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Kirchhoff's Voltage Law (Three-Element Loop)

Also known as KVL · loop rule · second law · sum of voltage drops · mesh equation

Vs=V1+V2+V3V_{s} = V_{1} + V_{2} + V_{3}

Worked example: 5.5 V + 12.1 V + 6.4 V → 24 V source — press Try an example to run it live, then adjust anything.

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See also
Kirchhoff's Current Law (Node with Three Branches) — The companion law for a node, where currents rather than voltages must balance.Voltage Divider — What KVL becomes once you know the resistances and want each drop directly.
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Kirchhoff's Voltage Law (Three-Element Loop) explained

V₁V₂V₃Vs

Gustav Kirchhoff published this in 1845 while still a student, and it is nothing more than energy conservation dressed for a circuit: carry a charge all the way around a loop and back to where it started, and it must return to the same potential. Every rise has to be paid for by drops. So a 24 V supply feeding three series elements that drop 5.5 V and 12.1 V must be dropping the remaining 6.4 V across the third, whatever that third element happens to be.

The trap is sign, not arithmetic. Two sources fighting each other in a loop — a battery charging another battery, a generator paralleled slightly out of phase — subtract rather than add, and a loop containing a reversed source needs Vs=V1+V2−V3V_s = V_1 + V_2 - V_3. This solver assumes all three elements drop in the same direction, so enter an opposing source as a negative number and the algebra still comes out right.

What people rarely notice is that KVL is an approximation, and a good one only because circuits are small. The law follows from the electric field being conservative, which stops being true the moment a changing magnetic field threads your loop. Move a scope probe's ground lead near a transformer and the "same" two points read different voltages — not a bad measurement, but a real induced EMF in the loop the probe lead just formed.

Kirchhoff's Voltage Law (Three-Element Loop) formula

Vs=V1+V2+V3V_{s} = V_{1} + V_{2} + V_{3}
Where
  • VsV_{s}= Source voltage (V)
  • V1V_{1}= Drop across element 1 (V)
  • V2V_{2}= Drop across element 2 (V)
  • V3V_{3}= Drop across element 3 (V)

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