2.1.6: Kirchhoff's Voltage Law

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Along with Ohm's law, the key law governing series circuits is Kirchhoff's voltage law, or KVL. Named after nineteenth century German physicist Gustav Kirchhoff, this law states that the sum of voltage rises and voltage drops around a series loop must equal zero (the rises and drops having opposite polarities). Alternately, it may be reworded as the sum of voltage rises around a series loop must equal the sum of voltage drops. As a pseudo formula:

\[\sum V\uparrow = \sum V\downarrow \label{3.4} \]

The Voltage Divider Rule (VDR)

An outgrowth of KVL is the voltage divider rule (VDR). In a series connection, the current is the same through each component. Thus, the voltage drops in a series connection must be directly proportional to the size of the resistances: the larger the resistor, the larger its voltage, and the larger its share of the total voltage applied to the series connection. Thus, the voltage across any resistor must equal the net supplied voltage times the ratio of the resistor of interest to the total resistance:

\[V_{Rx} = E \cdot R_X /R_{TOTAL} \label{3.5} \]

In fact, Equation \ref{3.5} is just a combination of two Ohm's law calculations into a single formula. The circulating current is equal to \(E/R_{TOTAL}\). This current is then multiplied by the resistor of interest (\(R_X\)) to arrive at the voltage across that resistor (\(V_{Rx}\)). It is worth pointing out that “the resistor of interest” can, in fact, be the sum of multiple resistors in series. While VDR is not required for any particular analysis, it serves two purposes: first, it saves some time because it skips over the computation of current, and second, it reinforces the ideal of a proportional division of voltage in a series connection. For example, if there are two resistors in series and one of them is twice the size of the other, then it must be the case that the larger resistor sees twice the voltage of the smaller resistor.