Nodal analysis

Nodal analysis with current source

Nodal analysis is the method to determine voltage or current using nodes of the circuit. In node analysis we choose node voltage instead of element voltages and hence the equations reduces in this process. We have to consider voltage source is not in this circuit. We have to solve a circuit with n nodes without voltage sources. To solve a circuit using nodal analysis method you must have good knowledge about node branch loop in a circuit. If you have no clear idea read the article then come back here other wise what is nodal analysis it will be difficult for you. There are three steps to solve a circuit using nodal analysis

1. Select a node as a reference node. Give names v₁, v_2,…. v_n-1 to remaining n-1 nodes. All the voltages are the referenced voltages respecting to the reference node.
2. Apply KCL to each non reference node. To express the branch currents in terms of node voltages use ohm’s law.
3. Solve the equations to get unknown node voltages.

First step is to select a reference node. It is also called datum node. The reference node commonly called the ground. It has zero potential.

In circuit the reference node is denoted by any of the three symbols in figure 1. Figure 1 (c) is called a chassis ground because it is used in the case chassis act, enclosure as a reference point in the circuit. Figure 1 (a) and (b) are used when the potential of the earth taken as reference. I use symbol (b).

Consider the figure 2 circuit. We will apply nodal analysis in this circuit. See there is node o is reference node. Node 1 and node 2 are non-reference nodes. v₁ and v₂ are assigned voltages of node 1 and node 2. One thing keep it mind that assigned voltages are defined with the respect to the reference node. In figure 2 each node voltage is the voltage rise from the reference node to the corresponding non reference node or simply the voltage of that node with respect to the reference node.

We have redrawn figure 2 circuit to avoid so much information in one circuit. We get figure 3 after redrawing

I₁, I_2, I₃ currents are flowing through R₁, R₂ and R₃ resistors.

Now second step is to apply KCL.

At node 1 applying KCL,

                                           I = I₁ + I₂ + I₄    ———————- (i)

At node 2,

                                         I₂ + I₄  = I₃        ———————— (ii)

To determine I₁, I_2, I₃ currents we have to apply ohm’s law. We know that current flows from a higher potential to a lower potential in a resistor.

In figure 3 applying this method we get,

Substituting the values of I₁, I_2, I₃ in equation (i) & (ii),

Putting the values of given elements in circuit from equation (iii) and (iv) we will get node voltages.

 Let’s solve nodal analysis problem given below circuit.

In figure 4 circuit has three resistors and two current source. Each element has value. We will find node voltages applying nodal analysis.

First step select a node as reference node. Here we get two nodes as non-reference nodes that are node 1 and node 2 and reference node is o.

We have to find node voltages v₁ and v₂.

Applying KCL and Ohm’s law at node 1 we get,

From node 2 we get,

Using the elimination method from equation (a) and (b),

If you want to learn how to find node voltages with voltage source click here read the article.