Nodal Analysis: Know Definition, Types, Steps, Properties & Examples

In physics, circuit analysis means understanding a circuit. In circuit analysis, we figure out voltages, current passing through each element of the circuit and try to find out the type of and total resistance. There are three methods available for Circuit Analysis and nodal analysis is one of them. It was introduced by Willy McAllister. It is an application of Kirchhoff’s Circuit Law.

Read on, to know more about nodal analysis, its types, properties and steps to apply nodal analysis. You will also learn about the concept of super node in nodal analysis and how mesh analysis differs from it with FAQs.

Nodal Analysis

Nodal Analysis or Node Voltage Mode is a systematic method used for analyzing circuits using node voltage as circuit variables. In simple words, this method is used for determining the voltage (potential difference) between nodes. Nodes are the points where branches or elements connect with each other. Nodal analysis can be applied for both planar and non-planar networks since each node, whether it is planar or non-planar, can be assigned a voltage. The nodal method of circuit analysis is based on Kirchhoff’s Current Law. 

There are three laws that are used to define equations for voltage measured between each of the circuit nodes:

• Ohm’s Law-  It states that the current through a conductor between two points is directly proportional to the voltage across the two points.
• Kirchhoff’s Voltage Law- It states that the algebraic sum of the potential differences in any loop must be equal to zero as: ΣV = 0. It is used for calculating current.
• Kirchhoff’s Current Law- It is used for the calculation of voltage and states that all the currents entering and leaving a junction must be equal to zero as: .

Types of Nodes in Nodal Analysis

There are two types of nodes in nodal analysis. These include:

Non-reference node

​The node which has a definite node voltage is known as a non-reference node.

Reference node

​The node which acts as the reference point for all the other nodes is known as a reference node. It is also known as a datum node. Reference nodes are further of two types-

• Chassis Ground– A node that acts as a common node for more than one circuit is known as chassis ground.

• Earth Ground– Whenever the potential of the earth is used as a reference, it is known as earth ground.

Since we have covered the basics of nodal analysis, let us discuss the steps to apply nodal analysis.

Steps to Apply Nodal Analysis

The following steps are to be followed while solving any electrical circuit using nodal analysis:

Step 1: Identify all the nodes i.e. connected wire segments and select one node as a reference node. This reference node is considered as ground reference. For this take a node that has a maximum number of connections.

Step 2: Label all the node voltages with respect to the ground from all principal nodes except for the reference node.

Step 3: Using Ohm’s law and Kirchhoff’s current law, derive a nodal equation for all the non-reference nodes.

Step 4: To find out current, the current between two nodes is equal to the voltage of the node where the current exits minus the voltage of the node where the current enters the node divided by the resistance between the two nodes.

Step 5: If between two unknown voltages, there is a voltage source, join the two nodes to form a super node. The current for these two nodes can be combined in a single equation and a new voltage equation is formed.

Nodal Analysis with Current Source

• Identify node voltages and label the directions of branch circuit wrt reference node V3.
• Apply KCL to all nodes except reference node to obtain node equations. Then apply Ohm’s law to obtain KCL equations. 

By applying KCL to node 1, we get

By applying KCL to node 2, we get

• Apply Ohm’s law to the KCL equations of nodes 1 and 2.

Applying Ohm’s law to node 1 KCL equation, we get,

Applying Ohm’s law to node 2 KCL equation, we get,

• Solve node equations 3 and 4 obtained by Ohm’s law to determine the node voltages V1 and V2

3V1 – V2= 20

-3V1 + 5V2 = 60

By using the elimination method, we get,

4V2 = 80

Therefore, V2 = 20 Volts

Substitute the value of V2 in eq 3, we get

3V1 – V2 = 20

3V1 – 20 = 20

3V1= 40

V1 = 40/3=13.33 Volts

Nodal Analysis with Voltage Sources

There are two cases to calculate node voltages using nodal analysis with a voltage source.

• Case 1: When a voltage source is connected directly between a non-reference node and the reference node, the node voltage at the non-reference node will be equal to the specified voltage of that source.
• Case 2: When a voltage source connects two previously separate non-reference nodes, it causes those nodes to be combined into a single super node. To analyse the super node, both Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) must be applied at/around the super node.

In the above circuit, O is the reference node and and V1, V2, and V3 are the voltages of non-reference nodes.

The node voltage V3 can be calculated using Ohm’s law.

[(V1 – V2) /10 ]+ [(V3 – V2) /20] = V2 / 40

Substitute the values of V1 and V3 in the above equation,

(1- V2 / 10) + (1- V2 / 20) = V2 / 40

2= V2 (1/40 + 1/20 + 1/10)

V2= 80/7 Volts

Substituting the value of V2 in V=IR equation, 

80/7= I2 (40)

From the circuit resistance R = 40ohms

I2= 2/7Amps= 0.286 Amps

What is a Super node?

Whenever there is a voltage source connected between two unknown voltage or non-reference nodes, the two nodes are joined to form a generalized node, which is known as a super node.

In the above image, the 5V source is connected between two non-reference nodes that are node 2 and node 3. Therefore, node 2 and node 3 combine to form a super node.

Properties of Super node

Some key properties of super node are discussed below.

• A super node itself does not have a well-defined voltage.
• To solve for the voltages within a super node, both Kirchhoff's Current Law (KCL) and Kirchhoff's Voltage Law (KVL) must be applied.
• Any circuit element, such as a resistor, can be placed in parallel within a super node since the nodes are effectively shorted together.
• KCL is satisfied at a super node just like at a regular node.
• Within a super node, the voltage difference between any two nodes is zero.

Difference between Mesh and Nodal Analysis

The difference between mesh and nodal analysis is discussed in the table below. 

Aspect	Mesh Analysis	Nodal Analysis
Main Focus	Voltage loops (meshes)	Node voltages
Basis	KVL (Kirchhoff's Voltage Law)	KCL (Kirchhoff's Current Law)
Variables	Mesh currents	Node voltages
Equations	One equation per mesh	One equation per node
Independent Variables	Mesh currents	Node voltages
Dependent Variables	Branch currents	Branch voltages
Applicability	Suitable for planar circuits	Suitable for any circuit
Number of Equations	Equal to the number of meshes	Equal to the number of nodes
Number of Unknowns	Equal to the number of meshes	Equal to the number of nodes
Solving Techniques	Simultaneous equations	Simultaneous equations or matrix methods
Complexity	Simpler for some circuits, e.g., ladder networks	Can handle complex circuits more easily
Example Applications	Circuits with current sources	Circuits with voltage sources

Properties of Nodal Analysis

The properties of nodal analysis have been stated in these points:

• Nodal analysis is an application of Kirchhoff’s current law, used for the calculation of voltage.
• If there are ‘n’ number of nodes present in the given circuit, then there will be ‘n-1’ number of equations formed to solve. For example, if there are 10 nodes, 10-1 = 9 number of equations are required to solve.
• The number of non-reference nodes is equal to the number of nodal equations to be solved. For example, if there are 10 nodes, there will be 1 reference node and 10-1=9 non-reference nodes which is equal to, 10-1 = 9 number of equations are required to solve.

Nodal Analysis Problems

For better understanding let us try to solve the problems below:

Example 1. Apply analysis to find node voltage V in the following circuit.

Solution:

(V_1) = 7.17V

Example 2. Use analysis to compute the voltage across the 18 A current source in the circuit of this figure:

Solution: By replacing the parallel conductances with their equivalents, the circuit simplifies as follows:

Applying analysis at Nodes 1, 2, and 3 we get:

Node 1:

Node 2:

Node 3:

On simplifying the equations above, we get:

…1

…2

…3

Add the equation 1 and 2 and group with 3. We get:

For the problem, v_2 = v_{18A}

So we use Cramer’s rule, and solve for . Therefore,

,

= -21 + 40 = 19

= 42 -25 = 17

And v_2 = v_{18A} = 19/17 = 1.12 V

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