19. Voltage dividers

A voltage divider is one of the most commonly encountered simple circuits. Consider the circuit shown below.

voltage divider with resistors

The symbol at the bottom with three lines indicates ground. We ask how does \(V_\mathrm{out}\) compare to \(V_\mathrm{in}\). We can derive the result from three key properties of circuits. First, Ohm’s law, which says that \(V = I R\), where \(V\) is a voltage, \(I\) is a current, and \(R\) is a resistance. Second, resistors in series add. Third, the current in to any point in a circuit is equal to the current out of it. This third rule means that in the circuit above, the current is the same everywhere. There is a voltage of \(V_\mathrm{out}\) across R₂ and a voltage of \(V_\mathrm{in}\) across R₁ and R₂ together. Thus,

\begin{align} I = \frac{V_\mathrm{in}}{R_1 + R_2} = \frac{V_\mathrm{out}}{R_2}, \end{align}

which yields

\begin{align} \frac{V_\mathrm{out}}{V_\mathrm{in}} = \frac{R_2}{R_1 + R_2}. \end{align}

So, the above circuit works as a voltage divider, giving \(V_\mathrm{out}\) as a fraction \(R_2/(R_1+R_2)\) of \(V_\mathrm{in}\).

Voltage dividers are one of the more ubiquitous circuits, as they allow delivery of a an arbitrary voltage that is less than the supplied voltage.

Follow-along exercise 14: Empirical voltage division

Wire up the circuit below.

schematic for voltage divider

You don’t need to upload any sketch; simply plug in your Arduino. Use your multimeter to verify that the resistors R₁ and R₂ function as a voltage divider. To do so, make sure you plug the red lead of your multimeter into the 500 mA port (the center one on the WeePro), and the black one on the COM port. Turn the dial to 20V DC (roughly at 10 o’clock on the dial). To measure the input voltage, touch the multimeter leads to the positive lead of R₁ and the negative lead of R₂. To measure the output voltage, touch the multimeter to the leads of R₂. Make sure you understand why touching those leads gives you the measurements they do.