac{\text{1}\text{ Ω} + \text{5}\text{ Ω} + \text{10}\text{ Ω}}{\text{10}\text{ Ω}} \right) \\ & = \left( \frac{\text{16}\text{ Ω}}{\text{10}\text{ Ω}} \right) \\ R_{p} & = \text{0.625}\text{ Ω} \end{align*}

Optional Videos by Khan Academy on Resistors in Series