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### Kurs: Fizyka - program rozszerzony I>Rozdział 11

Lekcja 5: Oporniki połączone szeregowo i równolegle

# Resistors in series and parallel review

Review how to find the equivalent resistance for resistors in parallel and series configurations. Recall the current and voltage properties of series and parallel configurations of resistors.

## Pojęcia kluczowe

Term (symbol)Meaning
Equivalent resistance (${R}_{\text{eq}}$)The total resistance of a configuration of resistors.

## Równania

EquationSymbol breakdownMeaning in words
$\begin{array}{rl}{R}_{s}& =\sum _{i}{R}_{i}\\ & \text{or}\\ {R}_{s}& ={R}_{1}+{R}_{2}+\text{…}\end{array}$${R}_{s}$ is equivalent series resistance and $\sum _{i}{R}_{i}$ is the sum of all individual resistances ${R}_{i}$.Equivalent series resistance is the sum of all the individual resistances.
$\begin{array}{rl}\frac{1}{{R}_{p}}& =\sum _{i}\frac{1}{{R}_{i}}\\ & \text{or}\\ \frac{1}{{R}_{p}}& =\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}+\text{…}\end{array}$${R}_{p}$ is equivalent parallel resistance and $\sum _{i}\frac{1}{{R}_{i}}$ is the sum of all individual resistances ${R}_{i}$ reciprocals.The reciprocal of the equivalent parallel resistance is the sum of all the individual resistance reciprocals.

## Resistors in series and parallel

### Series resistor properties

Any time we have more than one resistor in a row, the configuration is described as having the resistors in series or series resistors (Figure 1).
Resistors in series have some special characteristics worth remembering. Any configuration of resistors in a series will have the following properties.
• The same current flows through each resistor: ${I}_{1}={I}_{2}=\dots ={I}_{n}$
• Potential difference is distributed among series resistors: $\mathrm{\Delta }{V}_{s}=\mathrm{\Delta }{V}_{1}+\mathrm{\Delta }{V}_{2}+\dots +\mathrm{\Delta }{V}_{n}$
• The resistor with the biggest resistance has the greatest voltage.
• The equivalent resistance ${R}_{s}$ is always more than any resistor in the series configuration.

### Parallel resistor properties

Another possible way to arrange resistors in a circuit is to have multiple resistors branch off from a single junction in the circuit (Figure 2).
Resistors in parallel also have some special characteristics:
• The current is distributed across resistors: $I={I}_{1}+{I}_{2}+\dots +{I}_{n}$
• Potential difference is the same across all resistors in parallel: $\mathrm{\Delta }{V}_{1}=\mathrm{\Delta }{V}_{2}=\dots =\mathrm{\Delta }{V}_{n}$
• The smallest resistance gets the most current.
• The equivalent resistance ${R}_{p}$ is always less than any resistor in the parallel configuration.
Keep in mind that not all circuits are strictly series or parallel. Sometimes they can be a combination of both. We will learn how to analyze more complicated circuits in the next few lessons.

## How to calculate equivalent resistance

Resistors in series or parallel can be replaced by a single resistor of equivalent resistance. This strategy is helpful for solving complex circuit problems because it let’s us simplify the circuit.

### Equivalent series resistance

We can redraw the circuit with the resistors in series replaced by a single equivalent resistor (Figure 3).
We can calculate ${R}_{s}$ from the resistances of the individual resistors in series. If ${R}_{1}=4\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$ and ${R}_{2}=8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$, then the equivalent resistance is the sum of ${R}_{1}$ and ${R}_{2}$:
$\begin{array}{rl}{R}_{s}& ={R}_{1}+{R}_{2}\\ \\ & =4\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }+8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }\\ \\ & =12\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }\end{array}$

### Equivalent parallel resistance

We can redraw a circuit with all resistors in parallel replaced by a single equivalent resistor (Figure 4).
We can calculate ${R}_{p}$ from the resistances of the individual resistors in parallel. If ${R}_{1}=4\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$ and ${R}_{2}=8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$, then the equivalent resistance ${R}_{p}$ is:
$\begin{array}{rl}\frac{1}{{R}_{p}}& =\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}\\ \\ & =\frac{1}{4\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}+\frac{1}{8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}\\ \\ & =\frac{2}{8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}+\frac{1}{8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}\\ \\ \frac{1}{{R}_{p}}& =\frac{3}{8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}\end{array}$
Now, let’s be careful here. Lots of people make a mistake here: $\frac{3}{8}$ is not the equivalent parallel resistance ${R}_{p}$ yet, it is the reciprocal. To solve for ${R}_{p}$, we need to take the reciprocal of both sides:
$\begin{array}{rl}{\left(\frac{1}{{R}_{p}}\right)}^{-1}& ={\left(\frac{3}{8\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}\right)}^{-1}\\ \\ {R}_{p}& =\frac{8}{3}\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }\\ \\ & \approx 2.7\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }\end{array}$

## Dowiedz się więcej

For deeper explanations, see our video on series resistors and video on parallel resistors.
To check your understanding and work toward mastering these concepts, check out the exercise on calculating equivalent resistance for series and parallel resistors.

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