CR · DC analysis

Kirchhoff’s Voltage Law

Go once around a closed loop and you always end up where you started, and the same holds for electric potential. This obvious fact becomes a rule that ties together the voltages in the loop, and this unit shows how to use it to pin down an unknown voltage.

Once around, the voltages sum to 0

One source on the left and three resistors form a single loop. The source raises the voltage and the resistors drop it, but the drop across the bottom resistor is still unknown. Change V3 until the voltages around one full trip of the loop add up to exactly 0.

Unknown voltage drop V3V3 = 8 V

Algebraic sum around the loop

ΣV = −5 V

(+12) + (−5) + (−4) + (−8)

Unbalanced

The potential returns to itself

Every point in a circuit has its own potential. Starting from some point and moving along the loop to the next point, the potential rises or falls at each step, yet after a full trip back to the starting point, its potential must be exactly what it was when you left. One point cannot have two potentials at once. So all the rises and falls along the way add up to 0.

Rises are +, drops are −

Going around the loop in one direction, write a rise in potential as + and a fall as −. Passing through a source (EMF) from its − terminal to its + terminal raises the voltage, and passing through a resistor in the direction of the current drops it. Adding every term with its sign gives the single line ΣV = 0. Put another way, the total voltage the sources raise equals the total voltage the resistors drop.

One equation solves an unknown voltage

If a loop has only one unknown voltage, the single equation ΣV = 0 gives its value at once. In this figure the source raises +12V and two resistors drop 5V and 4V, so the third resistor's drop must be 12 − 5 − 4 = 3V. When solving a whole circuit, writing a KVL equation for every loop and a KCL equation for every node and solving them together is the core of mesh analysis and nodal analysis.

ObserveΣV = 0

The algebraic sum of the loop voltages is 0.

ChooseΣ(rise) ? Σ(drop)

Total rise equals total drop.

Fill in12 − 5 − 4 = ?

The source voltage minus the two drops is 3.

On your ownV3 = ?

The remaining drop is the unknown voltage.

Back to the first screen

As you changed the unknown drop, the loop's voltage sum started with one sign, crossed 0, and passed to the opposite sign, and at the point where the sum was exactly 0, V3 was 3V. The 12V raised by the source was shared by two resistors as 5V and 4V, and the third resistor took the remaining 3V. A single simple fact, that a trip around the loop comes back to where it started, pinned down the unknown voltage in one step.

Kirchhoff's voltage law (KVL) states that the algebraic sum of the voltages around any closed loop is 0. Taking rises as + and drops as −, ΣV = 0: the total voltage raised by sources equals the total voltage dropped by resistors. Because each point has only one potential (conservation of energy), the law always holds, and writing this equation for every loop is the foundation of mesh analysis.