# Series resistors:

## Introduction

If the total voltage difference across a set of series resistors is known, there is an easy way to determine the voltage across any individual resistor in the series combination. The appropriate formula is called the voltage divider formula, since the total voltage is divided among the individual resistors. For this reason, a pair of series resistors is commonly referred to as a voltage divider. Voltage dividers are an extremely useful tool in electrical circuits, and are commonly used in circuit design. Familiarity with voltage dividers is essential for electrical engineers. This section derives the voltage divider formula and provides examples of its use.

A basic voltage divider is a pair of resistors in series, as shown in Fig. 1. The resistors have resistance R1 and R2, and their individual voltage differences are v1 and v2, respectively. The total voltage across the series combination is v, and the current through both resistors is i1.

• For resistors in series, Ohm's law gives:

(1)

${v_1} = {R_1}i$

(2)

${v_2} = {R_2}i$
• Applying KVL around the loop:

(3)

$- v + {v_1} + {v_2} = 0 \Rightarrow v = {v_1} + {v_2}$
• Substituting equations (1) and (2) into equation (2), and solving for the current i results in:

(4)

$i = \frac{v}{{{R_1} + {R_2}}}$
• We can rearrange equations (1) and (2) to write the individual voltages in terms of the current:
• (5)

$i = \frac{{{v_1}}}{{{R_1}}}$

and

(6)

$i = \frac{{{v_2}}}{{{R_2}}}$
• Combining equations (5) and (6) with equation (4) results in the following expressions for v1 and v2:

(7)

${v_1} = \frac{{{R_1}}}{{{R_1} + {R_2}}}v$

(8)

${v_2} = \frac{{{R_2}}}{{{R_1} + {R_2}}}v$

These results are commonly called voltage divider relationships, because they state that the total voltage drop across a series combination of resistors is divided among the individual resistors in the combination. The ratio of each individual resistor's voltage drop to the overall voltage drop is the same as the ratio of the individual resistance to the total resistance.

## Important Point:

This result can be generalized to a series combination of N resistances as follows:

• The voltage drop across any resistor in a series combination of N resistances is proportional to the total voltage drop across the combination of resistors. The constant of proportionality is the same as the ratio of the individual resistor value to the total resistance of the series combination. For example, the voltage drop of the kth resistance in a series combination of resistors is given by:
• ${v_k} = \frac{{{R_k}}}{{{R_1} + {R_2} + \cdots + {R_N}}}v$
• Where v is the total voltage drop across the series combination of resistors.

#### Example 1:

For the circuit below, determine the voltage across the 5Ω resistor, v, and the current supplied by the source, i.

#### Solution:

• The voltage across the 5Ω resistor can be determined from our voltage divider relationship:

• $v = \left[ {\frac{{5\Omega }}{{5\Omega + 15\Omega + 10\Omega }}} \right] \cdot 15V = \frac{5}{{30}} \cdot 15V = 2.5V$
• The current supplied by the source can be determined by dividing the total voltage by the equivalent resistance:

• $i = \frac{{15V}}{{{R_{eq}}}} = \frac{{15V}}{{5\Omega + 15\Omega + 10\Omega }} = \frac{{15V}}{{30\Omega }} = 0.5A$
• We can double-check the consistency between the voltage v and the current i with Ohm's law. Applying Ohm's law to the 5Ω resistor, with a 0.5 A current, results in $v = (5\Omega )(0.5A) = 2.5V$ , which agrees with the result obtained using the voltage divider relationship.

## Important Points:

• If the total voltage difference across a set of series resistors is known, the voltage differences across any individual resistor can be determined by the concept of voltage division.
• The term voltage division comes from the fact that the voltage drop across a series combination of resistors is divided among the individual resistors.
• The ratio between the voltage difference across a particular resistor and the total voltage difference is the same as the ratio between the resistance of that resistor and the total resistance of the combination. If vk is the voltage across the kth resistor, vTOT is the total voltage across the series combination, Rk is the resistance of the kth resistor, and RTOT is the total resistance of the series combination, the mathematical statement of this concept is:
• $\frac{{{v_k}}}{{{v_{TOT}}}} = \frac{{{R_k}}}{{{R_{TOT}}}}$

1. What is the voltage V1 in the circuit below?
2. We want to make the voltage V in the circuit below 4V. What resistance, R, should we use?
3. What is the voltage V1 in the circuit below?
4. In the circuit below, we want V1 to be 5V. What should we set the voltage source VS to be?
5. In the circuit of problem 4, what should VS be if we want V1 = 3V?

6. In the circuit below, we want V1 = 2V and I1 to be less than 10mA. Using only resistors in from the Digilent® Analog Parts Kit, choose values of R1 and R2 to meet these criteria.

1. 3V
2. 4kΩ
3. 2V. (Note: the 4kΩ resistor on the left has no effect on the voltages in the voltage divider to the right of the source!)
4. 10V
5. 6V
6. I1 = V1/R1 (by Ohm's law), so If V1 = 2V, then I1=2V/R1. If I1 < 0.01A, this means we need R1 > 200Ω.
• Also, from the voltage divider relation:

• ${V_1} = 5V \cdot \frac{{{R_1}}}{{{R_1} + {R_2}}} = 2V$
• So:
• $\frac{{{R_1}}}{{{R_1} + {R_2}}} = \frac{2}{5}$
• Or:
• ${R_2} = \frac{3}{2}{R_1}$
• One possible solution is: R1 = 1kΩ and R2 = 1.5kΩ.

• 1 Since the resistors are in series, then by definition they both have the same current.