# Parallel Circuits

## Resistance, voltage and current in parallel networks

The sum of currents in a parallel network can be calculated as

I = I_{1}+ I_{2}+ ... + I_{n}(1)

where

I_{n}= parallel current (amps)

I = sum of currents (amps)

The total resistance in a parallel circuit can be calculated as

1 / R = 1 / R_{1}+ 1 / R_{2}+ ... + 1 / R_{n}(2)

where

R_{n}= parallel resistance (ohms, Ω)

R = total resistance in the parallel network (ohms, Ω)

### Example - Parallel Network

The total resistance in a network with three resistors *R _{1} = 10 Ω, R_{2} = 20 Ω and R_{3} = 30 Ω *can be calculated as

1 / R = 1 / (10 Ω) + 1 / (20 Ω) + 1 / (30 Ω)

= 0.183 (1/Ω)

R = 1 / 0.183

= 5.46 (Ω)

Connected to a *12 V* battery the sum of currents can be calculated

I = U / R

= (12 V) / (5.46 Ω)

= 2.2 Amps)

The current through each resistor can be calculated as

I_{1}= U / R_{1}

= (12 V) / (10 Ω)

= 1.2 (Amps)

I_{2}= U / R_{2}

= (12 V) / (20 Ω)

= 0.6 (Amps)

I_{3}= U / R_{3}

= (12 V) / (30 Ω)

= 0.4 (Amps)

### Resistors in Parallel and resulting Resistance

Two resistors in parallel - resistance ranging 1 - 100 ohm

Resulting Resistance (Ω) | ||||||||||||

R2 ( Ω) | R1 (Ω) | |||||||||||

1 | 1.5 | 2.2 | 3.3 | 4.7 | 6.8 | 10 | 15 | 22 | 33 | 47 | 68 | |

1 | 0.5 | 0.6 | 0.69 | 0.77 | 0.83 | 0.87 | 0.91 | 0.93 | 0.95 | 0.97 | 0.98 | 0.99 |

1.5 | 0.6 | 0.75 | 0.89 | 1.03 | 1.14 | 1.22 | 1.30 | 1.36 | 1.40 | 1.43 | 1.45 | 1.46 |

2.2 | 0.69 | 0.89 | 1.1 | 1.32 | 1.50 | 1.66 | 1.82 | 1.92 | 2.0 | 2.06 | 2.10 | 2.13 |

3.3 | 0.77 | 1.03 | 1.32 | 1.65 | 1.94 | 2.22 | 2.48 | 2.70 | 2.87 | 3.00 | 3.08 | 3.14 |

4.7 | 0.83 | 1.14 | 1.50 | 1.94 | 2.35 | 2.78 | 3.20 | 3.58 | 3.87 | 4.12 | 4.27 | 4.39 |

6.8 | 0.87 | 1.22 | 1.66 | 2.22 | 2.78 | 3.40 | 4.05 | 4.68 | 5.19 | 5.64 | 5.94 | 6.18 |

10 | 0.91 | 1.30 | 1.82 | 2.48 | 3.20 | 4.05 | 5.0 | 6.0 | 6.9 | 7.7 | 8.3 | 8.7 |

15 | 0.93 | 1.36 | 1.92 | 2.70 | 3.58 | 4.68 | 6.0 | 7.50 | 8.9 | 10.3 | 11.4 | 12.2 |

22 | 0.95 | 1.40 | 2.00 | 2.87 | 3.87 | 5.19 | 6.9 | 8.9 | 11.0 | 13.2 | 15.0 | 16.6 |

33 | 0.97 | 1.43 | 2.06 | 3.0 | 4.12 | 5.64 | 7.7 | 10.3 | 13.2 | 16.5 | 19.4 | 22.2 |

47 | 0.98 | 1.45 | 2.1 | 3.08 | 4.27 | 5.94 | 8.3 | 11.4 | 15.0 | 19.4 | 23.5 | 27.8 |

68 | 0.99 | 1.46 | 2.13 | 3.14 | 4.39 | 6.18 | 8.7 | 12.2 | 16.6 | 22.2 | 27.8 | 34.0 |