Kirchhoff Two-Loop Circuit Solver
Solve the classic two-loop circuit — three branches between two nodes, each a resistor with an optional source — for node voltage, currents, powers and KCL/KVL.
Node voltage, the current in each branch with direction, the drop and power in every resistor, each source’s delivered or absorbed power, and Kirchhoff current and voltage law sums.
Example: 12 V with 2 Ω, 6 V with 4 Ω and a 6 Ω load between the same nodes: node 8.18 V; 1.909 A from the 12 V branch, 0.545 A into the 6 V source (charging), 1.364 A through the 6 Ω; 19.6 W in and out.
One node voltage
solves both loops.
The circuit, the node-voltage shortcut, and how the two laws are checked.
The circuit
Three branches join two nodes, A on top and B below. Each branch is a resistor in series with an optional voltage source whose + terminal points toward A (enter a negative voltage to reverse it). This covers the classic two-loop textbook problem: two sources feeding a shared load, a battery charging another, or a bridge of three resistors.
Solving it
Taking B as the reference, Kirchhoff’s current law at A gives one equation: Σ (V_k − V_A)/R_k = 0, so V_A = Σ(V_k/R_k) ÷ Σ(1/R_k). Each branch current toward A is then (V_k − V_A)/R_k. This is the node-voltage method; it reaches the same answer as writing two loop (mesh) equations, in one line.
Checks and signs
The page sums the currents at A (KCL, should be zero) and walks loops 1–2 and 2–3 (KVL, should be zero), and confirms the power the sources deliver equals what the resistors dissipate. A negative current flows from A down to B; a source whose current enters its + terminal absorbs power — it is being charged. Ideal elements: add internal resistance to a branch resistor if you need it. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Node-voltage method: V_A = Σ(V_k/R_k) ÷ Σ(1/R_k), I_k = (V_k − V_A)/R_k; Kirchhoff’s current and voltage laws verified numerically; ideal elements
Last reviewed 21 September 2026. How results are checked: How we verify.