This Kirchhoff calculator solves the classic three-branch parallel circuit — two source branches and a load branch between the same two nodes — using Kirchhoff’s current law at the node and checking every loop with the voltage law. KCL and KVL are the two conservation rules all circuit analysis is built from; this is them doing real work on a circuit you can build on a bench.

Node voltage (top rail vs bottom rail) –
Branch 1 current (through V1 + R1) –
Middle branch current (through R3) –
Branch 2 current (through V2 + R2) –
KCL check at the top node –

Circuit: three branches in parallel between a top and bottom rail — [V1 in series with R1], [R3 alone], [V2 in series with R2], both sources’ + terminals toward the top rail. Currents are signed: positive flows downward through R3’s branch convention — read each branch’s direction note in the worked example; a negative branch current simply means that source is being charged by the other.

Two Laws, Both Conservation Laws in Disguise

Kirchhoff’s current law (KCL): the currents entering any node sum to zero — charge doesn’t accumulate in a junction, so what flows in flows out. Kirchhoff’s voltage law (KVL): the voltages around any closed loop sum to zero — energy per charge gained from sources equals energy dropped across components, so a trip around the loop returns you to the same potential. Everything else in circuit analysis — nodal analysis, mesh analysis, Thévenin, superposition — is bookkeeping built on these two rules. This calculator uses the cleanest instance: with only two nodes, KCL at the top node gives one equation in one unknown (the node voltage VT), and every branch current then falls out of Ohm’s law applied branch by branch. The formula VT = (V1/R1 + V2/R2)/(1/R1 + 1/R2 + 1/R3) is Millman’s theorem — KCL already solved and packaged.

The KVL story is verified by construction: each branch independently satisfies VT = (source) ± (I×R drop), so any loop you trace — say, around branches 1 and 3 — sums to zero identically: rise V1, drop I1R1, drop I3R3 going down the middle, back to start. When your own hand analysis of a circuit fails a KVL loop check, the error is never in Kirchhoff; it’s in a sign. Which brings up the calculator’s most educational output: the signed currents. A negative branch current isn’t a mistake — it’s the circuit telling you a source you expected to supply power is actually absorbing it.

Worked Example (Verified)

  • 12 V battery (R1 = 2 Ω) and 6 V battery (R2 = 3 Ω) sharing a 4 Ω load (defaults): node voltage VT = (12/2 + 6/3)/(1/2 + 1/4 + 1/3) = 8/1.0833 = 7.385 V. Branch currents: through R1, (12 − 7.385)/2 = 2.308 A supplied by the 12 V source; through the load R3, 7.385/4 = 1.846 A; through R2’s branch, (7.385 − 6)/3 = 0.462 A flowing INTO the 6 V source’s positive terminal. KCL: 2.308 in = 1.846 + 0.462 out. ✔ And the physics headline: the 6 V battery is being charged by the 12 V battery through the load — parallel unequal sources fight, and the smaller one loses. This is why you never parallel mismatched batteries without isolation.
  • Equal sources: set V2 = 12 V — VT rises to 9.23 V and both sources share the load current (1.385 A and 0.923 A, inversely as their series resistances). Stiffer branch (smaller R) carries more — the load-sharing rule of paralleled supplies, visible in one edit.
  • Source removed: set V2 = 0 and R2 becomes a second load branch: VT = 6/(0.5+0.25+0.333) = 5.54 V with both resistor branches draining the single source — the laws don’t care whether a branch contains a source at all.

From Two Nodes to Real Circuits

Real circuits have more nodes and loops, and the method scales mechanically: assign each unknown node a voltage (nodal analysis, KCL at every node) or each loop a mesh current (mesh analysis, KVL around every loop), solve the linear system, done. Computers do exactly this — SPICE, the simulator behind essentially all modern electronics design, is nodal analysis industrialized. The three-branch circuit here is the largest circuit solvable by pure inspection, which is why textbooks (and this page) live in it: every conceptual trap of the general method — sign discipline, sources being charged, supermodes when sources link nodes — already appears in miniature.

Kirchhoff’s laws themselves have limits worth knowing: KCL assumes no charge accumulation (true until antenna/plasma physics), KVL assumes no changing magnetic flux through the loop (when there is, the “loop voltage” genuinely depends on the path — the Lewin paradox, discussed on our EMF page). Within lumped circuits at ordinary frequencies, the laws are as reliable as arithmetic.

Kirchhoff Discipline

  • Pick reference directions first, write them down. Signs are chosen, then obeyed; a negative answer is information, not an error to flip away.
  • Every loop check must close at zero. After solving any circuit by hand, trace two independent loops; a nonzero sum localizes your algebra mistake better than re-solving does.
  • Power balance is the third check: Σ supplied = Σ absorbed (with the defaults: 12 V × 2.308 A supplied = 27.7 W; absorbed: I²R losses 10.65 + 13.64 + 0.64 W plus 6 V × 0.462 A = 2.77 W charged into the small battery = 27.7 W ✔).
  • Series resistances of sources are real: R1 and R2 here model battery internal resistance and wiring — ideal voltage sources in parallel at different voltages are a mathematical explosion (infinite current), and the resistances are what keep the problem physical.

Related Tools

Frequently Asked Questions

What are Kirchhoff’s two laws?

The current law (KCL): currents entering a node sum to currents leaving — charge is conserved and doesn’t pile up in junctions. The voltage law (KVL): voltages around any closed loop sum to zero — energy per charge is conserved on a round trip. Stated in 1845 by Gustav Kirchhoff, they remain the complete foundation of lumped circuit analysis; this calculator applies KCL at the node to solve a three-branch circuit and verifies KVL on every branch automatically.

How do you solve a circuit with Kirchhoff’s laws?

Assign unknowns (node voltages or loop currents), write one KCL equation per node (or KVL per loop), add Ohm’s law for each component, and solve the linear system. For the two-node circuit here it collapses to one equation — VT = Σ(V/R)/Σ(1/R) over the branches (Millman’s theorem) — then each branch current follows from its own voltage drop. Our worked example verifies the solution three ways: KCL balances, KVL closes on every loop, and total power supplied equals power absorbed.

Can a battery be charged by another battery in the same circuit?

Yes — and this calculator’s default circuit demonstrates it: the 6 V source has 0.462 A pushed backward through it because the shared node sits at 7.385 V, above its own voltage. Whether that’s harmful depends on the battery (a rechargeable cell just charges; a primary cell may leak or rupture; paralleled power supplies need OR-ing diodes or active sharing). The general lesson: in parallel-source circuits, always check the sign of every source’s current, not just the load’s.

What is Millman’s theorem?

A packaged consequence of KCL for circuits with one pair of nodes: the node voltage equals the sum of each source branch’s V/R divided by the sum of all branches’ 1/R (conductances). It’s not a third law — it’s KCL pre-solved for the two-node topology, and it generalizes: any number of source branches and load branches between two rails reduces to this single division. This calculator is Millman’s theorem with the branch currents and checks filled in.

When do Kirchhoff’s laws fail?

KVL in its simple form assumes the loop encloses no changing magnetic flux; with significant induction through the loop, “the voltage between two points” becomes path-dependent and lumped analysis needs the flux made explicit (as an inductor component). KCL assumes nodes don’t store charge — at very high frequencies, stray capacitance makes that approximate too. Engineers restore exactness by adding those effects as components (stray C, mutual L), which is a modeling choice, not a refutation: within lumped-circuit assumptions, the laws are exact.

How do I check my own circuit solution?

Three independent audits, all used in this page’s example: (1) KCL at every node — currents balance; (2) KVL around at least two loops — voltages sum to zero; (3) power balance — total power delivered by sources (counting any source with reversed current as absorbing) equals total I²R heating plus power absorbed elsewhere. An answer that passes all three is almost certainly right; one that fails any is certainly wrong, and the failing check usually points at the faulty branch.

Why do the series resistors matter in this circuit?

Because paralleling two ideal voltage sources of different voltages is a contradiction — KVL around their loop would demand the impossible, and the implied current is infinite. Real sources always have internal resistance (and wires have some), so R1 and R2 model reality: they limit the cross-current, set how the load current is shared, and dissipate the difference as heat. Any honest parallel-source problem includes them; this calculator requires them for the same reason nature does.

Is this Kirchhoff calculator free?

Yes — free, instant, no sign-up. Enter two source branches and a load branch to get the node voltage, all branch currents with directions, and the KCL balance check.