Kirchhoff's Voltage Law states that the sum of all voltages around any closed loop is zero. Voltage rises (sources) must equal voltage drops (across components).

What if the sum is not zero?

Either a voltage is missing from the loop, a value is incorrect, or there is a measurement error. The residual shows the missing voltage needed to balance the loop.

Can I use this for AC?

Yes. KVL applies to instantaneous voltages and to AC phasor voltages. Enter the magnitudes with appropriate signs for DC, or use complex values for AC analysis.

How many voltages can I enter?

Up to 8 voltages per loop. For multi-loop circuits, analyse each loop separately and verify KVL for each.

KVL Circuit Calculator

ΣV_loop = 0

V_source = V_R1 + V_R2 + …

KVL — Enter Voltages Around a Loop

Enter positive values for voltage rises (sources) and negative for voltage drops (resistors).

Enter voltages around the loop

KVL Verification

Sum of Voltages
—
V

Residual Error

—

KVL Status

—

Total Rise

—

Total Drop

—

Missing V (if any)

—

Figure 1: The sum of all voltages around any closed loop is zero. Source voltages (rises) are positive; resistor voltage drops are negative. If the sum is not zero, a voltage is missing or incorrect.

Table of Contents

Fundamentals

What Is Kirchhoff’s Voltage Law?
KVL Rules

Worked Examples

Single Loop with Three Resistors
Two-Loop Circuit
Finding an Unknown Voltage

Deep Dive

KVL and Mesh Analysis
KVL vs KCL

Reference

Frequently Asked Questions
Related Circuit Analysis Calculators

What Is Kirchhoff’s Voltage Law?

Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of all voltages around any closed loop in a circuit is zero. In other words, the total voltage rise (from sources) must equal the total voltage drop (across components). This is a consequence of energy conservation — a charge travelling around a complete loop returns to its starting point with the same energy.

KVL is one of the two fundamental circuit laws (along with KCL) used to analyse any circuit. The Series Circuit Calculator is a direct application of KVL: the source voltage equals the sum of all resistor voltage drops.

KVL Rules

ΣV = 0 around any closed loop.
Convention: voltage rises are positive, voltage drops are negative (or vice versa — be consistent).
V_source = V_R1 + V_R2 + ... + V_Rn

Worked Example — Single Loop with Three Resistors

Given: V_s = 12 V, R₁ = 100 Ω, R₂ = 220 Ω, R₃ = 330 Ω

I = 12 / (100+220+330) = 12/650 = 18.46 mA

KVL check: −12 + (0.01846×100) + (0.01846×220) + (0.01846×330) = −12 + 1.846 + 4.062 + 6.092 = 0 ✓

Worked Example — Two-Loop Circuit

Given: Loop 1 has V_s1 = 10 V and R₁ = 100 Ω, shared R₂ = 200 Ω. Loop 2 has R₃ = 300 Ω and V_s2 = 5 V.

Loop 1 KVL: −10 + 100I₁ + 200(I₁−I₂) = 0

Loop 2 KVL: 200(I₂−I₁) + 300I₂ + 5 = 0

Solving: I₁ = 43.5 mA, I₂ = 7.39 mA

Multi-loop KVL creates simultaneous equations. The Thevenin Equivalent Calculator can simplify multi-source circuits for single-load analysis.

Worked Example — Finding an Unknown Voltage

Given: V_s = 24 V, V_R1 = 8 V, V_R2 = 10 V. Find V_R3.

V_R3 = 24 − 8 − 10 = 6 V

KVL is especially useful when you can measure some voltages and need to deduce others. The Voltage Drop Calculator applies this principle to cable voltage drops.

KVL and Mesh Analysis

Mesh analysis (or loop analysis) applies KVL systematically to every independent loop in a circuit, creating a set of simultaneous equations that can be solved for all loop currents. It is the standard method for analysing planar circuits with multiple sources and is taught in every electrical engineering course. The Circuit Current Calculator handles the simpler single-loop case.

KVL vs KCL

KVL deals with voltages around loops; KCL deals with currents at nodes. Together they provide enough equations to solve any linear circuit. For circuits with many nodes and few loops, mesh analysis (KVL) is more efficient. For circuits with many loops and few nodes, nodal analysis (KCL) is better. The Norton Equivalent Calculator is naturally suited to KCL-based analysis.

Frequently Asked Questions

Does KVL work for AC circuits?

Yes. KVL applies to instantaneous voltages in any circuit. For AC steady-state analysis, KVL is applied to phasors (complex voltages) instead of real numbers, but the principle is identical.

What if my KVL equation doesn’t sum to zero?

Either you missed a voltage (check all components in the loop), made a sign error (check rise/drop convention), or there is a measurement error. KVL is a physical law — it always holds in any closed loop.

How many KVL equations do I need?

The number of independent KVL equations equals the number of independent loops, which is B − N + 1, where B is the number of branches and N is the number of nodes. For a simple series circuit, one equation suffices.

Can KVL handle dependent sources?

Yes. Dependent (controlled) sources are treated like independent sources in the KVL equation, but their values depend on other circuit variables. This creates additional equations that must be solved simultaneously.

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Last updated: March 2026