Nodal and mesh analysis

Series-parallel reduction works beautifully on simple circuits, but it soon runs out. A circuit with several sources, a bridge, or dependent sources cannot be reduced step by step, and writing KCL and KVL equations at random produces too many equations, some of them redundant. Engineers need a method that works on every circuit, produces exactly the right number of independent equations, and can be carried out by a computer.

Two such methods exist. Nodal analysis uses node voltages as the unknowns and writes KCL at each node. Mesh analysis uses mesh currents as the unknowns and writes KVL around each mesh. Both turn any linear circuit into a set of simultaneous linear equations, which is exactly what circuit simulators such as SPICE do internally, using an extension of nodal analysis. This chapter develops both methods, their special cases (supernodes and supermeshes) and their use with dependent sources, and shows how to solve the resulting equations with matrices.

Nodal analysis

Choose one node as the reference node (also called ground or datum), usually the one with the most branches connected, or the bottom of the diagram. Its voltage is defined as zero. Every other node gets a node voltage: the voltage of that node relative to the reference. In a circuit with \(n\) nodes there are \(n - 1\) node voltages.

Node voltages make KVL automatic. The voltage across any element connecting nodes \(j\) and \(k\) is simply \(v_j - v_k\), so the voltages around any loop sum to zero by construction. Only KCL remains to be written, once at each non-reference node. The procedure is:

• Pick the reference node and label the node voltages \(v_1, v_2, \dots, v_{n-1}\).
• At each non-reference node, write KCL, expressing every resistor current in terms of node voltages: the current leaving node \(j\) through a resistor \(R\) to node \(k\) is \((v_j - v_k)/R\).
• Solve the \(n - 1\) equations. Then find any current or power from the node voltages.

Writing the equations by inspection

For circuits with only resistors and independent current sources, the nodal equations have a pattern that can be written down directly in matrix form, \(\mathbf{G}\mathbf{v} = \mathbf{i}\):

• The diagonal entry \(G_{jj}\) is the sum of the conductances connected to node \(j\).
• The off-diagonal entry \(G_{jk}\) is minus the sum of the conductances connecting nodes \(j\) and \(k\).
• The right-hand entry \(i_j\) is the total source current entering node \(j\).

For the example above:

The conductance matrix is symmetric, which is a quick check on hand-written equations. Large circuits are solved this way by computer. The program below uses NumPy (introduced in AI-110) to find the resistance of the unbalanced bridge from Lesson 2 by injecting a 1 A test current and solving for the node voltages.

import numpy as np

# Bridge from Lesson 2: nodes a, b, c (d is the reference). Inject a 1 A test current at a.
G = lambda r: 1 / r
A = np.array([[G(30) + G(90), -G(30),                -G(90)],
              [-G(30),         G(30) + G(60) + G(20), -G(60)],
              [-G(90),        -G(60),                 G(90) + G(60) + G(40)]])
i = np.array([1.0, 0.0, 0.0])
v = np.linalg.solve(A, i)
print("node voltages (V):", v.round(3))
print("R_ad =", round(v[0] / 1.0, 3), "ohms")

node voltages (V): [36. 14. 12.]
R_ad = 36.0 ohms

The result, 36 Ω, matches the wye-delta calculation of Lesson 2. The test-source idea used here (apply 1 A, measure the voltage) returns in Lesson 6 for finding Thévenin resistances.

Voltage sources in nodal analysis

A voltage source connected between a node and the reference simply fixes that node's voltage: one unknown fewer, and no KCL equation is needed at that node. A voltage source between two non-reference nodes is harder, because its current is unknown and cannot be written from Ohm's law. The solution is a supernode: enclose the voltage source and its two nodes in a boundary, and write KCL for the whole boundary (by the generalized KCL of Lesson 2). The source supplies the second equation, a constraint between the two node voltages.

Dependent sources in nodal analysis

Dependent sources are handled like independent ones, with one extra step: express the controlling variable in terms of the node voltages and substitute.

Mesh analysis

Mesh analysis is the dual of nodal analysis. It applies to planar circuits (those that can be drawn without crossing wires). Assign a mesh current to each mesh, conventionally circulating clockwise. A mesh current is a fictitious loop current; the actual current in a branch is the mesh current if the branch belongs to only one mesh, or the difference of two mesh currents if it is shared between meshes. Mesh currents satisfy KCL automatically, since each one enters and leaves every node it passes through. Only KVL remains, written around each mesh.

The procedure:

• Assign clockwise mesh currents \(i_1, i_2, \dots\) to the \(b - n + 1\) meshes.
• Around each mesh, write KVL. For a resistor shared by meshes \(j\) and \(k\), the voltage drop in the direction of \(i_j\) is \(R(i_j - i_k)\).
• Solve, then find branch currents as mesh currents or their differences.

As with nodal analysis, a resistive circuit with only independent voltage sources gives a symmetric matrix by inspection: each diagonal entry is the total resistance around the mesh, each off-diagonal entry is minus the resistance shared by the two meshes, and the right-hand side is the net source voltage rise around each mesh.

Current sources in mesh analysis

A current source on the outer edge of a circuit, belonging to one mesh only, fixes that mesh current: one unknown fewer. A current source shared between two meshes fixes the difference of their mesh currents, but its voltage is unknown, so KVL cannot be written around either mesh directly. The answer is a supermesh: write KVL around the larger loop formed by the two meshes with the shared current source removed, and use the source as the constraint equation.

Choosing a method

Both methods work on any planar circuit; nodal analysis also works on non-planar circuits. The usual rule is to choose the method with fewer equations, after accounting for sources that fix unknowns (Lesson 5 shows how computers solve either form as a matrix equation):

• Nodal analysis needs \(n - 1\) equations, minus one for each voltage source (each removes an unknown, directly or through a supernode).
• Mesh analysis needs \(b - n + 1\) equations, minus one for each current source.

Circuits with many parallel branches and current sources favor nodal analysis; circuits with many series elements and voltage sources favor mesh analysis. Nodal analysis also fits the way voltages are measured in practice (relative to a common ground), and it is the basis of computer simulation, using a formulation called modified nodal analysis that adds the currents of voltage sources as extra unknowns.

Checking answers

Three habits catch nearly every error in nodal and mesh analysis. First, check that the conductance or resistance matrix is symmetric when the circuit has no dependent sources. Second, substitute the solution back into an equation you did not use, for example KCL at the reference node or KVL around an outer loop. Third, check the power balance. Each takes a minute and saves hours of chasing a sign error later.