4. Nodal Analysis: Systematic KCL // Network Analysis // bhaswanth

The dual of mesh analysis. Pick nodes, write KCL at each, solve.

4.1 The procedure

Identify all the nodes in the circuit. Pick one as the reference node (ground, voltage = 0).
Assign a node voltage variable to each remaining node ( $V_a$ , $V_b$ , ...).
Write KCL at each node, expressing branch currents in terms of node voltages and component values.
You have $n - 1$ equations in $n - 1$ unknowns. Solve.
Recover branch currents from node voltages using Ohm's law.

4.2 Worked example: three-node circuit

Consider this circuit:

plaintext

       Va     Vb
        ●─[R1]─●─[R2]─ GND
        │       │
        │      [R3]
       (Vs)     │
        │       GND
       GND

Let $V_s = 10$ V, $R_1 = 1$ kΩ, $R_2 = R_3 = 2$ kΩ. Assume $V_s$ acts at node $V_a$ , say there is an ideal voltage source from GND to $V_a$ that fixes $V_a = 10$ V.

KCL at node $V_b$ : currents flowing out of $V_b$ sum to zero. Through $R_1$ to $V_a$ : $(V_b - V_a)/R_1$ . Through $R_2$ to ground: $V_b/R_2$ . Through $R_3$ to ground: $V_b/R_3$ . Sum = 0:

$\frac{V_b - V_a}{R_1} + \frac{V_b}{R_2} + \frac{V_b}{R_3} = 0$

$\frac{V_b - 10}{1000} + \frac{V_b}{2000} + \frac{V_b}{2000} = 0$

Multiply by 2000:

$2(V_b - 10) + V_b + V_b = 0 \implies 4V_b = 20 \implies V_b = 5 \text{ V}$

That is the node voltage. From here, every branch current follows by Ohm's law: $I_{R_1} = (10-5)/1000 = 5$ mA, etc.

4.3 Why nodal scales better than mesh

In a circuit with $n$ nodes and $b$ branches, mesh requires $b - n + 1$ equations; nodal requires $n - 1$ . For typical electronics circuits ( $b$ much larger than $n$ ), nodal has fewer equations. Plus, nodal generalizes naturally to non-planar circuits and to circuits with op-amps (where node voltages are the natural variables).

Modern circuit simulators (SPICE and its derivatives) all use modified nodal analysis (MNA), a refinement that handles voltage sources elegantly. When you press "simulate" in any EDA tool, MNA is what runs under the hood. Solve a few thousand simultaneous equations representing every node in your schematic, and out pops the answer. The matrix is sparse (most entries are zero, because most pairs of nodes share no component) and SPICE exploits sparsity to handle million-node netlists in real time.

4.4 Supernodes

When an ideal voltage source connects two non-ground nodes, neither node's voltage is independent (they are forced to differ by the source voltage). The trick: treat both nodes as a single "supernode" for KCL purposes, then add the constraint equation $V_a - V_b = V_{src}$ to fix the relationship.

For example, if a 5 V source connects node $V_a$ to node $V_b$ (with + on $V_a$ ), and the rest of the circuit has a few resistors hanging off both nodes, write KCL summing all currents leaving the combined supernode (i.e., ignoring the source between the two), then add $V_a - V_b = 5$ as a separate equation. The system has the same number of unknowns and equations as before; you have just shuffled where the constraint lives.