Mesh Current Analysis Calculator: Three-Loop Circuits by Kirchhoff's Voltage Law

Solve multi-loop circuits with the mesh current method.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Draw a clockwise current in each loop. Enter the voltage source in each loop, positive if it drives that current.

Enter the resistance that is only in each loop and the resistors that are shared between two loops.

The calculator writes the three mesh equations, solves them, and gives the mesh currents, the shared resistor currents and the power balance.

Formulas used

  • [R][I] = [E]
  • R_{kk} = \text{sum of resistors in mesh }k
  • R_{jk} = -\text{shared resistance}

Worked example

With the default numbers in the calculator above:

  1. Mesh equations (Kirchhoff's voltage law round each loop): the loop resistance on the diagonal, minus the shared resistances off it: Mesh 1: 3 I1 - 1 I2 = 12; Mesh 2: -1 I1 + 6 I2 - 2 I3 = 0; Mesh 3: -2 I2 + 6 I3 = -6
  2. Mesh currents (solved by Gaussian elimination): I1 = 4.13 A, I2 = 0.40 A, I3 = -0.87 A (clockwise positive)
  3. Current and voltage in the shared resistor R12: 3.73 A, 3.73 V across it
  4. Current and voltage in the shared resistor R23: 1.27 A, 2.53 V across it
  5. Power supplied by the sources and power lost in the resistors: 54.80 W supplied and 54.80 W in the resistors (equal)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What is mesh analysis?

A method that applies Kirchhoff's voltage law round each loop using loop currents, which gives one equation for each loop.

How do I find the current in a shared resistor?

It is the difference of the two mesh currents that pass through it.

What does a negative mesh current mean?

That the current really flows anticlockwise.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year electrical circuits text (mesh and nodal analysis).

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