Resistors in Parallel

By Ritesh|Updated : October 4th, 2022

Resistors are connected in series or parallel in a complex network. An important circuit analysis technique involves replacing the resistors in parallel with an equivalent resistance in parallel for simplification. Resistors in parallel are basically current dividers circuits.

The voltage across the resistors in parallel is the same. Resistors in parallel have multiple paths for the current to flow. This article elaborates on the resistors in parallel and equivalent resistance of resistors in parallel, along with some examples.

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Resistors in Parallel

When resistors are in parallel, the current gets divided between the resistors in parallel, and the voltage across all the resistors will be the same. If the resistors in parallel have the same resistance, then the current divides equally among all resistors. Resistors in parallel decrease the equivalent resistance of the circuit.

Resistors in parallel are treated as resistance current divider circuits. The power dissipated in the parallel circuit is much less, and maximum power is transmitted through these circuits.

Need for Parallel Combination of Resistors

In a parallel combination of resistors, the current gets divided between the resistors in parallel, and the voltage across all resistors is equal to the source voltage. A parallel combination can easily remove a resistor from the circuit without affecting the other resistors. Resistors are said to be in parallel combination when their two terminals are connected to the same two nodes, as shown below:

Resistors in Parallel

Resistors are connected in parallel for the following purposes:

  • To decrease the overall resistance.
  • To divide the source current between all the resistors.
  • Resistors are connected in parallel combinations to deliver maximum power to the load.

Equivalent Resistance of Resistors in Parallel

For simplification, let us calculate the equivalent resistance for the two resistors connected in parallel, as shown below:

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Resistors in Parallel

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Where Req is the effective resistance of the circuit and can be modeled as:

Resistors in Parallel

The source current i is divided between R1 and R2. This is known as the principle of current division.

Similarly, let us calculate the equivalent resistance for the three resistors connected in parallel, as shown below:

3 Resistors in Parallel

Resistors in Parallel

Where Req is the effective resistance of the circuit and can be modeled as:

3 Resistors in Parallel Equivalent Circuit

Resistors in Parallel

Resistors in Parallel Formula

When N number of resistors are connected in parallel, then the overall equivalent resistance is given by:

Resistors in Parallel

The equivalent resistance of resistors in parallel combination is always smaller than the resistance of the smallest resistor present in the circuit.

If there are N number of resistors connected in parallel and the values of all resistors are the same, i.e., R, then the overall resistance of the circuit is given by:

Req = R/N

In this case, the current also divides equally between all the resistors. If I am the total current and In is the current across the nth resistor, then:

In = I/N

Resistors in Parallel Examples

Example 1: Find the equivalent resistance seen from the terminal ab of the circuit shown below

Resistors in Parallel

Solution: Resistance 2 Ω and 4 Ω are connected in series so that they will form an equivalent resistance of 6 Ω. The reduced circuitry be,

Resistors in Parallel

These resistors are in parallel, so

Resistors in Parallel

Example 2: Find the current i1 and i2 in the circuit shown below,

Resistors in Parallel

Solution: From the figure, resistance 2 Ω and 3 Ω are connected in series, so the circuit can be reduced as shown below,

Resistors in Parallel

By using the current division rule,

Resistors in Parallel

So, the answers are correct.

Important GATE Topics

Determinate And Indeterminate StructuresAstable Multivibrators
Bistable MultivibratorTruss And Frame
Network LayerStatically Determinate
Anomalies In DbmsEulers Equation Of Motion
Dalembert's PrincipleStatically Indeterminate

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FAQs on Resistors in Parallel

  • When resistors are connected in parallel combination, the voltage across each resistor will remain the same and the current divides between the resistors. The equivalent resistance is always smaller than the smallest resistance present in the circuit.

  • No. The resistors in parallel don't have the same current. Although this condition is possible only when all the values of resistance are equal. In that case, the current divides equally between all the resistors.

  • The voltage across all resistors is the same on connecting resistors in parallel combination. The equivalent resistance decreases because more paths exist for the current to pass through. The total current increases, and hence the equivalent resistance decreases.

  • Resistors are said to be in parallel when their two terminals are connected to the same two nodes. Because the nodes are common, the voltage drop across each resistor will also be the same.

  • We can remove or add any other branch in a parallel connection without overloading the circuit. Also, a separate individual switch can be implemented for each branch. If any fault exists in one of the branches, it can be easily removed. So a parallel circuit is preferred over a series circuit.

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