Nodal Method

NODAL METHOD

 20-200.1-8071-1-

The nodal method is used to analyze multi source circuits. In this method, we solve the simultaneous equations using Kirchhoff's current law (KCL) applied at various nodes in an electric circuit. Node is defined as junction or joining point of two or more component terminals.

In nodal method, we select one node as a reference mode, with respect to the voltages at all other nodes are measured. Thus, the reference node acts as a ground or common for the circuit.

The first step in nodal method is to convert all voltage sources to current sources. The second step is to identify all the principal nodes in the circuit and to choose a reference node. The choice is arbitrary, but it is usually convenient to select the reference node as the one having the most of the components connected to it. All the nodes except the reference node are then numbered and their corresponding voltages are designated as V₁, V₂... etc. The reference node and other nodes are indicated as shown in circuit.

Steps

1. Select one major node as the reference node and assign each of the (n-1) remaining nodes with its own unknown potential (with respect to the reference node).

2. Assign branch current to each other branch.

The arrow is drawn from V₁ towards V₂ because current always flows from higher potential to the lower potential.

3. Express the branch currents in terms of the node potentials.

I = V₁ - V₂ / R

4. Write the current equation at each of the (n - 1) unknown nodes.

5. Substitute the current expressions (Step-3) into the current equations (Step-4), which then become a set of simultaneous equations of unknown node voltages.

6. Solve for the unknown voltages and the branch currents.

We shall consider the circuit shown in figure 1.25 for nodal analysis.

In many circuits the reference node is most conveniently chosen as a common terminal or the ground terminal. The above circuit diagram consists of three nodes. It is possible to write (n - 1) equations for the circuit having 'n' nodes. Applying KCL at node 1 gives

Next consider node 2,

We get I₂ + I_R3 - I_R2 = 0

which in terms of node voltages and resistances

The generalised node equations can be written as

[G] [V] = [ I]  ...(43)

where the square matrix G is called the node conductance matrix, V is the column matrix of the node voltages with respect to the reference node and I is the column matrix of input currents.

Nodal equations by inspection method

The following steps are used in writing the node equations in matrix form.

Step 1: First, convert all the voltage sources to equivalent current sources.

Step 2: The conductances of all branches connected to node 1 are added and denoted by G₁₁ G₁₁ is called the self conductance of node 1.

Step 3: All the conductances connected to nodes 1 and 2 are added and denoted by G₁₂. G₁₂ is called mutual conductance of node 1 and 2. This G₁₂ is written with negative sign. If no conductance is connected between nodes 1 and 2, then G₁₂ = 0, G₁₂ = G₂₁.

Step 4: I₁ denotes the value of the current source current to node 1 and is written on the right hand side of the equation. The sign of I, is positive if it is flowing towards node I₁, otherwise it is negative. If no current source is connected to node 1, then

I₁ = 0.

For example, consider the circuit given below.

First convert voltage sources into equivalent current sources.

This circuit consists of two nodes 1 and 2 and common (ref) node.

At node 1,

By solving this matrix, we can find out nodal voltages V₁, and V₂.

 

EXAMPLE 100: Using nodal analysis, determine the current in the 20 Ω resistor.

Solution :

Convert all the voltage sources into their equivalent current sources.

 

EXAMPLE 101: Find V₁, V₂, V₃ by the nodal method for the given circuit.

Solution:

Matrix representation by inspection method

V1 = 66.66 V

 

EXAMPLE 102: Find the voltages of the nodes 1 and 2 in the network shown in figure by nodal method.

Solution:

Convert all the voltages sources into their equivalent current sources.

At node 1, the current is 5A. At node 2, the current is 10 - 19 A

Nodal equation in matrix form by inspection method

V₁ = 6.93 V

V₂ = 8.6 V

EXAMPLE 103: In the circuit of figure, find the node voltages V₁ and V₂.

Solution:

Using inspection method, the matrix form for given network is given by

 

EXAMPLE 104: Using nodal analysis, obtain the currents flowing in all the resistors of the circuit shown in figure.

Solution :

Convert all the voltage sources into their equivalent current sources.

 

EXAMPLE 105: Find the current through 15 2 resistor in the network shown in figure by nodal method.

Solution :

Convert all the voltage sources into their equivalent current sources.

 

EXAMPLE 106: Calculate the voltage across the 152 resistor in the network shown in figure below using nodal analysis.

Solution:

Convert all the voltage sources into their equivalent current sources.

 

EXAMPLE 107: Find the current through 2 2 resistor in the network using loop current analysis and node voltage analysis. (AU/CSE - Dec 2004)

Solution:

Loop current analysis

First, we should mark current direction in this network. Refer the figure shown below.

Let us assume three mesh currents, I₁, I₂ and I₃ as shown in figure.

Here, we have to solve the mesh current 12. By applying Cramer's rule, we can easily find out mesh current.

The negative sign indicates that the direction of I₂ is anticlockwise.

I₂ = 0.617 A

Nodal Analysis

Convert all the voltage sources into their equivalent current sources.

The negative sign indicates the current direction of 12 is anticlockwise

I₂ = - 0.618 A

 

EXAMPLE 108: For the circuit given find the node voltages V₁ and V2 using nodal analysis.

Solution :

Convert all the voltage sources into their equivalent current sources.

I₁ = 6/5 = 1.2 A

I₂ = 10/5 = 2 A

Then, the circuit becomes

At node 1, the current is 1.2 A

At node 2, the current is 2A

Node equations in matrix form by inspection method

 

EXAMPLE 109: For the circuit of figure, find the current in each branch by nodal method.

Solution :

Convert all the voltage sources into its equivalent current sources.

 

EXAMPLE 110: Determine the voltage at each node for the given circuit figure.

Solution :

Convert the voltage source into their equivalent current source

I₂ = 10 / 10 = 1 A

Now the circuit becomes

 

EXAMPLE 111: In the circuit shown in figure, what are the values of R₁ and R₂, when the current flowing through R₁ is 1A and R₂ is 5A? What is the value of R₂ when the current flowing through R₁ is zero?

Solution :

The current in the 5Ω resistance I₅ = I₁ + I₂ =  1+ 5 = 6A

Voltage across 5 Ω, V₅ = 5 × 6 = 30 V

The voltage at node A, V_A = 100 - 30= 70 V