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 *R _{1}* and

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:
- Combining equations (5) and (6) with equation (4) results in the following
expressions for
*v*and_{1}*v*:_{2}(7)

\[{v_1} = \frac{{{R_1}}}{{{R_1} + {R_2}}}v\](8)

\[{v_2} = \frac{{{R_2}}}{{{R_1} + {R_2}}}v\]

(5)

\[i = \frac{{{v_1}}}{{{R_1}}}\]and

(6)

\[i = \frac{{{v_2}}}{{{R_2}}}\]
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.

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
*k*resistance in a series combination of resistors is given by:^{th} - Where
*v*is the total voltage drop across the series combination of resistors.

\[{v_k} = \frac{{{R_k}}}{{{R_1} + {R_2} + \cdots + {R_N}}}v\]

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

The voltage across the 5

**Ω**resistor can be determined from our voltage divider relationship:The current supplied by the source can be determined by dividing the total voltage by the equivalent resistance:

We can double-check the consistency between the voltage

*v*and the current*i*with Ohm's law. Applying Ohm's law to the 5**Ω**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.

\[v = \left[ {\frac{{5\Omega }}{{5\Omega + 15\Omega + 10\Omega }}} \right] \cdot
15V = \frac{5}{{30}} \cdot 15V = 2.5V\]

\[i = \frac{{15V}}{{{R_{eq}}}} = \frac{{15V}}{{5\Omega + 15\Omega + 10\Omega }} = \frac{{15V}}{{30\Omega }} = 0.5A\]

- 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
*v*is the voltage across the_{k}*k*resistor,^{th}*v*is the total voltage across the series combination,_{TOT}*R*is the resistance of the_{k}*k*resistor, and^{th}*R*is the total resistance of the series combination, the mathematical statement of this concept is:_{TOT}

\[\frac{{{v_k}}}{{{v_{TOT}}}} = \frac{{{R_k}}}{{{R_{TOT}}}}\]

- What is the voltage
*V*in the circuit below?_{1} - We want to make the voltage
*V*in the circuit below 4V. What resistance,*R*, should we use? - What is the voltage
V _{1}in the circuit below? - In the circuit below, we want
*V*to be 5V. What should we set the voltage source_{1}*V*to be?_{S} In the circuit of problem 4, what should

*V*be if we want_{S}*V*= 3V?_{1}- In the circuit below, we want
*V*= 2V and_{1}*I*to be less than 10mA. Using only resistors in from the Digilent_{1}^{®}Analog Parts Kit, choose values of*R*and_{1}*R*to meet these criteria._{2}

- 3V
- 4k
**Ω** - 2V. (Note: the 4k
**Ω**resistor on the left has no effect on the voltages in the voltage divider to the right of the source!) - 10V
- 6V
*I*=_{1}*V*/_{1}*R*(by Ohm's law), so If_{1}*V*= 2V, then_{1}*I*=2V/_{1}*R*. If_{1}*I*< 0.01A, this means we need_{1}*R*_{1}**>**200**Ω**.Also, from the voltage divider relation:

- So:
- Or:
- One possible solution is:
*R*= 1k_{1}**Ω**and*R*_{2}= 1.5k**Ω**.

\[{V_1} = 5V \cdot \frac{{{R_1}}}{{{R_1} + {R_2}}} = 2V\]\[\frac{{{R_1}}}{{{R_1} + {R_2}}} = \frac{2}{5}\]\[{R_2} = \frac{3}{2}{R_1}\]

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

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