# Voltage Dividers

## Using Voltage Dividers

The voltage divider equation is arguably the most important equation for an electrical engineer to know.  At the very least, it is one of the most fundamental.  Although the voltage divider technique becomes cumbersome when applied to larger circuits, no other method is faster when it comes to finding voltages in smaller circuits.  This beefy voltage divider explanation was kindly donated by Ryan Eatinger. The simplest form of a voltage divider circuit is shown in Figure 1.  V1 and V2 can be found using the following equations.

$V1 = Vs ( \frac{R1}{R1 + R2}) = 9 ( \frac{1k}{1k + 2k}) = 3V$ $V2 = Vs ( \frac{R2}{R1 + R2}) = 9 ( \frac{2k}{1k + 2k}) = 6V$
The voltage divider equation applies to series circuits where the current remains constant throughout the circuit.  If current is constant for all resistors, then it can be taken out of the equation.  This is the true advantage of the voltage divider.  If there is a choice between working with currents and voltages and working only with voltages, the choice becomes obvious. Here’s how it works.  Once again, we have the circuit in Figure 1.  The current remains constant throughout the circuit, meaning that the current through the source equals the current through R1 equals the current through R2.
$Is = I1 = I2$
Recalling Ohm’s law, write the equation above in terms of voltage and resistance.

Ohm’s Law: $I = \frac{V}{R}$ –> $\frac{Vs}{R1 + R2} = \frac{V1}{R1} = \frac{V2}{R2}$

These equations can now be used to find V1 and V2.

$V1 = Vs( \frac{R1}{R1 + R2})$ $V2 = Vs(\frac{R2}{R1 + R2})$

The general equation for a voltage divider is given below, where Vo is the measured voltage, Vs is the source voltage, Ro is the resistance across which the voltage is measured, and RT is the equivalent resistance of the circuit.  Figure 2 shows the corresponding circuit.

General Voltage Divider Equation$V0 = Vs(\frac{R0}{Rt})$

A voltage divider is not always in the simple form shown so far.  Recognizing a voltage divider is a skill that takes time to develop.  This article introduces some variations on the basic voltage divider circuit that you may encounter.  The best way to solve a voltage divider is to simplify it to the basic form shown in Figure 2.  Once in this form, apply the general voltage divider equation to find the desired voltage.

### Series Circuits

Applying the voltage divider equation to a series circuit is a fairly straightforward process.  It’s simply a matter of identifying which resistors make up Ro and then adding all the resistors together to find the equivalent resistance.

Example 1: In Figure 3, there are four resistors and you’re trying to find the voltage across one of them.  The resistors are all in series, making the equivalent resistance of the circuit 10 kΩ.

$V0 = Vs(\frac{R0}{Rt}) = Vs(\frac{R4}{R1 + R2 + R3 + R4}) = 20(\frac{4k}{1k + 2k + 3k + 4k}) = 8V$

Example 2: When analyzing a circuit, pay attention to the orientation and location of the plusses and minuses.  Voltages aren’t always across one resistor.  In Figure 4, the terminal voltage is across the combination of resistors R2, R3, and R4.  Adjust the Ro accordingly to include the equivalent resistance of the three resistors.

$V0 = Vs(\frac{R0}{Rt}) = Vs(\frac{R2 + R3 + R4}{R1 + R2 + R3 + R4}) = 20(\frac{2k + 3k + 4k}{1k + 2k + 3k + 4k}) = 18V$

Example 3: You should also remember that voltages aren’t always measured to ground.  In Figure 4, Vo is measured across resistors R2 and R3 only.  Ro only includes the resistance between the plus and minus of Vo.

$V0 = Vs(\frac{R0}{Rt}) = Vs(\frac{R2 + R3}{R1 + R2 + R3 + R4}) = 20(\frac{2k + 3k }{1k + 2k + 3k + 4k}) = 10V$

### Circuits with Parallel Resistors:

Voltage dividers apply to resistors in series.  If you encounter a circuit with resistors in parallel, you must combine any parallel resistors before applying the voltage divider equation.

Only after combining the parallel resistors will the voltage divider equation work.  For this example, the parallel combination of R2, R3, and R4 combine to form the Ro in the general voltage divider equation.

$V1 = Vs(\frac{R0}{Rt}) = Vs(\frac{R1}{R1 + R1||R3||R4}) = 10(\frac{1k}{1k + 1k}) = 5V$ $V1 = Vs(\frac{R0}{Rt}) = Vs(\frac{R2||R3||R4}{R1 + R1||R3||R4}) = 10(\frac{1k}{1k + 1k}) = 5V$

Well, that should just about cover voltage dividers.  As always, if you have any questions feel free to make a comment or send me or another one of the admins a message and it will be taken care of!  Thanks again to Ryan Eatinger (reatinge@ksu.edu) for the article.