## What is nodal analysis?

In the analysis of any electric circuit, nodal analysis is the method of determining the potential difference or voltage between different nodes in an electric circuit. A node is a point where two or more branches connect each other. In the nodal analysis, the voltages are determined in the terms of branch currents. The nodal analysis is also known as the node voltage method or branch current method. The nodal analysis method is one of the several applications of Kirchhoff's Current Law (KCL).

## Nodal analysis of a circuit

The electric circuit can be analyzed either by nodal analysis or mesh analysis. For analyzing the circuit using the nodal analysis method, Kirchhoff's Current Law (KCL) is used, whereas to analyze the circuit using mesh analysis, Kirchhoff's Voltage Law (KVL) is used.

### Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law states that for any node in an electric circuit, the sum of currents flowing into the node is equal to the sum of currents flowing out of the node. The KCL is also known as Kirchhoff's first law. A general circuit consisting of various circuit elements is shown below. The circuit consists of 4 branches 1, 2, 3, and 4, carrying currents i_{1}, i_{2}, i_{3}, and i_{4} as given below. A resistor R_{1} is connected across branch 1, voltage source v_{g} is connected across branch 4, and a bulb is connected across branch 2. Let the node formed by the four branches be named node k. The currents i_{2} and i_{3} are flowing into node k from their respective current sources and the currents i_{1} and i_{4} are flowing out from node k.

Hence, as per Kirchhoff's law, i_{2} + i_{3} = i_{1} + i_{4}

## Types of nodes

In a nodal analysis, there are two types of nodes- non-reference nodes and reference nodes. Non-reference nodes are the nodes that have well-defined node voltages. A reference node is a node that acts as a reference point to the other nodes in a circuit. Reference nodes are also known as Datum nodes. The reference node is further categorized into two types - Chassis ground and Earth ground. The Chassis ground type reference node is a node that acts as a common node for multiple circuits. The reference node is known as the Earth ground type reference node when Earth potential is used as a reference for a circuit. Another type of node called supernode is constructed in the circuit to solve the circuits. A supernode is a voltage source that is connected between two non-reference nodes. A supernode is a theoretical construction made to solve a circuit, primarily used to determine the voltages of two non-reference nodes.

## Procedure for nodal analysis

The nodal analysis can be solved either manually or by using certain software like Matlab. The manual procedure of nodal analysis done is discussed here. During nodal analysis, nodal equations are written at each electric node. The basic requirement for the nodal analysis is that the algebraic sum of all the branch current at a node should be equal to zero. Before nodal analysis, all the currents in the branches of the circuits are converted into node voltages. This can be achieved by proper conversion. For example, for a resistor in a particular branch, using ohm's law, V_{branch} = I_{branch} . R_{branch} or V_{branch} = I_{branch} . G_{branch} where, V_{branch}, I_{branch}, and R_{branch} are the voltage, current, and resistance of a particular branch and G_{branch} is the conductance of the resistor in that branch. If there are 'n' number of nodes in a circuit then 'n-1' simultaneous equations need to be solved to obtain the node voltages of the circuit. Once, all the conversions are done, the following procedure is then adopted for nodal analysis:

- The nodes in a circuit are identified first. The connected wire segments in a circuit are the nodes in a circuit.
- One of the nodes is then selected as a reference node. The reference node is treated as ground. To simplify the analysis, it is preferred to select a node having the most number of connections.
- Next, all the nodes except the reference nodes are labeled using variables. Some other nodes except the reference node whose node voltages are known are also excluded from labeling.
- Now, to find voltages at each node, the nodal equation for all the nodes having unknown node voltages is developed using Kirchhoff's Current Law. To achieve this, KCL equations are developed for each node. For developing the KCL equations, all the currents leaving the node are added and equated to zero.
- Next, the current at each node is obtained by dividing (nodal voltage when current exits the node - nodal voltage when current enters the node) to the Resistance between the nodes.
- In this case, if there are voltage sources between two unknown voltages, two nodes are joined to form a supernode.
- The simultaneous equations formed using the nodal analysis can also be written in a matrix equation form. The equations developed will form a matrix that will be non-singular.

## Modified nodal analysis of a circuit

Modified nodal analysis is an improvised version of the standard nodal analysis method. The Modified Nodal Analysis determines the branch currents along with the nodal voltages. The method uses voltage-current characteristics, also known as Branch Constitutive Equations (BCE), and Kirchhoff's Current Law to analyze the circuit. The following procedures are adopted in analyzing the circuit using the nodal analysis:

- First, the KCL equations are written for the circuit. The current coming into the node and going out of the node are written on each node. The current of independent voltage sources is considered to be flowing from positive to negative.
- Next, the Branch Conservative Equations are written in the terms of node voltages, and the maximum possible branch currents are eliminated. Then the remaining linear equations are written down and solved simultaneously.
- Instead of writing the BCEs in terms of node voltages, they can be written in the terms of branch voltages. In that case, another step of converting the branch voltages to the node voltages gets added to the procedure.
- The linear equations obtained can also be written in the terms of matrices and solved.

## Context and Applications

The nodal analysis is useful for the students undergoing the following courses:

- Bachelors in Technology (Electrical Engineering)
- Masters in Technology (Electrical Engineering)
- Masters in Technology (Power System and Power Electronics)

## Practice Problems

1. Which of the following is a voltage source that is connected between two non-reference nodes?

- Supernode
- Amplifier
- Resistor
- Capacitor

Answer: Option a

Explanation: Supernode is a voltage source that is connected between two non-reference nodes.

2. What is determined in the terms of branch currents in a nodal analysis?

- Current sources
- Resistances
- Voltages
- Passive elements

Answer: Option c

Explanation: The voltages are determined in the terms of branch currents in a nodal analysis.

3. What is the alternative name of the reference node?

- Constructive node
- Datum node
- Resistive node
- Conductive node

Answer: Option b

Explanation: Datum node is an alternative name of the reference node.

4. Which of the following is the basic requirement for a nodal analysis?

- The algebraic sum of all the branch current at a node should be equal to unity.
- The current sources of all the branch currents should be present outside the circuit.
- The algebraic sum of all the branch current at a node should be equal to zero.
- The voltage sources of the circuit should be present outside the circuit.

Answer: Option c

Explanation: The basic requirement for a nodal analysis is that the algebraic sum of all the branch current at a node should be zero.

5. Which of the following nodes has well-defined node voltage?

- Chassis ground
- Earth ground
- Reference node
- Non-reference node

Answer: Option d

Explanation: Non-reference node has well-defined node voltage.

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