Nodal Analysis

Nodal Analysis is a systematic method for performing circuit analysis.

Nodal analysis is a formalized procedure based on KCL equations.

Steps:

1. Identify all nodes.
2. Choose a reference node. Identify it with reference (ground) symbol. A good choice is the node with the most branches, or a node which can immediately give you another node voltage (e.g., below a voltage source).
3. Assign voltage variables to the other nodes (these are node voltages.)
4. Write a KCL equation for each node (sum the currents leaving the node and set equal to zero). Rearrange these equations into the form A*V1+B*V2=C (or similar for equations with more voltage variables.)
5. Solve the system of equations from step 4. There are a number of techniques that can be used: simple substitution, Cramer's rule, the adjoint matrix method, etc.

Example

Given the Circuit below, find the voltages at all nodes.

node 0:  (defined as ground node)
node 1:  (free node voltage)
node 2: 

node 3: 

which results in the following system of linear equations: 

therefore, the solution is: 

Another solution with KCL would be to solve node in terms of node 2;