Mesh Analysis Solver

Physics Electricity and Magnetism • Current and Circuits

Written by STEM Calculators Team Published January 14, 2026 Updated February 24, 2026

10. Mesh Analysis Solver

Solve multi-loop circuits using mesh currents (KVL in each loop). This tool builds the mesh resistance matrix $\mathbf{Z}$ and solves $\mathbf{Z}\mathbf{I}=\mathbf{V}$ for the mesh currents $I_1,I_2,\dots$ .

Inputs support: pi, e, sqrt(), sin, cos, tan, exp, log/ln, abs, pow(a,b), and ^. Use * for multiplication.

Model + Inputs

Circuit template: Schematic preview: Show branch results:

Two meshes: $(R_1+R_s)I_1 - R_s I_2 = V_1$, $-R_s I_1 + (R_2+R_s)I_2 = V_2$

Source $V_1$ (V):

Clockwise mesh sign convention.

Source $V_2$ (V):

Positive = rise along loop direction.

Exclusive $R_1$ (Ω):

Left loop total resistance (exclusive).

Exclusive $R_2$ (Ω):

Right loop total resistance (exclusive).

Shared $R_s$ (Ω):

Shared current is $I_1-I_2$.

Three meshes (ladder): shared $R_{12}$ and $R_{23}$

Source $V_1$ (V):

Clockwise meshes.

Source $V_2$ (V):

Use 0 if none.

Source $V_3$ (V):

Positive = rise along loop.

Exclusive $R_1$ (Ω):

Loop 1 only.

Exclusive $R_2$ (Ω):

Loop 2 only.

Exclusive $R_3$ (Ω):

Loop 3 only.

Shared $R_{12}$ (Ω):

Between meshes 1 and 2.

Shared $R_{23}$ (Ω):

Between meshes 2 and 3.

Pattern: $Z_{ii}=$ sum in mesh i, $Z_{ij}=-$(shared between i and j).

Custom system: enter $\mathbf{Z}$ (Ω) and $\mathbf{V}$ (V)

Number of meshes $n$: Auto-symmetrize $Z_{ij}=Z_{ji}$:

Tip: For mesh resistance matrices, diagonals are usually positive and off-diagonals are usually non-positive.

Ready

Steps

Choose a template, enter values, and click Solve.

Schematic (loop-current arrows)

Clean, scalable diagram

Clockwise mesh currents are assumed by default.

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Mesh Analysis Solver — Theory

Mesh analysis is a systematic way to solve planar circuits by assigning one current to each loop (mesh) and writing Kirchhoff’s Voltage Law (KVL) around every mesh. The result is a linear system whose unknowns are the mesh currents $I_1, I_2, \dots, I_n$.

1) Core idea

Choose a consistent direction (usually clockwise) for each mesh current. Then write KVL around mesh $i$: the algebraic sum of voltage drops equals the algebraic sum of source rises in that loop.

\[ \mathbf{Z}\mathbf{I}=\mathbf{V} \] \[ \mathbf{I}= \begin{bmatrix} I_1\\ I_2\\ \vdots\\ I_n \end{bmatrix}, \qquad \mathbf{V}= \begin{bmatrix} V_1\\ V_2\\ \vdots\\ V_n \end{bmatrix} \]

2) How to build the mesh matrix $ \mathbf{Z} $

Diagonal entries $Z_{ii}$: sum of all resistances that belong to mesh $i$.
Off-diagonal entries $Z_{ij}$ (for $i\neq j$): negative of the total resistance shared by meshes $i$ and $j$. (Shared branches depend on current differences such as $R(I_i-I_j)$.)
Source vector $ \mathbf{V} $: net source voltage in the direction of the mesh current (sign depends on how you traverse the sources).

3) Two-mesh example (shared resistor)

Consider two meshes with exclusive resistors $R_1$ and $R_2$, and one shared resistor $R_s$. With clockwise currents $I_1$ and $I_2$, the shared resistor drop is $R_s(I_1-I_2)$ in mesh 1 and $R_s(I_2-I_1)$ in mesh 2.

\[ (R_1+R_s)I_1 - R_s I_2 = V_1 \] \[ -R_s I_1 + (R_2+R_s)I_2 = V_2 \] \[ \begin{bmatrix} R_1+R_s & -R_s\\ -R_s & R_2+R_s \end{bmatrix} \begin{bmatrix} I_1\\ I_2 \end{bmatrix} = \begin{bmatrix} V_1\\ V_2 \end{bmatrix} \]

Sample values: $V_1=10\,\mathrm{V}$, $V_2=5\,\mathrm{V}$, $R_1=4\,\Omega$, $R_2=6\,\Omega$, $R_s=2\,\Omega$.

\[ \begin{bmatrix} 6 & -2\\ -2 & 8 \end{bmatrix} \begin{bmatrix} I_1\\ I_2 \end{bmatrix} = \begin{bmatrix} 10\\ 5 \end{bmatrix} \] \[ I_1=\frac{45}{22}\approx 2.045\,\mathrm{A}, \qquad I_2=\frac{25}{22}\approx 1.136\,\mathrm{A} \] \[ I_{\text{shared}} = I_1-I_2 \approx 0.909\,\mathrm{A} \]

4) What changes at higher difficulty?

3+ meshes: same matrix idea, just larger $n\times n$ systems.
Current sources: you often use a supermesh or transform sources (Norton/Thévenin) to keep KVL equations consistent.
Dependent sources (university level): include the control relationship as an additional equation, or substitute it into the mesh equations.

5) Mesh vs node analysis

Mesh analysis is usually best for planar circuits with fewer meshes than nodes. Node analysis is often best when there are many loops but fewer essential nodes. Knowing both lets you pick the fastest method for a given circuit.

Website tip: show loop arrows (mesh currents) on the diagram, and compare with a “node vs mesh” mini-note to reinforce the difference.

Steps

Schematic (loop-current arrows)

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