After discussing series resistors and voltage divider, let us learn about parallel resistors and current division.

*Make sure to read what is dc circuit first.*

## Parallel Resistors

Consider the circuit in Figure.(1),

Figure 1. Parallel resistor connection |

From Ohm's law

Theequivalent resistanceof two parallel resistors is equal to the product of their resistances divided by their sum.

Must be noted that the Equation.(5) only works for two resistors in parallel.

We can expand the Equation.(4) to the general case of a circuit with

*N*resistors in parallel.
The equivalent resistance is

(6) |

*R*=

_{1}*R*= ... =

_{2}*R*= R, then

_{N}(7) |

It is easier to use conductance than resistance when dealing with resistors connected in parallel. From Equation.(6) the equivalent conductance for

*N*resistors is(8) |

*= 1/*

_{eq}*R*, G

_{eq}*= 1/*

_{1}*R*, G

_{1}*= 1/*

_{2}*R*, G

_{2}*= 1/*

_{3}*R*, G

_{3}*= 1/*

_{N}*R*

_{N}
Equation.(8) states :

Theequivalent conductanceof resistors connected in parallel is the sum of their individual conductances.

It means we can redraw Figure.(1) with (2) where we replace the resistances to conductances. The equivalent conductances of parallel resistors is obtained the same way as equivalent resistances of series resistors. In opposite, the equivalent conductances of series resistors is obtained the same way as equivalent resistances of parallel resistors.

Figure 2. Equivalent resistance or conductance |

*of*

_{eq}*N*resistors in series is

(9) |

*i*entering node

*a*in Figure.(1) with same values of voltage, we get

(10) |

## Current Divider

Combining Equations.(1) and (10) we get(11) |

*principal of current division*, and the circuit in Figure.(1) is known as

*current divider*. Take a note that the larger current flows through the smaller resistance.

Figure 3. Short and open circuit in parallel connection |

*R*; so

_{2}= 0*R*is a short circuit, as can be seen in Figure.(3a). From Equation.(11),

_{2}*R*implies that

_{2}= 0*i*

_{1}*= 0*,

*i*=

_{2}*i*. This means that the entire current

*i*bypasses

*R*and flows through the short circuit

_{1}*R*= 0, the path with least resistance. When a circuit is short-circuited as can be seen in Figure.(3a), take note that :

_{2}- The equivalent resistance
*R*_{eq}= 0 - The entire current flows through the short circuit.

For another extreme example where

*R*= ∞ , that is,_{2}*R*is an open circuit as can be seen in Figure.(3b). The current still flows through a path with least resistance,_{2}*R*._{1}
Equation.(11) becomes

(12) |

*N*conductors in parallel with source current

*i*, the

*n*th conductor will have current

(13) |

*R*. Such equivalent resistance must have the same values of current and voltage as the original network at the terminal.

_{eq}## Parallel Resistor Example

Let us review the example below for better understanding

Find

*R*for the circuit in Figure.(4)_{eq}Figure 4. Parallel connection example |

1 Ω and 5 Ω in series

2 Ω and 2 Ω in series

6 Ω and 4 Ω in parallel

Three resistors in series

Have you understood what is parallel resistor? Don't forget to share and subscribe! Happy learning!

*Reference: Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku*

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