# Voltage Divider Calculator

Created by Wojciech Sas, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Dec 21, 2022

This is a voltage divider calculator – a comprehensive but simple tool that helps you evaluate the output signal (i.e., voltage) that we obtain in a single voltage divider, often used in voltage regulators.

Read on to learn what is a voltage divider, find out the basic voltage divider formula, and how it extends to various equations for different voltage divider types. Also, learn how it is possible to obtain some fraction of input voltage by applying the voltage divider rule.

Additionally, check out the numerous applications of a voltage divider in basic and incredibly complex systems, and convince yourself that electronic circuits are nothing to worry about!

🙋 You may also be interested in using our voltage regulation calculator.

## What is a voltage divider?

A simple voltage divider is a part of a linear circuit that changes input voltage ($V_1$) into an output voltage ($V_2$), which is a different value. Since the circuit is a passive one, the ratio $V_2 / V_1$ is never greater than 1. You can see the generic voltage divider in this simple diagram:

where $Z_1$ and $Z_2$ are some impedances. The impedances can be related to the resistance $R$, the capacitance $C$, or the inductance $L$. We can distinguish a few basic types of voltage dividers for which we can apply the voltage divider rule:

1. Resistive divider, RR;

2. Capacitive divider, CC;

3. Inductive divider, LL;

4. RC and CR dividers (also known as RC, CR filters);

5. RL and LR dividers (also known as RL, LR filters); and

6. CL and LC dividers (also known as CL, LC filters).

## Voltage divider rule

The principle behind voltage dividers is that the current passing through several elements connected in series is constant, but the voltage divides somehow between them. We need to apply Ohm's law to our circuit to find the exact values. Before doing that, we must state one crucial point:

RR composition is the only one applicable to DC (direct current) circuits. In these cases, we can treat any impedance as a wire with zero resistivity, and capacitances work as a gap in a circuit, so they have infinite resistance. For the rest, they are all used with AC (alternating current) circuits, and the voltage divider rule is applicable for the maximum value of the potential difference. It may also be helpful to find the phase shift for these voltages.

## Voltage divider formula

The general voltage divider equation (or formula) for impedances is as follows:

$V_2 = \frac{Z_2}{Z_1+Z_2}V_1$

A reminder: in general, $V_1$ and $V_2$ correspond to the amplitudes of signals, e.g., sinusoidal ones.

If we are considering only resistances, the voltage divider formula naturally changes to:

$V_2 = \frac{R_2}{R_1+R_2}V_1$

Since the resistance doesn't affect the signal phase, the formula is the same for both AC and DC cases. We compare voltage values at a given moment. As mentioned before, we consider the remaining kinds of dividers for AC circuits, so let's take a look at some examples.

## Capacitive and inductive voltage divider equations

For a CC divider, we need to use the impedances of capacitors: $Z\!=\!\frac{1}{j\omega C}$, where $j$ stands for an imaginary number, and $\omega$ is the angular frequency of alternating voltage. Substituting the original voltage divider equation with this expression, we obtain:

$\begin{split} V_2 &= \frac{Z_2}{Z_1\!+\!Z_2}V_1\\[1.5em] &= \frac{\frac{1}{j\omega C_2}}{\frac{1}{j\omega C_1}\!+\!\frac{1}{j\omega C_2}}V_1, \end{split}$

and multiplying each term by $j\omega C_1C_2$ the result is:

$V_2 = \frac{C_1}{C_1+C_2}V_1.$

We can proceed with a similar procedure for LL dividers, where $Z = j\omega L$. This time the outcome voltage is:

$\begin{split} V_2 &= \frac{Z_2}{Z_1\!+\!Z_2}V_1\\[1.5em] &= \frac{j\omega L_2}{j\omega L_1\!+\!j\omega L_2}V_1 \end{split}$

Dividing numerator and denominator by $j\omega$ the final formula is:

$V_2 = \frac{L_2}{L_1+L_2}V_1.$

In both cases, the output voltage is in phase with respect to the input phase.

## RC and CR filters

Voltage divider circuits consisting of more than one type of element are not as simple to evaluate as the previous examples. We have to deal with complex number algebra, but trust us, it looks more frightening than it actually is.

For an RC divider, we can extend the voltage divider formula to:

$\begin{split} V_2 &= \frac{Z_2}{Z_1\!+\!Z_2}V_1 = \frac{\frac{1}{j\omega C}}{R\!+\!\frac{1}{j\omega C}}V_1 \\[1.5em] &= \frac{V_1}{j\omega RC\!+\!1} . \end{split}$

The result is a complex number, so in order to estimate the amplitude of the output voltage, we have to find its module:

$\begin{split} |V_2| &= \left| \frac{V_1}{j\omega RC\!+\!1} \right|\\[1.5em] &= \frac{|V_1|}{\sqrt{(\omega RC)^2\!+\!1}}. \end{split}$

If the frequency increases, the output amplitude of the voltage decreases, which is why we call this circuit a low pass filter. We can work out the phase shift can as the inverse tangent of the imaginary part divided by the real part of our complex number:

$\Delta\varphi = \arctan(-\omega RC).$

Analogically, we can find the amplitude and the phase shift for the CR circuit. The first step is to evaluate the general formula for output voltage:

$\begin{split} V_2 &= \frac{Z_2}{Z_1\!+\!Z_2}V_1 = \frac{R}{R\!+\!\frac{1}{j\omega C}}V_1 \\[1.5em] &= \frac{j\omega RC}{j\omega RC\!+\!1}V_1. \end{split}$

We can work out its amplitude as:

$\begin{split} |V_2| &= \left| \frac{j\omega RC}{j\omega RC\!+\!1}V_1 \right|\\[1.5em] &= \frac{\omega RC}{\sqrt{(\omega RC)^2\!+\!1}}|V_1|, \end{split}$

and the phase shift is given as:

$\Delta\varphi = \arctan\left(\frac{1}{\omega RC} \right).$

This time we can see that if the frequency tends towards 0, the same happens to the amplitude of $V_2$, and, for high values of $\omega$, it remains the same as the input voltage. The conclusion is that CR can be treated as a high pass filter.

## RL and LR filters

Circuits containing resistant and inductive elements aren't much different from RC and CR ones when it comes to calculations. Still, it's worth repeating each step to realize all the subtle differences.

In the case of the RL filter, we can start, as usual, with the general formula of the voltage divider:

$V_2 = \frac{Z_2}{Z_1\!+\!Z_2}V_1 = \frac{j\omega L}{R\!+\!j\omega L}V_1.$

To find the amplitude of the output voltage, we need to estimate the module of that value:

$\begin{split} |V_2| &= \left| \frac{j\omega L}{R\!+\!j\omega L}V_1 \right|\\[1.5em] &= \frac{\omega L}{\sqrt{R^2\!+\!(\omega L)^2}}|V_1|, \end{split}$

and its phase shift is:

$\Delta\varphi = \arctan\left(\frac{R}{\omega L} \right).$

For an LR divider, we just replace these elements, so the voltage divider equation results in the following:

$V_2 = \frac{Z_2}{Z_1\!+\!Z_2}V_1 = \frac{R}{R\!+\!j\omega L}V_1.$

Once again, we can determine the amplitude of the output voltage and the phase shift:

$\begin{split} |V_2| &= \left| \frac{R}{R\!+\!j\omega L}V_1 \right|\\[1.5em] &= \frac{R}{\sqrt{R^2\!+\!(\omega L)^2}}|V_1|, \end{split}$
$\Delta\varphi = \arctan\left(\frac{-\omega L}{R} \right).$

Take a look at the outcomes; the $V_2$ amplitude for the RL filter is very similar to CR, and that of LR resembles the voltage amplitude of RC. So do their phase shifts. It's a very valuable outcome because it turns out that we can use RC and RL filters interchangeably when designed in an appropriate configuration, and the values of conductance and inductance are adequately adjusted. It's especially useful for circuits that have to be resized to the nanometer scale since applying really small capacitors is much easier than creating tiny coils.

Anyway, suppose you have trouble calculating the output signal's properties. In that case, you can always come back to these chapters or try our voltage divider calculator! Choose a suitable option, and the result will be displayed instantly.

We described several passive filters above, but we can also apply the voltage divider rule with active ones.

## CL and LC as resonant circuits

We could perform the same calculations for CL and LC systems. However, some ridiculous outcomes may occur if we simply apply the voltage divider rule. We can cause the amplitude of the output voltage can go into infinity! This outcome is caused by the fact that the connected LC elements are sometimes called resonant circuits. We use them for generating and receiving radio waves, which is most efficient at a resonant frequency, given as:

$\omega = \frac{1}{\sqrt{LC}}.$

In a more realistic picture, we have to consider also some non-zero resistance, so the problem of the resonant circuits is described in the RLC circuit calculator.

## Usage of a voltage divider in more complex circuits

What we have already done shows how voltage dividers work for the simplest systems possible. You can obviously imagine that they are used practically nowhere in real life, and generally, more sophisticated circuits are applied.

However, all the results obtained above can be helpful while simplifying more complex ones. For instance, whenever you can spot resistors in either series or juxtaposed parallel, you can treat them as a single resistance. Similarly, capacitors and solenoids work in much the same way. We evaluate mixed components in practically the same way, but we have to take into account impedances $Z$ instead of $R$, $C$, or $L$.

## Applications of voltage dividers

Here are a few applications of voltage dividers:

1. One of the most commonly used devices which works thanks to the voltage divider concept is a potentiometer. Another word describing this element is a rheostat. They are usually made out of only resistive components. We can distinguish between analog and digital ones, but, in any case, we can set the resistance with high precision. Some of the most popular types of potentiometers are slide pots, trimpots, or thumb pots, which vary by size and structure. The critical element is a sliding contact, adjusting the output resistance possible.

2. High voltage measurement devices – It turns out that it is possible to measure high voltages even if the direct measurement can be destructive for the device. In that case, it's convenient to use the voltage divider to lower the voltage to a safe region. For exceptionally high voltages (let's say above 100 kV), it is better to use capacitive instead of resistive ones.

3. Finding an unknown resistance – Is it possible to find some unknown resistance if you only have a voltage source and another resistor with known resistance? You're lucky if you can read its color code, but what if there isn't one? Well, you can just create a simple circuit with both resistors arranged in series, set an input voltage, and measure the voltage over the desired resistor. Afterward, just insert all these values into the voltage divider calculator, and that's it – the mystery is revealed. You can always transform the general voltage divider equation to find $R₂$ as an unknown parameter:

$\qquad R_2 = \frac{V_2}{V_1-V_2}R_1$

💡 You may also face the opposite challenge: Which external resistors to choose to get your desired output voltage? In such case, we recommend checking Omni's LM317 calculator!