Potential Dividers and Sensor Circuits

Potential Dividers and Sensor Circuits

A potential divider is a circuit that uses two or more resistors in series to produce an output voltage that is a fraction of the input voltage. They are fundamental to sensor circuits and signal conditioning.

The Potential Divider Equation

For two resistors R₁ and R₂ in series across a supply voltage V_in:

V_out = V_in × R₂ / (R₁ + R₂)

where V_out is the voltage across R₂.

Derivation

The current through the series circuit is I = V_in / (R₁ + R₂)

The voltage across R₂ is V_out = IR₂ = V_in × R₂ / (R₁ + R₂)

This shows that the output voltage is proportional to the ratio of R₂ to the total resistance.

Worked Example

A 9.0 V supply is connected across a 3.0 kΩ and a 6.0 kΩ resistor in series. Find the voltage across each resistor.

V across 6.0 kΩ = 9.0 × 6.0/(3.0 + 6.0) = 9.0 × 6.0/9.0 = 6.0 V

V across 3.0 kΩ = 9.0 × 3.0/(3.0 + 6.0) = 3.0 V

Check: 6.0 + 3.0 = 9.0 V (correct, by Kirchhoff's voltage law).

Using a Potentiometer

A potentiometer is a three-terminal variable resistor that acts as a continuously adjustable potential divider. Turning the wiper varies V_out smoothly from 0 to V_in. Used for volume controls, brightness controls, and calibration.

Sensor Circuits Using Potential Dividers

By replacing one of the fixed resistors with a sensor (whose resistance changes with an environmental variable), the output voltage becomes dependent on that variable.

Thermistor Circuits (Temperature Sensing)

A negative temperature coefficient (NTC) thermistor has resistance that decreases as temperature increases.

Configuration 1 — Thermistor as R₂ (bottom resistor):

• As temperature rises → R_thermistor falls → V_out falls
• V_out decreases with increasing temperature

Configuration 2 — Thermistor as R₁ (top resistor):

• As temperature rises → R_thermistor falls → fraction R₂/(R₁ + R₂) increases → V_out rises
• V_out increases with increasing temperature

Worked Example — Thermistor Circuit

A 5.0 V supply is connected across a 4.7 kΩ fixed resistor (R₁) and an NTC thermistor (R₂) in series. The thermistor resistance is 10 kΩ at 20°C and 2.0 kΩ at 60°C. Find V_out at each temperature.

At 20°C: V_out = 5.0 × 10/(4.7 + 10) = 5.0 × 10/14.7 = 3.40 V

At 60°C: V_out = 5.0 × 2.0/(4.7 + 2.0) = 5.0 × 2.0/6.7 = 1.49 V

The voltage change (3.40 − 1.49 = 1.91 V) can be used to trigger an alarm or control system.

LDR Circuits (Light Sensing)

A light-dependent resistor (LDR) has resistance that decreases as light intensity increases. Typical values: ~1 MΩ in darkness, ~1 kΩ in bright light.

For an automatic light that turns on in darkness: Put the LDR as R₁ (top). In darkness, R_LDR is high, so V_out across R₂ is high → triggers the light.

Choosing the Fixed Resistor Value

The fixed resistor should be chosen to give the best sensitivity around the operating point. Sensitivity is greatest when the fixed resistor value is comparable to the sensor resistance at the transition point.

If the thermistor varies between 1 kΩ and 10 kΩ, a fixed resistor of about 3–5 kΩ gives the largest voltage swing across the operating range.

Loading Effects

A potential divider only gives the expected output if no significant current is drawn from the output. If a load resistance R_L is connected across V_out:

The effective bottom resistance becomes: R₂_eff = R₂R_L / (R₂ + R_L)

This parallel combination is always less than R₂, so the output voltage drops below the unloaded value.

Rule of thumb: The load resistance should be at least 10 times the output resistance of the divider for the loading effect to be negligible (< 10% error).

Worked Example — Loading

A potential divider with R₁ = R₂ = 10 kΩ on a 12 V supply gives V_out = 6.0 V unloaded.

If a 10 kΩ load is connected: R₂_eff = 10 × 10 / (10 + 10) = 5.0 kΩ

V_out = 12 × 5.0/(10 + 5.0) = 4.0 V (not 6.0 V)

This 33% drop shows why loading must always be considered in circuit design.

Comparison: Potential Divider vs Variable Resistor

Feature	Potential divider	Variable resistor in series
Output range	0 to V_in	Cannot reach 0 V
Linearity	V_out proportional to ratio	Non-linear with load
Sensitivity	Can be optimised	Limited
Use case	Sensor circuits, signal conditioning	Simple current control

Applications

• Wheatstone bridge: Four resistors in a diamond arrangement — a null-deflection method for precise resistance measurement; balanced when R₁/R₂ = R₃/R₄
• Strain gauges: Resistance changes with deformation; used in Wheatstone bridges for precise measurement
• Audio volume controls: Potentiometer varies output to amplifier
• Digital-to-analogue converters: Networks of precision resistors create specific voltage levels