Potential Dividers

A potential divider is a simple but powerful circuit that uses two or more resistors in series to produce an output voltage that is a fraction of the input voltage. Potential dividers are used extensively in sensor circuits and electronic control systems.

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The Potential Divider Principle

When two resistors R₁ and R₂ are connected in series across a supply voltage V_in, the voltage is divided between them in proportion to their resistances.

Circuit description:

• V_in is applied across R₁ and R₂ in series
• V_out is measured across R₂ (the output voltage)

The Potential Divider Equation

$V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}$Vout​=Vin​×R1​+R2​R2​​

Derivation

The current through both resistors (they are in series) is:

I = V_in / (R₁ + R₂)

The voltage across R₂ is:

V_out = IR₂ = V_in × R₂ / (R₁ + R₂)

Similarly, the voltage across R₁ is:

V₁ = IR₁ = V_in × R₁ / (R₁ + R₂)

Note: V₁ + V_out = V_in (conservation of energy) ✓

Key Point: The potential divider equation only works when no current is drawn from the output (i.e., the output is connected to a device with very high resistance, or is unloaded). If a load is connected in parallel with R₂, the effective resistance of the bottom section changes and the simple formula no longer applies.

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Worked Examples

Worked Example 1 — Basic Potential Divider

A potential divider consists of a 4.0 kΩ resistor (R₁) and a 6.0 kΩ resistor (R₂) connected in series across a 10 V supply. Calculate V_out across R₂.

Solution:

V_out = V_in × R₂ / (R₁ + R₂) = 10 × 6000 / (4000 + 6000) = 10 × 6000 / 10000 = 10 × 0.60 = 6.0 V

Worked Example 2 — Designing a Potential Divider

You need to produce a 3.0 V output from a 9.0 V supply. If R₂ = 10 kΩ, calculate the required value of R₁.

Solution:

V_out = V_in × R₂ / (R₁ + R₂)

3.0 = 9.0 × 10000 / (R₁ + 10000)

3.0(R₁ + 10000) = 90000

3.0R₁ + 30000 = 90000

3.0R₁ = 60000

R₁ = 20 kΩ

Check: V_out = 9.0 × 10000 / (20000 + 10000) = 9.0 × 10000/30000 = 9.0 × 1/3 = 3.0 V ✓

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Sensor Circuits Using Potential Dividers

By replacing one of the fixed resistors with a sensor (LDR or thermistor), the output voltage varies automatically in response to environmental changes.

Thermistor in a Potential Divider

Configuration 1: Thermistor as R₂ (bottom resistor)

V_out = V_in × R_thermistor / (R_fixed + R_thermistor)

• Temperature increases → R_thermistor decreases (NTC) → V_out decreases

Configuration 2: Thermistor as R₁ (top resistor)

V_out = V_in × R_fixed / (R_thermistor + R_fixed)

• Temperature increases → R_thermistor decreases (NTC) → denominator decreases → V_out increases

Worked Example 3 — Thermistor Sensor Circuit

A thermistor (NTC) is connected as R₁ in a potential divider with a fixed 10 kΩ resistor as R₂. The supply is 5.0 V. At 20°C, the thermistor has a resistance of 15 kΩ. At 80°C, it has a resistance of 2.0 kΩ. Calculate V_out at each temperature.

Solution:

At 20°C: R₁ = 15 kΩ, R₂ = 10 kΩ

V_out = 5.0 × 10000 / (15000 + 10000) = 5.0 × 10000 / 25000 = 2.0 V

At 80°C: R₁ = 2.0 kΩ, R₂ = 10 kΩ

V_out = 5.0 × 10000 / (2000 + 10000) = 5.0 × 10000 / 12000 = 4.17 V

The output voltage increases from 2.0 V to 4.17 V as the temperature rises. This voltage change could be used to trigger a warning system or control a cooling fan.

LDR in a Potential Divider

Configuration 1: LDR as R₂ (bottom resistor)

• Light intensity increases → R_LDR decreases → V_out decreases

Configuration 2: LDR as R₁ (top resistor)