# Kirchhoff’s Voltage Law (KVL) Kirchhoff’s Circuit Laws

**“Kirchhoff’s voltage law states that the algebraic sum of the voltages around a closed loop is zero.” **

In a closed circuit, Kirchhoff’s voltage law (KVL) means that the sum of the voltage rises is equal to the sum of the voltage drops. In our methodology it is not necessary to specify whether there is a **“voltage rise”** or a **“voltage drop.” **

We have seen that voltages can be expressed in either double-subscript or sign notation. The choice of one or the other is a matter of individual preference. We will double-subscript notation, followed later by the sign notation.

## KVL And Double-Subscript Notation

Consider an image shown below in which six circuit elements A,B,C,D,E and F are connected together. The element may be sources or loads and the connections (nodes) are labeled 1 to 4. In going around a circuit loop, such as the loop involving elements A,E and D. We can start with any node and move in either a clockwise or anticlockwise direction until we come back to the starting point. In so doing, we encounter the labeled nodes one after the other. This ordered set of labels is used to establish the voltage subscripts. We write the voltage subscripts in sequential fashion, following the same order as the nodes we meet.

Let’s start with node 2 and moving clockwise around loop ABCD, we successively encounter nodes 2-4-3-1-2. The resulting KVL equation is therefore written as

**E _{24}+E_{43}+E_{31}+E_{12} = 0**

If we select loop CEF and start with node 4 and move counter clockwise, we successively encounter nodes 4-2-3-4. The resulting KVL equation is

**E _{42}+E_{23}+E_{34} = 0**

The set of voltages designated by the KVL equations may be AC or DC. If they are AC, the voltages will usually be expressed as phasors, having certain magnitudes and phase angle. In some cases the set magnitudes and phase angles. In some cases the set of voltages can even represent instantaneous values. In order to prevent errors, it is essential to equate all KVL equations to zero as we have done so far and will continue to do. We do not recommend attempts to equate voltage rises to voltage drops.

In finding the solution to such double-subscript equations, it is useful to remember that a voltage expressed as E_{XY} can always be expressed as -E_{YX} and vice versa.