Nodal and Mesh Analysis
Both methods are the same move, and the choice between them is a count rather than a preference. Every difficulty in one has an exact dual in the other.
Skip to the animationNodal and mesh analysis are the same move applied to dual quantities: choose node voltages or mesh currents as unknowns, write one Kirchhoff law per unknown, and the other law is satisfied automatically — with the choice between them decided by which count is smaller.
Why a method is needed
Kirchhoff's laws hold at every node and around every loop, generating far more equations than unknowns — most of them linear combinations of the others. A method is a recipe for writing exactly enough independent equations and no more.
The two methods
| Nodal | Mesh | |
|---|---|---|
| Unknowns | Node voltages | Mesh currents |
| Law written | KCL at each node | KVL around each mesh |
| Law satisfied free | KVL — node voltages are single-valued | KCL — a loop current enters and leaves every node |
| Equation count | N − 1 | M |
| Reference | A chosen ground node | A chosen circulation direction |
| Needs planar circuit | No | Yes |
Count N−1 against M and take the smaller — that is the whole selection rule. A ladder network has few nodes; a bridge has few meshes.
Awkward sources
A voltage source between two nodes has an unknown current, so KCL cannot be written at either end alone. Enclose both in a supernode, write KCL around the pair, and add the source's voltage as the second equation. The count is preserved.
A current source shared by two meshes has an unknown voltage, breaking KVL the same way. Take a supermesh loop around both, avoiding the source, and add the source's current as the constraint. The exact dual.
Duality runs through the whole subject — voltage against current, node against mesh, series against parallel. It is a structural fact rather than a mnemonic, which is why learning one method properly nearly gives you the other.
The matrix form
Both methods produce a linear system. For a resistive network the matrix is symmetric and can be written by inspection: diagonal entries are the sum of conductances at a node, off-diagonal entries the negative of the conductance between a pair.
Requiring no insight is exactly what makes it programmable. SPICE has been a nodal solver since 1973, and every simulator descended from it still is — partly because nodal analysis does not require the circuit to be planar.
The assumptions
- Lumped elements — fails when the wavelength approaches the circuit's size, at which point a wire becomes a transmission line and KVL around a loop stops being well defined.
- Linear elements — diodes and transistors need linearisation and iteration, which is what SPICE does internally at each operating point.
- No coupling outside the wires — stray capacitance and mutual inductance are invisible to the model unless drawn in explicitly.
The methods are exact within their model. Knowing where the model stops — which is roughly where RF design begins and S-parameters take over — is the real skill.
The numbers you will be asked for
- Nodal equation count
N − 1
N nodes including the reference
- Mesh equation count
M = B − N + 1
B branches, N nodes
- Nodal matrix form
[G]·[V] = [I]
- Mesh matrix form
[R]·[I] = [V]
- Conductance matrix, by inspection
G_ii = Σ conductances at node i · G_ij = −G between i and j
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.