Learning Objectives of Nodal Analysis

a. Apply Ohm’s law using nodal voltages. 

b. Apply the nodal analysis method to determine multiple unknown node voltages and branch currents in a simple dc circuit.

What is 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.

 Nodal analysis is based on a systematic application of KCL. The technique allows us to obtain a set of simultaneous equations to solve for voltages and current in any linear circuit. It is a powerful technique but it is just a multi-step procedure. It is not a formula (and thus it’s not on the formula sheet) and it is somewhat more difficult to grasp than the subjects studied up to this point until one has progressed through a number of problems.  

Steps for Nodal Analysis

Step 1: Select a reference node. 

Step 2: Assign voltages Va, Vb, Vc to the remaining nodes, identifying any known voltages. 

Step 3: Assume a direction for the current passing through each resistor adjacent to a node with an unknown voltage and express the branch currents in terms of node voltages. 

Step 4: Apply KCL to each node. 

Step 5: Solve the resulting simultaneous equations to obtain the unknown node voltages. 

Let us consider the circuit below:  

We first select a reference node and assign labels to the remaining nodes. The circuit above has four nodes. Now select the reference node (ground) to be the bottom node (Step 1 complete), and  label the remaining three nodes as a, b, and c, as shown below:

Now, each of the labelled nodes, a, b, and c will have a voltage associated with it:  Va ,Vb,Vc Now we are not quite done with Step 2; we have to identify any known voltages. Look at the figure above: Are any of the quantities Va , Vb ,Vc known? Va= 6 V,  Vb=12 V

Step 2 completed. Next, for Step 3, we assume a direction for the current passing through each resistor adjacent to a node with an unknown voltage (node b). Stated another way, we are going to write the branch currents. We have chosen to label the currents I1, I2,  I3 in the direction shown below. 

But we’re not done with Step 3: we now have to express each of the branch currents in terms of node voltages. This is done using Ohm’s law, but we have to be careful about the polarities. The way we have selected the direction of  I1 assumes that the voltage drop across the 4Ω resistor has its positive polarity at node b. Stated another way,  assuming that Vb is greater than Va . So, expressing the currents as I1 , I2,  I3 in terms of the node voltages, we have: 

Step 3 completed. Next, we apply KCL to node b.  

I1 + I2+  I3 =0

This is one equation in one unknown. You can solve this with some simple algebra, and we find out, Vb and any other quantity in the circuit like    I1, I2,  I3.