Kirchhoff’s voltage law states that, around any closed loop of a lumped circuit, the algebraic sum of the potential differences is zero. Equivalently, the source voltage equals the sum of the voltage drops around the loop.
Learning objectives
- State Kirchhoff’s voltage law and its energy interpretation.
- Apply a consistent sign convention around a closed loop.
- Verify voltage distribution in a series circuit.
- Explain when KVL needs an explicit induced EMF term.
Progressive explanation
Start with the plain-language idea, then move into the engineering detail. Each level stands on its own, so you can stop at the depth you need.
Beginner The core idea
In a series circuit, the source voltage divides between the components. If a 12 V battery feeds three resistors and they drop 4 V, 5 V and 3 V, the sum is 12 V, which matches the source. KVL is simply a systematic way of writing that balance.
Intermediate How it behaves in a circuit
For a series circuit, KVL gives V_source = V₁ + V₂ + … = I(R₁ + R₂ + …), which is the basis of series resistance addition. KVL is used together with KCL in mesh analysis: one equation per independent loop yields the loop currents. The number of independent loops equals the number of branches minus the number of nodes plus one for a connected network.
Advanced Limits, assumptions, engineering detail
This subtlety matters for transformer windings, current loops in strong magnetic fields, and high-frequency circuits. Nevertheless, KVL remains usable as ΣV = ΣEMF when the induced EMF is included explicitly. The lumped approximation also requires that the loop dimensions are small compared with the wavelength, so that radiation and propagation delay can be neglected.
Formulas and variables
Units of measurement
| Quantity | Symbol | Unit |
|---|---|---|
| Voltage / potential difference | V | volt (V) |
| Electric field | E | volt per metre (V/m) |
| Magnetic flux | Φ | weber (Wb) |
Worked examples
Series voltage distribution
Problem. A 12 V source feeds 1.0 kΩ and 2.0 kΩ in series. Find the current and verify KVL.
Solution. R_total = 1.0 + 2.0 = 3.0 kΩ, so I = 12/3,000 = 4.0 mA. Drops: V₁ = 4.0 mA × 1.0 kΩ = 4.0 V and V₂ = 4.0 mA × 2.0 kΩ = 8.0 V. KVL: 4.0 + 8.0 = 12 V.
Finding a missing voltage drop
Problem. A 9.0 V source feeds two series resistors. The first drops 3.0 V. What is the voltage across the second?
Solution. KVL: V_source = V₁ + V₂, so V₂ = 9.0 − 3.0 = 6.0 V.
Practical applications
- Series circuit analysis
- Mesh analysis of networks
- Voltage-drop checks in wiring
- Verifying sensor and reference circuits
Automotive applications
KVL is used in voltage-drop testing: the source voltage equals the sum of the drops across the cables, connectors, switch and load. If a 12 V circuit shows only 9 V at the load, the missing 3 V is distributed across unintended resistances between the battery and the load, or in the return path. Measuring each segment localises the loss.
Industrial applications
In industrial control panels, KVL explains why a series circuit of contacts, coils and indicators must have all elements accounted for. An apparently small resistance in a terminal or contactor contact can drop enough voltage to prevent a contactor from picking up reliably. Loop analysis is also the basis of verifying that protective and control circuits are complete.
Measurement procedure
- Measure each voltage with the voltmeter in parallel across the respective element.
- Keep the probe polarity consistent with the chosen loop direction to get signs right.
- Sum the measured drops and compare with the source voltage; a significant mismatch indicates a measurement error or an unaccounted element.
- For very small drops, use a meter with adequate resolution and account for lead resistance.
Common mistakes
- Forgetting the source term or adding drops twice.
- Using inconsistent signs when traversing the loop in different directions.
- Assuming KVL alone determines all currents; it must be combined with KCL or element equations.
- Ignoring induced EMF in a loop that encloses a changing magnetic flux.
Safety considerations
- When measuring around a live loop, keep the meter and leads rated for the highest voltage present.
- Do not create a short across a source while probing a loop; use the correct probe placement.
- Treat unexpected voltage drops in protective conductors as a fault and investigate before restoring power.