# Nodal Analysis

Nodal analysis is a method of network analysis particularly suited for the networks having many parallel circuits with a common ground connected node. This method of network analysis is based on Kirchhoff’s first law or KCL. Nodal analysis significantly reduces the computational work required by reducing the set of equations to be solved in a particular circuit or network. Nodal analysis is also termed as node-voltage theorem.

## Application of Nodal Analysis

Application of nodal analysis requires taking a node in the circuit as a reference or datum node. For simplicity and easy approach, the node which has highest number of circuit elements connected.

Each node I assigned with unknown voltage variable such as V_{1}, V_{2} and so on. The voltage for the reference node is taken as 0 Volts.

Kirchhoff’s current law is applied at each of these node to form a system of linear algebraic equations. For a circuit consisting of ‘N’ number of nodes, by nodal analysis the number of equation thus formed will be N-1.

These system of linear equations are solved to determine the unknown nodal voltages.

## Illustration of Steps Involved in Nodal Analysis

For the illustration of the analysis of the network using this method we will consider an electrical circuit as shown below.

__At first, we determine nodes in the given circuit and will take one of the node as a reference node.__ For simplicity and easy approach the node with highest number of circuit elements connected can be taken as a reference node. For above circuit we will take node C as reference node as it is junction for source EMF E_{1} and E_{2} along with resistor R_{4} and R_{5} making it the node with highest number of connections of circuit elements.

__Secondly, we will assign variables for the unknown node voltages.__ Such as V_{A} and V_{B} be the unknown node voltage for node A and B. While the voltage for reference node C is taken to be 0 Volts.

__Thirdly, we will apply the Kirchhoff’s Current Law for each of the node (excluding the reference node).__

Such as in above circuit.

Applying KCL for Node A which is the junction of resistor R_{1}, R_{2} and R_{4}.

Applying KCL for Node B which is the junction of resistor R_{2}, R_{3} and R_{5}.

__Finally, solve the linear equation to find the unknown node voltages.__ Such as in above case the two system of linear equations (i) and (ii) can be solved to find the unknown node voltages V_{A} and V_{B}.

## Examples

__Example-1:__

For the circuit shown below, we will determine the current flowing through 1 Ohm resistor using nodal analysis.

At first we will take a node as a reference node. For our simplicity we will take node C as a reference node as shown below.

Now, assigning the voltages at node as V_{A} and V_{B}.

Then applying KCL at node A and B.

For Node A,

For Node B,

Now, solving equation (i) and (ii)

Current through 1 Ohm resistor is given by

The negative sign indicates the current is flowing from node B to A.

__Example-2:__

For the circuit shown below we will determine the node voltages for node 1 and 2 using nodal analysis.

For our easiness we will arbitrarily assume the current distribution as shown below.

Now, applying KCL at node 2 we have

So, we have

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