Kirchhoff's Voltage Law (KVL)

KVL states that the sum of all voltages around any closed loop in a circuit equals zero. This is conservation of energy: if you travel around a loop and return to your starting point, the energy gained and lost must balance exactly. In practice, it means the voltage drops across components in a loop must sum to the supply voltage. Together with KCL, it forms the complete foundation for analysing any DC circuit.

The Law

∑V = 0   around any closed loop
ConventionVoltage riseVoltage drop
Travelling through a source from − to ++V (gain)—
Travelling through a source from + to −—−V (loss)
Travelling through a resistor in current direction—−IR (loss)
Travelling through a resistor against current direction+IR (gain)—

Pick a starting point, pick a direction (clockwise or counter-clockwise), and be consistent. The algebra takes care of itself.

Simple Series Loop

Simple Series Loop A closed loop with a 12V supply on the left and two resistors, R1 (4 ohms) and R2 (8 ohms), in series on the top branch. 12V R1=4Ω R2=8Ω
KVL applied to a simple series circuit with one loop.
Clockwise loop starting at battery positive terminal:
  +12 − V_R1 − V_R2 = 0

Total resistance: 4 + 8 = 12Ω
Current: I = 12/12 = 1A

V_R1 = 1A × 4Ω = 4V
V_R2 = 1A × 8Ω = 8V

Verify KVL: 12 − 4 − 8 = 0 ✓

Three-Resistor Series Circuit

Three-Resistor Series Loop A closed loop with a 9V supply on the left and three resistors in series on the top branch: R1 (100 ohms), R2 (220 ohms), and R3 (330 ohms). 9V R1=100Ω R2=220Ω R3=330Ω
A series loop with three resistors dividing the 9V supply.
R_total = 100 + 220 + 330 = 650Ω
I = 9V / 650Ω = 13.85 mA

V_R1 = 0.01385 × 100 = 1.38V
V_R2 = 0.01385 × 220 = 3.05V
V_R3 = 0.01385 × 330 = 4.57V

KVL: 9 − 1.38 − 3.05 − 4.57 = 0.00 ✓ (rounding)

Mesh Analysis — Multiple Loops

Mesh analysis applies KVL to every independent loop, with each loop assigned an assumed mesh current. The loop currents may overlap through shared components:

Mesh Analysis Circuit Two adjacent loops. The left loop has a 10V supply, R1 (2 ohms) on top, and shares R2 (4 ohms) in the middle. The right loop shares R2, has R3 (3 ohms) on top, and a 5V supply on the right. 10V R1=2Ω R2=4Ω R3=3Ω 5V I1 I2
Two-loop circuit analyzed with mesh currents I1 and I2.
Two meshes: I1 (left loop, clockwise), I2 (right loop, clockwise)

KVL left loop:
  10 − 2I1 − 4(I1 − I2) = 0
  10 − 2I1 − 4I1 + 4I2 = 0
  10 − 6I1 + 4I2 = 0   ... (1)

KVL right loop (note: 5V battery opposes clockwise direction):
  −4(I2 − I1) − 3I2 − 5 = 0
  4I1 − 4I2 − 3I2 = 5
  4I1 − 7I2 = 5          ... (2)

From (1): I1 = (10 + 4I2) / 6
Substitute into (2): 4 × (10 + 4I2)/6 − 7I2 = 5
  (40 + 16I2)/6 − 7I2 = 5
  40 + 16I2 − 42I2 = 30
  −26I2 = −10
  I2 = 0.385A

  I1 = (10 + 4×0.385)/6 = (10 + 1.54)/6 = 1.923A

Currents:
  R1: 1.923A
  R2: I1 − I2 = 1.923 − 0.385 = 1.538A
  R3: 0.385A

KVL and the Voltage Divider

The voltage divider formula is a direct consequence of KVL and Ohm's Law:

Voltage Divider A voltage source Vs connected to two resistors in series, R1 and R2. Vout is measured at the node between R1 and R2 relative to ground. Vs R1 Vout R2
The classic voltage divider derived directly from KVL.
KVL: Vs − V_R1 − V_R2 = 0     → Vs = V_R1 + V_R2
Ohm: Same current I through both → V_R1 = I×R1, V_R2 = I×R2

Solve for Vout = V_R2:
  I = Vs / (R1 + R2)
  Vout = I × R2 = Vs × R2 / (R1 + R2)

Supermesh

If a current source appears in a branch shared between two meshes, you can't write a KVL equation that includes it (current sources don't have a defined voltage drop). The trick is a supermesh: exclude the current source branch, write KVL around the outer perimeter of the combined loops, then add the current source as an additional constraint:

If a 2A current source shares a branch between mesh I1 and I2:

Constraint: I1 − I2 = 2A   (or I2 − I1 = 2A, depending on direction)
KVL: written around outer perimeter ignoring the shared branch

Summary — KVL vs KCL

KVLKCL
Governing principleConservation of energyConservation of charge
Applied toClosed loopsNodes (junctions)
Solves forVoltages and currents (mesh analysis)Node voltages (nodal analysis)
Equation type∑V = 0 around loop∑I = 0 at node
Number of equationsOne per independent loopOne per non-reference node

References