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.
Key Formula
Sum of all voltages in a closed loop = 0
LaTeX: \sum_{k=1}^{n} V_k = 0
| Symbol | Meaning | Unit |
|---|---|---|
| V_k | Voltage across the k-th element in the loop | Volt (V) |
| n | Total number of elements in the closed loop | dimensionless |
Worked Example
Problem
A series circuit has a 12 V battery, a 3 Ω resistor, and a 5 Ω resistor. Using KVL, find the voltage drop across the 5 Ω resistor.
Solution
Step 1: Find total resistance: R_total = 3 + 5 = 8 Ω. Step 2: Find current using Ohm's Law: I = V / R = 12 / 8 = 1.5 A. Step 3: Voltage across 5 Ω: V₅ = I × R = 1.5 × 5 = 7.5 V. Step 4: KVL check: 12 - (1.5 × 3) - 7.5 = 12 - 4.5 - 7.5 = 0 ✓
Answer
Voltage across 5 Ω resistor = 7.5 V
KVL Sign Convention for Voltage Drops and Rises
| Element | Direction of Traversal | Sign Convention | Effect on Sum |
|---|---|---|---|
| Battery (EMF source) | From − to + terminal | Positive (+V) | Adds to sum |
| Battery (EMF source) | From + to − terminal | Negative (−V) | Subtracts from sum |
| Resistor | In direction of current | Negative (−IR) | Subtracts from sum |
| Resistor | Against direction of current | Positive (+IR) | Adds to sum |
| Inductor | In direction of current | Negative (−L dI/dt) | Subtracts from sum |
| Capacitor | In direction of current | Negative (−Q/C) | Subtracts from sum |
Interactive Tools
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Related Terms
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.
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.
Thevenin's Theorem
Thevenin's Theorem states that any linear electrical network with voltage sources, current sources, and resistances can be replaced by an equivalent circuit consisting of a single voltage source (V_th) in series with a single resistance (R_th). This simplification makes it much easier to analyse the behaviour of a load connected to a complex network, as only the terminal behaviour matters. It is widely used in circuit design, power systems, and electronics to simplify analysis without solving the full network repeatedly.
Named after German physicist Gustav Kirchhoff who formulated the law in 1845. The word "voltage" derives from Alessandro Volta, the Italian physicist who invented the electric battery. "Law" is from Old English "lagu", meaning rule or ordinance.