Mesh analysis and superposition
Mesh analysis solves circuits using loop currents and Kirchhoff's voltage law. Superposition offers another route for linear circuits with several sources.
Meshes
A mesh is a loop in a planar circuit, one drawable without crossing wires, that contains no other loops inside it, like a window pane. Mesh analysis assigns a mesh current circulating around each mesh, usually clockwise. The actual current in a branch shared by two meshes is the difference of their mesh currents.
Writing mesh equations
For each mesh, apply KVL, expressing each resistor's voltage through Ohm's law using mesh currents. For a resistor shared with a neighbouring mesh, the voltage is R times the difference of the two mesh currents. A circuit with m meshes and only voltage sources and resistors gives m equations, often fewer than nodal analysis would.
Current sources and supermeshes
A current source on the outer edge of a mesh fixes that mesh current directly. A current source shared between two meshes creates a supermesh: write one KVL equation around the combined loop that avoids the source, and add a constraint that the difference of the two mesh currents equals the source current.
Choosing nodal or mesh
Use whichever gives fewer equations or simpler handling of the sources: nodal analysis suits circuits with many parallel branches and current sources, while mesh analysis suits circuits with many series elements and voltage sources. Nodal analysis works for non-planar circuits too; mesh analysis requires planar ones.
Linearity
A circuit made of resistors, linear dependent sources, and independent sources is linear: responses scale proportionally with sources, so doubling every source doubles every current and voltage. Linearity underlies superposition, Thevenin's theorem, and most circuit techniques, and it applies to capacitors and inductors too.
Superposition
In a linear circuit with several independent sources, any voltage or current equals the sum of the contributions from each source acting alone. To find one contribution, turn off the other independent sources: replace voltage sources with short circuits (0 V) and current sources with open circuits (0 A). Dependent sources stay in the circuit.
When to use superposition
Superposition gives insight, such as how much of a signal comes from each input, and simplifies circuits with sources of different kinds, for example a DC bias plus an AC signal. It does not apply directly to power, which depends on the square of current: powers from separate sources cannot simply be added.
Worked example
Two meshes share a 2 \(\Omega\) resistor. Mesh 1 has a 10 V source and a 3 \(\Omega\) resistor; mesh 2 has a 5 \(\Omega\) resistor and a 4 V source opposing the clockwise direction. KVL mesh 1: \(10 = 3i_1 + 2(i_1 - i_2)\), so \(5i_1 - 2i_2 = 10\). Mesh 2: \(-4 = 5i_2 + 2(i_2 - i_1)\), so \(-2i_1 + 7i_2 = -4\). Solving: \(i_1 = \frac{62}{31} = 2\) A and \(i_2 = 0\) A, so the shared resistor carries 2 A.
Key takeaways
Mesh analysis assigns circulating currents to the meshes of a planar circuit and writes KVL for each; shared branches carry the difference of mesh currents. Edge current sources fix mesh currents; shared current sources form supermeshes with a constraint. Choose nodal or mesh analysis by equation count and source types; mesh needs planar circuits. Linear circuits scale proportionally with their sources. Superposition sums each independent source's contribution, with others turned off: voltage sources shorted, current sources opened, dependent sources kept. Superposition does not apply directly to power.
Flashcard
Loop containing no other loops
Check your understanding
When applying superposition, how is an independent voltage source turned off?