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# Voltage divider

A voltage divider is a simple series resistor circuit. It's output voltage is a fixed fraction of its input voltage. The divide-down ratio is determined by two resistors. Written by Willy McAllister.
A very common and useful series resistor circuit goes by the nickname voltage divider. We will work out how this circuit operates, and you will see where the nickname comes from.
A voltage divider looks like this:
Our goal is to come up with an expression relating output v, start subscript, o, u, t, end subscript to input v, start subscript, i, n, end subscript. A good place to start is finding the current through start text, R, 1, end text and start text, R, 2, end text.
Assumption: Assume 0 current is flowing out of the divider. (Before we are done we will check to see what happens if this zero-current assumption doesn't hold).
With this assumption, start text, R, 1, end text and start text, R, 2, end text have the same current, and we can consider them to be in series.
i, start subscript, start text, 1, end text, end subscript, equals, i, start subscript, start text, 2, end text, end subscript and for now let's just call this i.
To find the current, we apply Ohm's law and what we know about resistors in series, (reminder: resistors in series add),
v, equals, i, start text, R, end text Ohm's Law
v, start subscript, i, n, end subscript, equals, i, left parenthesis, start text, R, 1, end text, plus, start text, R, 2, end text, right parenthesis
Rearranging to solve for i,
i, equals, v, start subscript, i, n, end subscript, start fraction, 1, divided by, start text, R, 1, end text, plus, start text, R, 2, end text, end fraction
We've solved for current i in terms of v, start subscript, i, n, end subscript and both resistors.
Next, we write an expression for v, start subscript, o, u, t, end subscript using Ohm's Law,
v, start subscript, o, u, t, end subscript, equals, i, start text, R, 2, end text
We can substitute for i in the previous equation to get,
v, start subscript, o, u, t, end subscript, equals, left parenthesis, v, start subscript, i, n, end subscript, start fraction, 1, divided by, start text, R, 1, end text, plus, start text, R, 2, end text, end fraction, right parenthesis, start text, R, 2, end text
and we have derived the voltage divider equation:
The output voltage equals the input voltage scaled by a ratio of resistors: the bottom resistor divided by the sum of the resistors.
The ratio of resistors is always less than 1 for any values of start text, R, 1, end text and start text, R, 2, end text. This means v, start subscript, o, u, t, end subscript is always less than v, start subscript, i, n, end subscript. Input voltage v, start subscript, i, n, end subscript is scaled down to v, start subscript, o, u, t, end subscript by a fixed ratio determined by the resistor values. This is where the circuit gets its nickname: voltage divider.

### Example - use the voltage divider equation to find $v_{out}$v, start subscript, o, u, t, end subscript

We want to find v, start subscript, o, u, t, end subscript using the voltage divider relationship.
v, start subscript, o, u, t, end subscript, equals, v, start subscript, i, n, end subscript, start fraction, start text, R, 2, end text, divided by, start text, R, 1, end text, plus, start text, R, 2, end text, end fraction
We insert the actual input voltage and resistor values into the equation, remembering the equation tells us the bottom resistor, start text, R, 2, end text, goes in the numerator.
v, start subscript, o, u, t, end subscript, equals, 12, start text, V, end text, dot, start fraction, 3, start text, k, end text, \Omega, divided by, 1, start text, k, end text, \Omega, plus, 3, start text, k, end text, \Omega, end fraction
v, start subscript, o, u, t, end subscript, equals, 12, start text, V, end text, dot, start fraction, 3, start text, k, end text, \Omega, divided by, 4, start text, k, end text, \Omega, end fraction
v, start subscript, o, u, t, end subscript, equals, 12, start text, V, end text, dot, start fraction, 3, divided by, 4, end fraction, equals, 9, start text, V, end text
Let's do an optional step to check the current.
i, equals, start fraction, v, start subscript, i, n, end subscript, divided by, start text, R, end text, 1, plus, start text, R, end text, 2, end fraction, equals, start fraction, 12, start text, V, end text, divided by, 1, start text, k, end text, \Omega, plus, 3, start text, k, end text, \Omega, end fraction, equals, start fraction, 12, start text, V, end text, divided by, 4, start text, k, end text, \Omega, end fraction, equals, 3, start text, m, A, end text
Now we know the current, so we can compute how much power is being dissipated by our voltage divider,
p, equals, i, v, equals, 3, start text, m, A, end text, dot, 12, start text, V, end text, equals, 36, start text, m, W, end text
Summary: Our voltage divider takes an input voltage (in this case 12, start text, V, end text, but it could be any value) and scales it down to create an output voltage that's 3, slash, 4 of the input voltage. The 3, slash, 4 ratio is determined by our choice of the two resistors. As long as v, start subscript, i, n, end subscript is turned on, a current of 3, start text, m, A, end text flows down through the voltage divider, so it dissipates 12, start text, V, end text, times, 3, start text, m, A, end text, equals, 36, start text, m, W, end text.

### Voltage divider practice problems

All of these problems use this circuit diagram,

#### Problem 1

Let v, start subscript, i, n, end subscript, equals, 6, start text, V, end text, start text, R, end text, 1, equals, 50, start text, k, end text, \Omega, and start text, R, end text, 2, equals, 10, start text, k, end text, \Omega
Find v, start subscript, o, u, t, end subscript, the voltage across start text, R, end text, 2.
v, start subscript, o, u, t, end subscript, equals
start text, V, end text

#### Problem 2

Let start text, R, end text, 1, equals, 90, start text, k, end text, \Omega, start text, R, end text, 2, equals, 10, start text, k, end text, \Omega,
let the output voltage v, start subscript, o, u, t, end subscript, equals, 1, point, 5, start text, V, end text,
Find v, start subscript, i, n, end subscript.
v, start subscript, i, n, end subscript, equals
start text, V, end text

#### Problem 3

Let v, start subscript, i, n, end subscript, equals, 5, start text, V, end text, v, start subscript, o, u, t, end subscript, equals, 2, start text, V, end text, and start text, R, end text, 1, equals, 30, start text, k, end text, \Omega
Find start text, R, end text, 2.
start text, R, end text, 2, equals
\Omega

#### Problem 4 - Challenge

Let v, start subscript, i, n, end subscript, equals, 1, start text, V, end text, v, start subscript, o, u, t, end subscript, equals, start fraction, v, start subscript, i, n, end subscript, divided by, 2, end fraction
Design a voltage divider that dissipates 10, mu, start text, W, end text.
start text, R, end text, 1, equals
\Omega
start text, R, end text, 2, equals
\Omega

A voltage divider doesn't do anything useful unless its output is connected to something. You should be aware of what happens when a divider is connected to a load. Remember we made an assumption at the beginning? We assumed the current flowing out of the output was 0. That let us treat start text, R, 1, end text and start text, R, 2, end text as if they were in series, and we developed the voltage divider equation. Let's check what happens if the assumption is not true.

### Operating the voltage divider near its mid-range

To start this discussion, we let start text, R, 1, end text, equals, start text, R, 2, end text. With matched resistors, the expected v, start subscript, o, u, t, end subscript of the voltage divider is the mid-point of the divider's range, 0, point, 5, v, start subscript, i, n, end subscript. To cause some current to flow out of the divider, we connect a resistor start text, R, end text, start subscript, start text, L, end text, end subscript. Does the divider still work? Does our voltage divider story collapse?
Resistor start text, R, end text, start subscript, start text, L, end text, end subscript acts as a load on the output of the voltage divider, meaning that it causes a current i, start subscript, start text, L, end text, end subscript to flow. The presence of start text, R, end text, start subscript, start text, L, end text, end subscript means start text, R, 1, end text and start text, R, 2, end text are no longer strictly in series. Let's make start text, R, end text, start subscript, start text, L, end text, end subscript fairly big, to make i, start subscript, start text, L, end text, end subscript fairly small relative to i, start subscript, 2, end subscript. Let start text, R, end text, start subscript, start text, L, end text, end subscript be ten times bigger than start text, R, 2, end text,
start text, R, end text, start subscript, start text, L, end text, end subscript, equals, 10, start text, R, 2, end text
start text, R, 2, end text and start text, R, end text, start subscript, start text, L, end text, end subscript are in parallel with each other. Combine the two parallel resistors using the parallel resistor formula to get start text, R, 2, end text, vertical bar, vertical bar, start text, R, end text, start subscript, start text, L, end text, end subscript,
start text, R, 2, end text, vertical bar, vertical bar, start text, R, end text, start subscript, start text, L, end text, end subscript, equals, start fraction, start text, R, 2, end text, dot, start text, R, end text, start subscript, start text, L, end text, end subscript, divided by, start text, R, 2, end text, plus, start text, R, end text, start subscript, start text, L, end text, end subscript, end fraction, equals, start fraction, start text, R, 2, end text, dot, 10, start text, R, 2, end text, divided by, start text, R, 2, end text, plus, 10, start text, R, 2, end text, end fraction, equals, start fraction, 10, divided by, 11, end fraction, start text, R, 2, end text, equals, 0, point, 91, start text, R, 2, end text
This is our loaded voltage divider circuit, redrawn to show the equivalent resistance of start text, R, end text, 2 in parallel with start text, R, end text, start subscript, start text, L, end text, end subscript,
The 10, times load resistor has the effect of reducing the resistance at the bottom of the voltage divider by roughly 9, percent. What is the impact of this additional load on the divider's output voltage? Without the load, the expected output is 0, point, 5, v, start subscript, i, n, end subscript. Now we figure out the output voltage in the presence of a load resistor.
v, start subscript, o, u, t, end subscript, equals, v, start subscript, i, n, end subscript, start fraction, 0, point, 91, start text, R, end text, 2, divided by, start text, R, end text, 1, plus, 0, point, 91, start text, R, end text, 2, end fraction
We designed our divider with start text, R, end text, 1, equals, start text, R, end text, 2, so they cancel out,
v, start subscript, o, u, t, end subscript, equals, v, start subscript, i, n, end subscript, start fraction, 0, point, 91, divided by, 1, plus, 0, point, 91, end fraction
v, start subscript, o, u, t, end subscript, equals, v, start subscript, i, n, end subscript, start fraction, 0, point, 91, divided by, 1, point, 91, end fraction, equals, 0, point, 48, v, start subscript, i, n, end subscript
The output voltage drops to 48, percent of the input voltage. How big an error is this?
start fraction, 0, point, 48, divided by, 0, point, 50, end fraction, equals, 0, point, 96, equals, 96, percent
The actual output of the voltage divider is low by 4, percent compared to the expected voltage. (Note the voltage error of 4, percent is significantly less than the 9, percent resistance change.) Does a few percent error matter? That's for you alone to decide. It depends on how accurate the voltage divider needs to be for your application.
The nugget to tuck away from this analysis: If the effective load resistance is 10, times greater than the bottom resistor in the voltage divider, you get roughly "one hand" of percent error left parenthesis, 4, minus, 5, percent, right parenthesis in the output voltage. This holds when the output voltage is near the center of its range (in the neighborhood of v, start subscript, start text, i, n, end text, end subscript, slash, 2).

### Operating the voltage divider near its extremes

If you design the voltage divider to operate near its extremes, with the output voltage close to 0 or v, start subscript, start text, i, n, end text, end subscript, the percentage error in output voltage will be different. We repeat the analysis with the output voltage set to 90, percent and 10, percent of the divider range. We keep the load resistor ten times the bottom resistor, so the parallel combination of start text, R, end text, 2 and start text, R, end text, start subscript, start text, L, end text, end subscript is still 0, point, 91, start text, R, end text, 2.

#### Case 1: $v_{out} = 90\%$v, start subscript, o, u, t, end subscript, equals, 90, percent of $v_{in}$v, start subscript, i, n, end subscript

Let v, start subscript, o, u, t, end subscript, equals, 90, percent of v, start subscript, i, n, end subscript. The expected output is 0, point, 90, v, start subscript, i, n, end subscript.
First we design a voltage divider that gives us the desired output. Figure out start text, R, end text, 2 in terms of start text, R, end text, 1 for a 90, percent voltage divider,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, 0, point, 90, equals, start fraction, start text, R, end text, 2, divided by, start text, R, end text, 1, plus, start text, R, end text, 2, end fraction
0, point, 90, left parenthesis, start text, R, end text, 1, plus, start text, R, end text, 2, right parenthesis, equals, start text, R, end text, 2
0, point, 90, start text, R, end text, 1, equals, start text, R, end text, 2, minus, 0, point, 90, start text, R, end text, 2
0, point, 90, start text, R, end text, 1, equals, 0, point, 10, start text, R, end text, 2
start text, R, end text, 2, equals, start fraction, 0, point, 90, start text, R, end text, 1, divided by, 0, point, 10, end fraction, equals, 9, start text, R, end text, 1
start text, R, end text, 2 is 9 times bigger than start text, R, end text, 1.
Now we load the circuit with start text, R, end text, start subscript, start text, L, end text, end subscript and see how the output voltage changes. The expression we derived above for the loaded voltage divider is,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, start text, R, end text, 2, divided by, start text, R, end text, 1, plus, 0, point, 91, start text, R, end text, 2, end fraction
We replace start text, R, end text, 2 with 9, start text, R, end text, 1,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, left parenthesis, 9, start text, R, end text, 1, right parenthesis, divided by, start text, R, end text, 1, plus, start text, 0, end text, point, 91, left parenthesis, 9, start text, R, end text, 1, right parenthesis, end fraction
All the start text, R, end text, 1's cancel out, leaving,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, left parenthesis, 9, right parenthesis, divided by, 1, plus, 0, point, 91, left parenthesis, 9, right parenthesis, end fraction, equals, start fraction, 8, point, 19, divided by, 9, point, 19, end fraction, equals, 0, point, 89
The actual output voltage is 89, percent of v, start subscript, i, n, end subscript instead of 90, percent.
The actual output voltage divided by the expected output is,
start fraction, 0, point, 89, divided by, 0, point, 90, end fraction, equals, 0, point, 99
So the actual voltage is lower than the expected by only 1, percent.

#### Case 2: $v_{out} = 10\%$v, start subscript, o, u, t, end subscript, equals, 10, percent of $v_{in}$v, start subscript, i, n, end subscript

Let v, start subscript, o, u, t, end subscript, equals, 10, percent of v, start subscript, i, n, end subscript. The expected output is 0, point, 10, v, start subscript, i, n, end subscript.
Express start text, R, end text, 1 in terms of start text, R, end text, 2 for a 10, percent voltage divider.
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, 0, point, 10, equals, start fraction, start text, R, end text, 2, divided by, start text, R, end text, 1, plus, start text, R, end text, 2, end fraction
0, point, 10, left parenthesis, start text, R, end text, 1, plus, start text, R, end text, 2, right parenthesis, equals, start text, R, end text, 2
0, point, 10, start text, R, end text, 1, equals, start text, R, end text, 2, minus, 0, point, 10, start text, R, end text, 2
0, point, 10, start text, R, end text, 1, equals, 0, point, 90, start text, R, end text, 2
start text, R, end text, 1, equals, start fraction, 0, point, 90, start text, R, end text, 2, divided by, 0, point, 10, end fraction, equals, 9, start text, R, end text, 2
start text, R, end text, 1 is 9 times bigger than start text, R, end text, 2.
Now we load the circuit with start text, R, end text, start subscript, start text, L, end text, end subscript and evaluate the change in output voltage. The expression we derived above for the loaded voltage divider is,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, start text, R, end text, 2, divided by, start text, R, end text, 1, plus, 0, point, 91, start text, R, end text, 2, end fraction
We replace start text, R, end text, 1 with 9, start text, R, end text, 2,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, start text, R, end text, 2, divided by, 9, start text, R, end text, 2, plus, start text, 0, end text, point, 91, start text, R, end text, 2, end fraction
All the start text, R, end text, 2's cancel out,
start fraction, v, start subscript, o, u, t, end subscript, divided by, v, start subscript, i, n, end subscript, end fraction, equals, start fraction, 0, point, 91, divided by, 9, plus, 0, point, 91, end fraction, equals, start fraction, 0, point, 91, divided by, 9, point, 91, end fraction, equals, 0, point, 092
The actual output voltage is 9, point, 2, percent of v, start subscript, i, n, end subscript instead of the expected 10, percent.
The actual output voltage divided by the expected output is,
start fraction, 0, point, 092, divided by, 0, point, 10, end fraction, equals, 0, point, 92
So the actual voltage differs by 8, percent from the expected. This is nearly twice the error compared to the mid-range divider.

### Lessons for a loaded voltage divider

With a 10, times, start text, R, end text, 2 load resistor connected to a voltage divider:
• Near mid-range, the output voltage is reduced by 5, percent.
• Near the top of its range, the error goes down substantially, to around 1, percent.
• Near the bottom of its range, the error roughly doubles compared to mid-range. The output voltage is 8, percent lower than expected.

### Controlling error in a loaded voltage divider

If your design requires the error to be significantly smaller, the load needs to be much larger than 10, times, start text, R, 2, end text, like an additional 10, times or more. You can get an additional 10, times two ways. Increase the load resistance. Or, redesign the voltage divider to have smaller start text, R, 1, end text and start text, R, 2, end text, (at the cost of more power dissipated by the voltage divider).

### Real-world resistor tolerance also impacts accuracy

Real-world resistors always have a plus minus tolerance on their value. If the accuracy of the voltage divider is critical to your application, use resistors with tight tolerances, and check for acceptable performance by analyzing the voltage divider at the anticipated tolerance extremes.

### What's in a nickname

We mentioned at the start that the nickname of this circuit is a voltage divider. In many situations, that is exactly what it does. However, we showed that under certain conditions when there is a load on the divider, the actual output voltage is slightly lower than the value predicted by the voltage divider equation. The lesson: Call a circuit by its nickname; just remember that it's only a nickname.

## Summary

Voltage divider:
v, start subscript, o, u, t, end subscript, equals, v, start subscript, i, n, end subscript, start fraction, start text, R, 2, end text, divided by, start text, R, 1, end text, plus, start text, R, 2, end text, end fraction
where start text, R, 2, end text is the resistor on the bottom of the divider.