Parallel Circuits

A kind of circuit in which current emerges from a node and branches off to different paths which eventually meet up at a common node. Due to the branching, the path appears to be in parallel thereby giving it the name parallel circuits. Due to the branching, different current flows in each branch but it is important to note that each branch has the same potential difference which is equal to the potential drop between the two node points.

In these circuits, first, the current is branched out and then it is recombined at the common point. In a parallel circuit, elements are not connected end-to-end.

Key Principles

Here are the three key principles of laws of the parallel circuit:

• Voltage : Voltage in a parallel circuit plays an important role. All components share the same voltage. This means that in any branch of a parallel circuit, the voltage drop is the same.
• Resistance : The equivalent resistance of a parallel circuit is always less than the smallest individual resistance. As more components are added in parallel, the overall resistance decreases because multiple paths are available for current flow.
• Current : The total current in a parallel circuit is the sum of currents flowing through each branch: I = I_1 + I_2 + I_3.... . This means that the current divides among the branches and recombines at the common node.

Working

The parallel circuit looks like any other circuit with the addition of branching as shown below

Voltage in a Parallel Circuit

In the Parallel circuit the voltage across the each parallel component is the same. This is because there are only two sets of the electrically common points in a parallel circuit and the voltage is measured between these sets of points that are same at any given time.

All resistors are connected between the same two nodes, hence the voltage across each resistor is equal.

Using Ohm’s Law for Parallel Circuits to Determine Current 

In a parallel circuit, the current flowing in the circuit is equal to the sum of current in the individual branch. We will apply this in the above circuit.

Hence I = I_1 + I_2 + I_3

Since the voltage across each branch is the same and using ohm's law, we write as I = \frac{V}{R_i}

I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}

How to Calculate Total Resistance in a Parallel Circuit

Let R_eq be the equivalent Resistance of the circuit then in the given current formula, Substituting the value of the resistor

\frac{V}{R_{eq}} = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}

Dividing V from both the side

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}

Hence equivalent resistance of the circuit is

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}

The total conductance of the parallel circuit can be given as the sum of the individual branch conductance. As we add more paths for the current to flow the circuit will becomes more conductive.

G_{total}=G_1+G_2+G_3

Conductance is the reciprocal of resistance G = \frac{1}{R}, so parallel circuits are easier to analyze using conductance.

Characteristics

• Common Nodes: All components are connected between the same two nodes, forming multiple parallel branches instead of an end-to-end connection.
• Equal Voltage: The voltage across each branch is the same and equal to the potential difference between the two common nodes.
• Current Division: The total current is the sum of currents through all branches: I = I_1 + I_2 + \cdots + I_n
• Reduced Equivalent Resistance: The total resistance decreases as more branches are added and is always less than the smallest individual resistance.
• Independent Operation: Each component operates independently, so failure of one branch does not affect the others.

Advantages

• Independent Components: In a parallel circuit, all components operate independently. Since the components have an independent voltage they work even when a component in any other branch is not working.
• Different Current Distribution: In the real world, every appliance has its current requirement depending on its rating. In a parallel circuit, every branch may have a different current flowing through it thereby allowing different current distribution.
• Stable Voltage: Each device in a parallel circuit receives constant voltage, this stable running voltage is the same across each branch ensuring a stable circuit.
• Low Resistance: In a parallel circuit, the equivalent resistance is less than the smallest individual resistance. As we know current is inversely proportional to resistance, This can result in increased current flow, hence being useful when a high current is required.
• Low Complexity: Parallel circuits are easy to design due to their low complexity and are reliable circuits.

Disadvantages

• Additional Cost: Due to additional components in parallel circuits as compared to series circuits, parallel circuits have an additional cost associated with purchasing components.
• Power Consumption: Due to more components in parallel circuits than series circuits, the power consumption of these circuits can increase drastically. This is common when multiple components are operating simultaneously.
• Complex Diagnosis: In a parallel circuit, Identifying the source of an error can be more difficult as compared to a series circuit because there are multiple paths for current to flow hence multiple sources of error.

Applications

There is a need for parallel circuits because they have various applications in different fields some of which are given below:-

• Household Wiring: Used in home electrical systems to provide equal and stable voltage to all appliances.
• Industrial Systems: Ensure independent operation of machines, so failure of one device does not affect others.
• Security Systems: Improve reliability by allowing security devices to function even if one component fails.
• Automobile Wiring: Used in vehicles so that failure of one light does not affect the operation of others.

Solved Example of Parallel Circuits

This example shows how you can mathematical concepts to calculate current and other parameters in a parallel circuit.

Calculate the total current in the circuit and the power across the 2 kΩ resistor.

Firstly we calculate the total resistance of the circuit to calculate the current. Let the total resistance be R_eq then

\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}

\frac{1}{R_{eq}} = \frac{1}{10} + \frac{1}{2} + \frac{1}{1}

\frac{1}{R_{eq}} = 0.1 + 0.5 + 1 = 1.6

Req = 1 / 1.6 = 0.625 kΩ

Now on applying ohms law

V = I * R_eq

I = V / Req = 9 / 0.625 = 14.4 mA

Hence total current in the circuit is 14.4 mA

Now we want to calculate power across 2kΩ resistor. Since voltage across each resistor is same we use the formula P = \frac{V^2}{R}

So Power across 2k is:

P = V² / R = 81 / 2000 = 0.0405 W = 40.5 mW

Hence power across 2k resistor is 40.5mW

Difference