Engineering Tutorial

What is Kirchhoff’s Voltage Law

Starting at any node in a circuit, we form a loop by traversing through elements (open-circuits included!) and returning to the starting node, never encountering any other node more than once.

Figure 1 : For example, the paths fabef and fdcef are loops:

Figure 2 : whereas the paths becba and fde are not

Kirchhoff’s Voltage Law (KVL) is essentially the law of conservation of energy. If voltages across elements traversed from – to + are positive in sense, and voltages across elements that are traversed from + to – are negative in sense (or vice versa), then KVL can be stated as follows:

KVL defined as

KVL: Around any loop in a circuit, the voltages algebraically sum to zero.

If there are n elements in the loop then, in symbols, KVL is:

KVL can also be stated as: In traversing a loop, the sum of the voltage rises equals the sum of the voltage drops.

Example 1

In the circuit shown in Figure 1, we select a traversal from – to + to be positive in sense. Then KVL around the loop abcefa gives:

– v1 – v2 + v3 – v8 + v6 = 0

and around loop bcdeb, we have:

– v2 – 3 – v4 + v7 = 0

In this last loop, one of the elements traversed (the element between nodes b and e) is an open-circuit; however, KVL holds regardless of the nature of the elements in the circuit.

Example 2

We want to find the current i, in the one-loop circuit shown below:

The polarities of v1 , v2 , v3 and the direction of i were chosen arbitrarily (the polarities of the 10 V and 34 V sources are given). Applying KVL we get:

10 – v1 – 34 – v2 + v3 = 0

Thus:

v1 + v2 – v3 = -24

From Ohm’s Law:

v1 = 2i

v2 = 4i

v3 = -6i

Substituting these into the previous equation yields:

(2i) + (4i) – (- 6i) = -24

2i + 4i + 6i = -24

12i = -24

i = -2 A