Voltage (and Current) Dividers

A voltage divider is a very simple circuit that allows you to obtain a lower voltage from an input voltage.

It is widely used in electronics and measurement (for example, to match a voltage to the input of a device).

The most common voltage divider consists of two resistors in series.

• The input voltage \(U_e\) is applied across the series \(R_1\)–\(R_2\).
• The output voltage \(U_s\) is obtained across one of the resistors (often \(R_2\)).

1. Two Resistors in Series

Consider two resistors \(R_1\) and \(R_2\) in series, powered by a voltage \(U_e\).

• The same current \(I\) flows through \(R_1\) and \(R_2\).

• The total voltage is: \(U_e = U_1 + U_2\)

  Using Ohm's law:

  • \(U_1 = R_1 \times I\)
  • \(U_2 = R_2 \times I\)

  Since \(I\) is the same in both resistors, the voltages are "divided" according to the resistances.

2. Voltage Divider Formula

The output voltage \(U_s\) is often defined as the voltage across \(R_2\). As shown in the diagram below:

The formula is derived by adding the voltages \(U_1\) and \(U_2\) (which give \(U_e\)), then substituting \(U_1\) and \(U_2\) with \(R_1 \times I\) and \(R_2 \times I\) using Ohm’s law.

A few algebraic steps then allow us to express \(U_s\) in terms of \(U_e\), \(R_1\), and \(R_2\).

Therefore:

• if \(R_2\) is small compared to \(R_1\), the output voltage is small;
• if \(R_2\) is large compared to \(R_1\), the output voltage approaches the input voltage.

We can also choose to measure the voltage across \(R_1\). In this case:

The principle remains the same: the voltage is distributed proportionally to the resistances.

Important: this formula is accurate when the output is measured ** "no-load" **, that is, with a device that draws almost no current (e.g., multimeter in voltmeter mode, high-impedance measurement input).

3. Numerical Example

A voltage divider is supplied with \(U_e = 12\ \mathrm{V}\):

• \(R_1 = 8\ \mathrm{k\Omega}\)
• \(R_2 = 4\ \mathrm{k\Omega}\)
• The voltage across \(R_2\) is measured.

Then:

The divider has "reduced" the voltage from \(12\ \mathrm{V}\) to \(4\ \mathrm{V}\).

4. Voltage divider formula (case with a load)

If we connect a load \(R_L\) to the output (a device that consumes current), this load is connected in parallel with \(R_2\); see the new diagram:

The "lower" resistance is then no longer \(R_2\), but an equivalent resistance:

The output voltage becomes:

We substitute \(R_{\mathsf{eq}}\) and expand: