Kirchhoff's Voltage Law (Three-Element Loop)

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 $V_s = V_1 + V_2 - V_3$. 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.