Nodal-Voltage Analysis & Supernodes

Nodal analysis makes node potential the primary unknown. Every resistor current becomes a voltage difference times conductance, so KCL produces one equation per non-reference node. The method scales well because each equation touches only neighboring nodes.

Conductance stamping #

For a resistor $R_{ab}$ between nodes $a$ and $b$ with conductance $G_{ab}=1/R_{ab}$, current from $a$ to $b$ is

$$I_{ab}=G_{ab}(V_a-V_b).$$

KCL at node $a$ collects all incident branches:

$$\left(\sum_k G_{ak}\right)V_a-\sum_k G_{ak}V_k=I_a,$$

where $I_a$ is the net independent current injected into the node. Stamping all nodes gives

$$GV=I.$$

Diagonal entries are sums of connected conductances; off-diagonal entries are negative mutual conductances.

Voltage sources and supernodes #

A voltage source between a node and reference sets that node voltage directly. A voltage source between two unknown nodes does not give its current, so enclose both nodes and the source in a supernode:

1. write one KCL equation for the supernode boundary;
2. add the source constraint $V_a-V_b=V_s$.

Dependent voltage sources are handled the same way, with their controlling relation added to the system.

Choosing the reference #

Any node may be ground, but a good choice simplifies source constraints and reduces awkward supernodes. The physical branch voltages do not depend on the choice.

Verification #

After solving, compute every branch current from the original element law, not from rounded intermediate values. Check KCL at the omitted reference node and calculate a power balance. These checks catch the two dominant errors: a source injection with the wrong sign and a conductance stamped between the wrong pair of nodes.