Node voltages as the unknowns

Nodal analysis picks the smallest useful set of unknowns and lets KCL generate the equations.

• Choose one node as the reference (ground). Its potential is defined as zero, and every other voltage is measured against it.
• Label the remaining n-1 nodes V1, V2, …. Those are your unknowns, not the branch currents.
• Ground is a choice, not a property of the circuit. Picking the node with the most connections usually gives the simplest algebra.

Writing and solving the equations

Every non-reference node gives you one equation, and Ohm's law turns each branch into a term.

• At each node write KCL: the algebraic sum of currents leaving is zero.
• Express each branch current from the node voltages: a resistor between V1 and V2 carries (V1-V2)/R leaving node 1.
• Solve the resulting simultaneous system by elimination for two unknowns, or by Cramer's rule when you have three or more.

Voltage sources: the two cases

A voltage source has no v = iR relation, so it needs special handling.

• Case 1, the source sits between the reference node and one other node: that node voltage is simply known, and the equation for it disappears.
• Case 2, the source sits between two non-reference nodes: enclose both in a supernode, apply KCL to the whole enclosure, and get the missing equation from KVL across the source.
• A supernode has no voltage of its own. It needs both KCL and KVL, which is exactly why it supplies one constraint equation.