Kirchhoff Voltage Loop Solver

Sum of voltages around a closed loop = 0. Solve a 2-loop circuit with two batteries and three resistors.

Inputs

Result

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How to use this calculator

  • Enter the two battery voltages and three resistor values.
  • Read both loop currents and the current through the shared middle resistor.

About this calculator

Kirchhoff's voltage law (KVL): the sum of voltage drops and rises around any closed loop is zero. For a circuit with two loops sharing a middle resistor, this gives two equations with two unknowns (the loop currents), solvable by determinant or substitution. The current through the shared resistor is the difference of the two loop currents.

Frequently asked

What is a "loop current"?+
An imagined current circulating around a single loop. Real branch currents are sums or differences of loop currents (the shared resistor between two loops sees I₁ − I₂).
Why sign conventions matter?+
KVL accounts for direction: voltage rises across batteries (going from − to +) are positive; voltage drops across resistors (in the direction of current) are also positive. Get them consistent.
Can I have more than 2 loops?+
Yes — the math scales as N×N matrices for N loops. Beyond ~5 loops, use computer algebra.
Is loop analysis the same as nodal analysis?+
They're duals: loop analysis uses KVL and currents as unknowns; nodal uses KCL and voltages. Equivalent answers; different convenience for different circuits.
What if a battery has internal resistance?+
Add it in series with that battery and treat as part of the loop's resistance.

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