Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) states that the algebraic sum of all currents entering and leaving any node (junction) in an electrical circuit equals zero. This law is a consequence of conservation of electric charge — charge cannot accumulate at a node under steady-state conditions. KCL is the basis for nodal analysis, a powerful technique for solving complex parallel and combined circuits.
Key Formula
Sum of all currents at a node = 0 (currents in = currents out)
LaTeX: \sum_{k=1}^{n} I_k = 0
| Symbol | Meaning | Unit |
|---|---|---|
| I_k | Current flowing into or out of the node (sign convention applied) | Ampere (A) |
| n | Total number of branches connected to the node | dimensionless |
Worked Example
Problem
At a node, three currents meet. I₁ = 5 A flows in, I₂ = 2 A flows in, and I₃ flows out. Find I₃.
Solution
Step 1: Apply KCL: sum of currents in = sum of currents out. Step 2: Currents in: I₁ + I₂ = 5 + 2 = 7 A. Step 3: Therefore I₃ = 7 A must flow out to satisfy KCL. Step 4: Verify: +I₁ + I₂ − I₃ = 5 + 2 − 7 = 0 ✓
Answer
I₃ = 7 A (flowing out of the node)
KCL Applied to Common Circuit Configurations
| Configuration | Number of Nodes | KCL Equations Needed | Typical Application |
|---|---|---|---|
| Series circuit | 2 | 1 | Single current path verification |
| Parallel circuit | 2 | 1 | Branch current distribution |
| T-network | 3 | 2 | Filter and attenuator design |
| Bridge circuit | 4 | 3 | Wheatstone bridge analysis |
| Ladder network | n+1 | n | Transmission line modelling |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Kirchhoff's Voltage Law
Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit equals zero. This principle is a direct consequence of the conservation of energy — as a charge traverses a complete loop, the energy gained from sources equals the energy lost across resistances. KVL is fundamental for analysing series circuits, mesh analysis, and determining unknown voltages in complex networks.
Ohm's Law
Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across those points, provided temperature and other physical conditions remain constant. It is one of the most fundamental relationships in electrical engineering and circuit analysis. The law applies to ohmic (linear) materials and is used to calculate unknown voltages, currents, or resistances in simple circuits.
Norton's Theorem
Norton's Theorem states that any linear electrical network can be replaced by an equivalent circuit consisting of a single current source (I_N) in parallel with a single resistance (R_N). It is the dual of Thevenin's Theorem and is particularly convenient when analysing circuits where current distribution is of primary interest. Norton and Thevenin equivalents are interconvertible, and choosing between them depends on whether the circuit is better suited to series or parallel analysis.
Named after Gustav Kirchhoff (1824–1887), a German physicist who published both circuit laws in 1845 while a student at the University of Königsberg. "Current" comes from Latin "currere" meaning "to run or flow", and "node" from Latin "nodus" meaning knot or junction.