Kirchhoff's Laws Reference
Interactive reference for Kirchhoff's Current Law (KCL) and Voltage Law (KVL). Example circuits with step-by-step node and mesh analysis.
How to Use
- Pick an example circuit (series, parallel, bridge).
- KCL: sum of currents entering a node = sum of currents leaving it.
- KVL: sum of voltage drops around any closed loop = 0.
- Use these to derive unknown currents and voltages in any DC circuit.
KCL + KVL Analysis
The Laws
History of Kirchhoff's Laws
Gustav Kirchhoff published his circuit laws in 1845 while still a 21-year-old student at the University of Königsberg. His paper formalized what practicing electrical experimenters had used implicitly for decades: that current must be conserved at any junction (KCL) and that voltage must be conserved around any closed loop (KVL). The same Kirchhoff went on to co-invent spectrum analysis with Robert Bunsen and formulate the thermal radiation law that bears his name — a remarkably productive scientific life.
Before Kirchhoff, circuit analysis was case-by-case. Ohm's 1827 law (V = IR) gave the relationship for one component; Kirchhoff's laws turned the analysis of entire networks into systems of linear equations. The modern matrix-based node/mesh analysis methods (which became practical after WWII with digital computers) are nothing but Kirchhoff's laws written as matrix equations solved by Gaussian elimination.
Today every SPICE simulator — the bedrock of circuit design since Berkeley's 1973 release — is a Kirchhoff-law solver at its core. SPICE writes KCL at each node, uses Newton-Raphson iteration to handle nonlinear devices, and solves the resulting sparse matrix equations. Every analog and mixed-signal IC from 1980 to today was born in a SPICE simulation — all of it ultimately KCL + Ohm's law.
About This Calculator
Pick a circuit topology (series, parallel, or mixed), enter the source voltage and three resistor values. The tool applies Kirchhoff's laws to derive total resistance, source current, and power dissipation, with a narrated step-by-step explanation of how KCL + KVL produce the result. This is a teaching tool — for arbitrary network analysis, use a SPICE simulator.
Series circuits illustrate KVL directly: V_source = V_R1 + V_R2 + V_R3. Parallel circuits illustrate KCL: I_total = I_R1 + I_R2 + I_R3. Mixed circuits require both laws. Once you can solve these three topologies by hand, any planar resistor network breaks down into combinations of these patterns. Everything runs client-side; no values leave your browser.
Frequently Asked Questions
What is Kirchhoff's Current Law?
KCL states that the algebraic sum of currents at any node is zero. Equivalently: current flowing into a node = current flowing out. It's a consequence of charge conservation — charge can\'t accumulate at a single point in a circuit.
What is Kirchhoff's Voltage Law?
KVL states that the algebraic sum of voltage drops around any closed loop is zero. Equivalently: energy gained from sources = energy dissipated in resistances (plus stored in reactive elements). It's a consequence of energy conservation.
When do these laws apply?
KCL/KVL apply in the quasi-static regime — where circuit dimensions are much smaller than the signal wavelength. For high-frequency RF or long transmission lines, you need full Maxwell\'s equations. For standard DC and audio/low-MHz electronics, KCL/KVL are exact.
Nodal vs. mesh analysis?
Nodal: pick node voltages as variables, write KCL at each node. Best for circuits with many branches converging on few nodes. Mesh: pick loop currents, write KVL around each mesh. Best for planar circuits with many loops. Both always work; pick whichever gives fewer equations.
What about AC circuits?
Same laws, just using phasors (complex numbers) instead of real values. Impedances replace resistances. The algebra becomes complex arithmetic, but the topology rules are identical.
Common Use Cases
Solving Unknown Current
3-branch circuit with known source and two resistor values — KCL at a node + KVL around a loop gives two equations, two unknowns.
Voltage Divider Derivation
KVL around the loop: Vin = V(R1) + V(R2). Equal current I through both: V(R2) = I × R2 = Vin × R2/(R1+R2).
Wheatstone Bridge
Balanced bridge condition derived by setting current through the galvanometer = 0 using KCL at the central node.
Circuit Debugging
Measure voltages at nodes and apply KVL around loops to identify component failures or wiring errors.
PCB Ground Analysis
Understand ground loops by applying KCL: ground currents must return somewhere. Careful layout minimizes unintended returns.
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