Nodal Analysis

As-Salaam-Alaikum, Welcome to my blogger "World of Knowledge".Here we discuss the topic which is very important in the field of electrical engineering i.e Nodal Analysis.

Points to be discuss in this Article are given below:

• Nodal Analysis
• Features of Nodal Analysis
• Types of Nodal Analysis
• Solving of Circuit Using Nodal Analysis
• Basic Steps Used in Nodal Analysis
• Nodal Analysis with Current Sources
• Nodal Analysis with Voltage Sources

Nodal Analysis:
                                    Nodal analysis is a method of analyzing circuits based on defining node voltages as the variables.

Solving circuits with a free floating voltage source using the nodal analysis technique can be a bit tricky at first.

Nodal analysis is a method of analyzing circuits based on defining node voltages as the variables.

Features of Nodal Analysis:

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 Nodal Analysis:

• Non Reference Node( It is a node which has a definite Node Voltage)
• Reference Node(It is a node which acts a reference point to all the other 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:

1. Select a node as the reference node. Assign voltages V₁, V₂… V_n-1 to the remaining nodes. The voltages are referenced with respect to the reference node.

2. Apply KCL to each of the non reference nodes.

3. Use Ohm’s law to express the branch currents in terms of node voltages.

Note:

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

Nodal Analysis with Current Sources:

Example:

In the following circuit,3 nodes 

one is reference node while other two are non reference nodes(Node 1 and Node 2)

Step 1:

Assign the nodes voltages as v1 and 2 and also mark the directions of branch currents with respect to the reference nodes

Step 2:

Apply KCL to Nodes 1 and 2
KCL at Node 1
KCL at Node 2
Step 3:

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 4: 

Now solve the equations 3 and 4 to get the values of v1 and v2 as,
Using elimination method

And substituting value v2 = 20 Volts in equation (3) we get-

Hence,

node voltages are v1 = 13.33 Volts and v2 = 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.
Case 2:

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

Supernode Analysis:

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.

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.

Example:

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 v2– v1 = 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, 

                                  v1 = – 7.333V  &  v2 = – 5.333V