Question

Find the current I in the circuit in the figure below using nodal analysis.
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Answer 1

Answer:

There are two basic methods that are used for solving any electrical network: Nodal analysis and Mesh analysis. In this chapter, let us discuss the Nodal analysis method.

In Nodal analysis, we will consider the node voltages with respect to Ground. Hence, Nodal analysis is also called a Node-voltage method.

Procedure of Nodal Analysis

Follow these steps while solving any electrical network or circuit using Nodal analysis.

Step 1 − Identify the principal nodes and choose one of them as the reference node. We will treat that reference node as the Ground.

Step 2 − Label the node voltages with respect to Ground from all the principal nodes except the reference node.

Step 3 − Write nodal equations at all the principal nodes except the reference node. The nodal equation is obtained by applying KCL first and then Ohm’s law.

Step 4 − Solve the nodal equations obtained in Step 3 in order to get the node voltages.

Now, we can find the current flowing through any element and the voltage across any element that is present in the given network by using node voltages.

Example

Find the current flowing through 20 Ω resistor of the following circuit using Nodal analysis.

Step 1 − There are three principle nodes in the above circuit. Those are labelled as 1, 2, and 3 in the following figure.

In the above figure, consider node 3 as a reference node (Ground).

Step 2 − The node voltages, V1 and V2, are labelled in the following figure.

In the above figure, V1 is the voltage from node 1 with respect to ground and V2

is the voltage from node 2 with respect to ground.

Step 3 − In this case, we will get two nodal equations, since there are two principal nodes, 1 and 2, other than Ground. When we write the nodal equations at a node, assume all the currents are leaving from the node for which the direction of current is not mentioned and that node’s voltage is greater than other node voltages in the circuit.

The nodal equation at node 1 is

V1−205+V110+V1−V210=0 V1−205+V110+V1−V210=0

⇒2V1−40+V1+V1−V210=0 ⇒2V1−40+V1+V1−V210=0

⇒4V1−40−V2=0 ⇒4V1−40−V2=0

⇒V2=4V1−40 ⇒V2=4V1−40 Equation 1

The nodal equation at node 2 is

−4+V220+V2−V110=0 −4+V220+V2−V110=0

⇒−80+V2+2V2−2V220=0 ⇒−80+V2+2V2−2V220=0

⇒3V2−2V1=80 ⇒3V2−2V1=80 Equation 2

Step 4 − Finding node voltages, V1 and V2

by solving Equation 1 and Equation 2.

Substitute Equation 1 in Equation 2.

3(4V1−40)−2V1=80 3(4V1−40)−2V1=80

⇒12V1−120−2V1=80 ⇒12V1−120−2V1=80

⇒10V1=200 ⇒10V1=200

⇒V1=20V ⇒V1=20V

Substitute V1 = 20 V in Equation1.

V2=4(20)−40 V2=4(20)−40

⇒V2=40V ⇒V2=40V

So, we got the node voltages V1 and V2 as 20 V and 40 V respectively.

Step 5 − The voltage across 20 Ω resistor is nothing but the node voltage V2 and it is equal to 40 V. Now, we can find the current flowing through 20 Ω resistor by using Ohm’s law.

I20Ω=V2R I20Ω=V2R

Substitute the values of V2 and R in the above equation.

I20Ω=4020 I20Ω=4020

⇒I20Ω=2A ⇒I20Ω=2A

Therefore, the current flowing through 20 Ω resistor of the given circuit is 2 A.

Answer 2

A nodal analysis is used to resolve any electrical network, and is defined as:

• A mathematical method to calculate the distribution of electrical energy between nodes in a circuit.

• This method is also known as the node-voltage method as ground node voltages.

The following are the three rules that define the number relative to the voltage measured between each circuit node:

•  Ohm's law

• Kirchhoff voltage regime and

• Kirchhoff's current law

• Nodal analysis features.

• Nodal analysis is an application of Kirchhoff's current law.

• If there are ‘n’ nodes in the power supply provided, there will be ‘n-1’ simultaneous calculations to be resolved.

• For all node voltages, 'n-1' must be resolved.

• The number of non-reference nodes and the number of nodal equations to be obtained are equal.