Potential Dividers and Sensor Components

A potential divider is a circuit of two or more resistors in series across a supply that provides a fraction of the supply voltage, V_out = V_in × R₂ ÷ (R₁ + R₂); when one of the resistors is a sensor such as a thermistor or light-dependent resistor (LDR), the output voltage changes with temperature or light. In brief: the voltage across a resistor in a series chain is proportional to its resistance. Replacing a fixed resistor by a variable resistor (potentiometer) makes the output adjustable. A load connected across the output reduces V_out, unless its resistance is much larger than that of the resistor it is connected across.

Volume controls, light-activated street lamps and thermostats use potential dividers. This guide derives the divider formula, explains potentiometers, thermistors and LDRs in dividers, discusses the effect of a load, compares sensor circuits in a table, works through numerical examples and corrects common mistakes.

How does a potential divider work?

Two resistors R₁ and R₂ in series across a supply V_in carry the same current I = V_in ÷ (R₁ + R₂). The voltage across R₂ is V₂ = IR₂, so

V_out = V_in × R₂ ÷ (R₁ + R₂).

The larger resistor gets the larger share of the voltage. If R₁ = R₂, V_out = V_in ÷ 2. A potentiometer is a resistor with a sliding contact (a wiper): the position of the wiper sets the ratio of the two parts, so the output can be varied continuously from 0 to V_in. It is used as a volume control, a dimmer or a position sensor.

How are sensors used in dividers?

NTC thermistor: its resistance falls as temperature rises. In a divider with the thermistor as R₁ (top) and a fixed resistor as R₂ (bottom), the output across R₂ rises when the temperature rises. Swapping the positions reverses the behavior.
LDR (light-dependent resistor): the resistance falls with increasing light (from megaohms in the dark to hundreds of ohms in bright light). With the LDR as the top resistor, the output across the fixed resistor rises in bright light; with the LDR at the bottom, the output rises in the dark, which is the arrangement used to switch on a lamp at night.
Strain gauges, pressure sensors and moisture sensors use the same idea.

The output voltage is fed to a transistor, comparator or microcontroller to switch devices when it passes a threshold (for example, a fan that turns on when the temperature exceeds a set value).

What is the effect of a load?

If a load R_L is connected across the output (in parallel with R₂), the resistance of the lower part falls to R₂R_L ÷ (R₂ + R_L), and V_out decreases. To keep the output close to the unloaded value, the load resistance should be much larger than R₂ (at least ten times larger). A divider is not an efficient way to supply power: the current through the resistors is wasted as heat.

Output across the 10 kΩ fixed resistor on a 9.0 V supply with the LDR as the top resistor, V_out = 9.0 × 10 ÷ (10 + R). In daylight (1 kΩ) the output is 8.2 V, and at night (100 kΩ) it falls to 0.82 V, as in the worked example.

Sensor circuits compared

Sensor positions in a potential divider (output across the bottom resistor)
Sensor	Position	V_out rises when
NTC thermistor	Top	Temperature rises
NTC thermistor	Bottom	Temperature falls
LDR	Top	Light level rises
LDR	Bottom	Light level falls (darkness)

Worked example: dividers with sensors

Fixed divider. A 9.0 V supply across a 6.0 kΩ and a 3.0 kΩ resistor. V_out (across the 3.0 kΩ resistor) = 9.0 × 3.0 ÷ 9.0 = 3.0 V. The current is 9.0 ÷ 9,000 = 1.0 mA.
Effect of a load. A load of 3.0 kΩ is connected across the 3.0 kΩ resistor. The parallel combination is 1.5 kΩ, so V_out = 9.0 × 1.5 ÷ (6.0 + 1.5) = 1.8 V, much lower than 3.0 V. With a load of 300 kΩ, the parallel resistance is 2.97 kΩ and V_out = 9.0 × 2.97 ÷ 8.97 = 2.98 V, almost unchanged.
LDR light switch. A fixed resistor of 10 kΩ is in series with an LDR across 9.0 V, and the output is taken across the fixed resistor. In daylight the LDR has a resistance of 1.0 kΩ: V_out = 9.0 × 10 ÷ 11 = 8.2 V. At night, its resistance is 100 kΩ: V_out = 9.0 × 10 ÷ 110 = 0.82 V. A circuit that switches on a lamp when V_out falls below 2 V would operate at night.
Thermistor. An NTC thermistor of resistance 10 kΩ at 25 °C and 2.0 kΩ at 60 °C is the top resistor, with a 10 kΩ fixed resistor below it, on a 5.0 V supply. At 25 °C: V_out = 5.0 × 10 ÷ 20 = 2.5 V. At 60 °C: V_out = 5.0 × 10 ÷ 12 = 4.2 V.
Potentiometer. A 10 kΩ potentiometer on a 12 V supply with the wiper at 30 percent of the way from the bottom: V_out = 12 × 0.30 = 3.6 V (unloaded).

How have dividers been used?

The potentiometer as a device for comparing voltages was developed in the 19th century, and the first variable resistors were used in telegraphy. Potential dividers in radio receivers controlled the volume from the early 1920s. Modern sensors based on divider circuits are in thermostats, camera exposure meters, automatic headlamps and the analog inputs of microcontrollers.

Which mistakes are common?

Mistake 1: Using the total resistance in the numerator

The numerator is the resistor that the output is taken across. The denominator is the total of both.

Mistake 2: Ignoring the load

A load in parallel with the output resistor lowers the output.

Mistake 3: Mixing up the sensor position

The behavior reverses when the sensor and the fixed resistor swap places.

Mistake 4: Thinking that a divider can power heavy loads efficiently

The current through the divider wastes energy and the output falls under load. Use a regulator.