The Node Method

In the next update to my book, AC/DC Principles and Applications, I plan to add a small section on the Node Method to Chapter Nine – Complex Network Analysis Techniques. Discussed in this chapter are Kirchoff’s voltage and current laws, superposition, Thevenin’s and Norton’s theorems. This post is an advance installment of the update.

The Node Method is a circuit analysis approach that uses the circuit element properties with Kirchoff’s voltage and current laws. This method helps to reduce analysis complexity as seen in the example.

Each element connection point is a circuit node. In using the Node Method, each node voltage is label with one node being selected as ground or 0 volts. Each node also has the current labeled showing the currents entering and leaving each node.

After all the node voltages and currents are labeled, KCL is written for each node. Based on Ohm’s Law, the current through a resistor is the voltage across the resistor divided by the resistance. The voltage across a resistor is the voltage difference between the terminals.

Using the set of KCL equations, solve for the missing information.

Node Method Steps

1. Select a reference node to be 0 volts (ground). A good choice for this reference node is one that has the most connections.
2. Label voltages and currents.
3. Write KCL equations for each node that has an unknown voltage.
4. Solve for missing values using the equations developed in step 3.

Example

Using Figure 9-3, what is the voltage drop for a 2Ω load resistor that is connected between a 12V closed-loop circuit (loop A) with a resistance of 4Ω and a 6V closed-loop circuit (loop B) with a resistance of 3Ω?

Write KCL
I_L = I₁ + I₂
e/2Ω = (12V – e)/4Ω + (6V – e)/3Ω

Solve for node voltage e
6e/12Ω = 3(12V – e)/12Ω + 4(6V – e)/12Ω
6e/12Ω = (60V – 7e)/12Ω
13e/12Ω = 60V/12Ω
e = 60V/13
e = 4.6V
I_L = 4.6V/2Ω
I_L = 2.3A

This solution was simpler than creating the KVL for each loop, solving for I₁, I₂, and then I_L, and then finding the node voltage e.