3.2 Voltage and Current Dividers
Voltage Dividers. Consider the circuit shown in figure 3.10 having three identical resistors connected in series to a battery. What is the voltage drop across each resistor? Intuition might correctly lead the reader to the answer of : the battery is connected in series with three identical resistors. The battery’s must be dropped across the series set of resistors; the resistors are identical; there should be of or across each resistor. We can readily obtain this result by applying KVL around the circuit loop and making use of the fact that the current is the same for all elements in the series-connected circuit, hence the voltage drop across each resistor is the same. Application of KVL around the loop gives:
(1)
which readily gives us .
This is a voltage divider circuit. Recall from the previous chapter that a series-connection of three resistors each having resistance is equivalent to a single resistor having resistance . In this circuit the resistance seen by the battery, that is, the equivalent resistance loading the battery, is .
Knowing this, we deduce that the current in the loop is
(2)
Having found the current, we can determine the voltage drop across each resistor via Ohm’s law:
(3)
Note that the resistance of each resistor is 1/3 of the total series resistance seen by the battery, and the voltage drop across each resistor is 1/3 of the battery voltage. This is the behavior of a voltage divider circuit which we will define below. We first extend the discussion to the case where the series-connected resistors have different values.
Consider the following network which has three series resistors each having a different value, and let us determine the voltages across , , and .
To begin, we note that the three resistors are all connected in series; this circuit is equivalent to the following circuit, in which the voltage source is applied across a single resistor having equivalent resistance
(4)
The current in the loop is
(5)
The current having been determined, we readily obtain the voltage across each of the resistors via Ohm’s law:
(6)
(7)
(8)
We thus see that the voltage drop across each series resistor is a fraction of the source voltage; the fraction is the ratio of the series resistance to the total resistance seen by the voltage source.
Note that the sum of equations (6), (7), and (8) gives:
(9)
Examples
Voltage Divider Example. Determine the voltages across the five resistors in the following circuit if , , and , and are all resistors.
Solution: This problem can be solved by combining resistances as shown in Figure 3.13b, then forming a single equivalent resistance, determining the current from the 6V battery, and applying KVL, KCL, and Ohm’s law to arrive at the results. Using voltage division arrives at the result more quickly when we realize that there is a 6V drop across the middle of the circuit and the right branch of the circuit (). Voltage division gives the voltage across as . Similarly, the