Kirchhoffs Voltage Law

Electrical Circuit Analysis > Series DC Circuits

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The law called Kirchhoff's voltage law (KVL) specifies that

The algebraic sum of the potential rises and drops around a closed path (or closed loop) is zero.

In symbolic form it can be written as

$$ \bbox[10px,border:1px solid grey]{\sum_{↻} V = 0} \tag{1}$$

where $ \sum$ represents summation, ↻ the closed loop, and V the potential drops and rises. The term algebraic simply means paying attention to the signs that result in the equations as we add and subtract terms.

Some questions that often arises are,
Which way should I go around the closed path?
Should I always follow the direction of the current?
How do I apply a sign to the various voltages as I proceed in a clockwise direction?

To simplify matters,

We will always try to move in a circuit by clockwise direction.

By selecting a direction, you eliminate the need to think about which way would be more appropriate. Any direction will work as long as you get back to the starting point.

For a particular voltage, we will assign a positive sign when proceeding from the negative to positive potential.

For a drop in voltage, we will assign a negative sign when proceeding from the positive to negative potential.

Fig. 1: Applying Kirchhoff's voltage law to a series dc circuit.

In Fig. 1 as we proceed from point d to point a across the voltage source, we move from a negative potential (the negative sign) to a positive potential (the positive sign), so a positive sign is given to the source voltage $E$. As we proceed from point a to point b, we encounter a negative sign, so a drop in potential has occurred, and a negative sign is applied. Continuing from b to c, we encounter another drop in potential, so another negative sign is applied. We then arrive back at the starting point d, and the resulting sum is set equal to zero as defined by Eq. (1).

Writing out the sequence with the voltages and the signs results in the following:

$$ +E - V_1 - V_2 = 0$$

which can be rewritten as

$$E = V_1 + V_2$$

The result is particularly interesting because it tells us that

The applied voltage of a series dc circuit will equal the sum of the voltage drops of the circuit.

Kirchhoff's voltage law can also be written in the following form:

$$ \bbox[10px,border:1px solid grey]{\sum_{↻} V_{rises} = \sum_{↻} V_{drops}} \tag{2}$$

This equation revealing that

The sum of the voltage rises around a closed path will always equal the sum of the voltage drops.

Example 1: Use Kirchhoff's voltage law to determine the unknown voltage for the circuit in Fig. 2.

Fig. 2: Dc Series Circuit to be examined for Ex.1.

Solution: When applying Kirchhoff's voltage law, be sure to concentrate on the polarities of the voltage rise or drop rather than on the type of element. In other words, do not treat a voltage drop across a resistive element differently from a voltage rise (or drop) across a source. If the polarity dictates that a drop has occurred, that is the important fact, not whether it is a resistive element or source.
Application of Kirchhoff's voltage law to the circuit in [Fig. 1] in the clockwise direction results in

$$ +E_1 - V_1 + V_2 - E_2 = 0$$

$$ V_1 = E_1 - V_2 - E_2 $$

$$ V_1 = 16V - 4.2V - 9 $$

$$ V_1 = 2.8V $$

The result clearly indicates that you do not need to know the values of the resistors or the current to determine the unknown voltage.

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