Nodal Analysis

What is Nodal Analysis?

Nodal analysis provides a general procedure for analyzing circuits using node voltages as the circuit variables.

In electric circuits analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

By using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). 

For instance, for a resistor, I_branch = V_branch * G, where G (=1/R) is the admittance (conductance) of the resistor.

 Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer.

While simple examples of nodal analysis focus on linear elements, more complex nonlinear networks can also be solved with nodal analysis by using Newton's method to turn the nonlinear problem into a sequence of linear problems.

In nodal analysis, we are about to find the node voltages. Given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following steps:

1. Select a node as the reference node. Assign voltages to the remaining nodes. The voltages are referenced with respect to the reference node. 
2. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
3. Solve the resulting simultaneous equations to obtain the unknown node voltages.
    
   Current flows from a HIGHER POTENTIAL to a LOWER POTENTIAL in a resistor.

i = vhigher - vlower / R