## Definition of Nodal Analysis

**Nodal analysis** is a method that provides a general procedure for analyzing circuits using node voltages as the circuit variables. **Nodal Analysis** is also called the **Node-Voltage Method**.

Some Features of Nodal Analysis are as

**Nodal Analysis**is based on the application of the Kirchhoff’s Current Law (KCL).- Having ‘n’ nodes there will be ‘n-1’ simultaneous equations to solve.
- Solving ‘n-1’ equations all the nodes voltages can be obtained.
- The number of non reference nodes is equal to the number of Nodal equations that can be obtained.

## Types of Nodes in Nodal Analysis

- Non Reference Node – It is a node which has a definite Node Voltage. e.g. Here Node 1 and Node 2 are the Non Reference nodes
- Reference Node – It is a node which acts a reference point to all the other node. It is also called the Datum Node.

### Types of Reference Nodes

- Chassis Ground – This type of reference node acts a common node for more than one circuits.
- Earth Ground – When earth potential is used as a reference in any circuit then this type of reference node is called Earth Ground.

## Solving of Circuit Using Nodal Analysis

#### Basic Steps Used in Nodal Analysis

- Select a node as the reference node. Assign voltages V
_{1}, V_{2}… V_{n-1}to the remaining nodes. The voltages are referenced with respect to the reference node. - Apply KCL to each of the non reference nodes.
- Use Ohm’s law to express the branch currents in terms of node voltages.

Node Always assumes that current flows from a higher potential to a lower potential in resistor. Hence, current is expressed as follows

IV. After the application of Ohm’s Law get the ‘n-1’ node equations in terms of node voltages and resistances.

V. Solve ‘n-1’ node equations for the values of node voltages and get the required node Voltages as result.

### Nodal Analysis with Current Sources

**Nodal analysis with current sources** is very easy and it is discussed with a example below.

Example: Calculate Node Voltages in following circuit

In the following circuit we have 3 nodes from which one is reference node and other two are non reference nodes – Node 1 and Node 2.

Step I. Assign the nodes voltages as v_{1} and _{2} and also mark the directions of branch currents with respect to the reference nodes

Step II. Apply KCL to Nodes 1 and 2

KCL at Node 1

KCL at Node 2

Step III. Apply Ohm’s Law to KCL equations

• Ohm’s law to KCL equation at Node 1

Simplifying the above equation we get,

• Now, Ohm’s Law to KCL equation at Node 2

Simplifying the above equation we get

Step IV. Now solve the equations 3 and 4 to get the values of v_{1} and v_{2} as,

Using elimination method

And substituting value v_{2} = 20 Volts in equation (3) we get-

Hence node voltages are as v_{1} = 13.33 Volts and v_{2} = 20 Volts.

## Nodal Analysis with Voltage Sources

Case I. If a voltage source is connected between the reference node and a non reference node, we simply set the voltage at the non-reference node equal to the voltage of the voltage source and its analysis can be done as we done with current sources. v_{1} = 10 Volts.

Case II. If the voltage source is between the two non reference nodes then it forms a supernode whose analysis is done as following

### Supernode Analysis

#### Definition of Super Node

Whenever a voltage source (Independent or Dependent) is connected between the two non reference nodes then these two nodes form a generalized node called the Super node. So, Super node can be regarded as a surface enclosing the voltage source and its two nodes.

In the above Figure 5V source is connected between two non reference nodes Node – 2 and Node – 3. So here Node – 2 and Node – 3 form the Super node.

#### Properties of Supernode

- Always the difference between the voltage of two non reference nodes is known at Supernode.
- A supernode has no voltage of its own
- A supernode requires application of both KCL and KVL to solve it.
- Any element can be connected in parallel with the voltage source forming the supernode.
- A Supernode satisfies the KCL as like a simple node.

#### How Solve Any Circuit Containing Supernode

Let’s take a example to understand how to **solve circuit containing Supernode**

Here 2V voltage source is connected between Node-1 and Node-2 and it forms a Supernode with a 10Ω resistor in parallel.

Note – Any element connected in parallel with the voltage source forming Super node doesn’t make any difference because v_{2}– v_{1} = 2V always whatever may be the value of resistor. Thus 10 Ω can be removed and circuit is redrawn and applying KCL to the supernode as shown in figure gives,

Expressing and in terms of the node voltages.

From Equation 5 and 6 we can write as

Hence, v_{1} = – 7.333V and v_{2} = – 5.333V which is required answer.