Potential Dividers

Spec mapping: OCR H556 Module 4.3 — Electrical circuits (potential divider circuits; the equation $V_{\text{out}} = V_{\text{in}} \times R_2/(R_1 + R_2)$ ; potentiometers and rheostats; sensor circuits using thermistors and LDRs; the open-circuit assumption and loading effect; PAG 4 anchor practical). Refer to the official OCR H556 specification document for exact wording.

A potential divider (or voltage divider) is the most elegant little circuit in introductory electronics. Two resistors in series across a fixed supply produce an output voltage that is a fixed fraction of the input — or, if one of the resistors is a sensor whose resistance varies with temperature, light, pressure, strain or moisture, an output voltage that varies with that physical quantity. Every sensor in an electronics project — from a fire alarm thermistor to a games-console joystick to a touchscreen — ultimately presents the user with a potential-divider output that an ADC then digitises.

This lesson derives the divider equation from Kirchhoff's laws, applies it to fixed dividers, potentiometers, and sensor circuits (thermistor, LDR, strain gauge), exposes the open-circuit caveat (the formula breaks when the load draws significant current), and works through the full Phase 2 mark-scheme moves: bicalibrated thresholds, loading-effect quantification, and the buffer-amplifier fix. By the end you should be able to design a divider for any sensor signal in your sleep.

The Basic Potential Divider — Derivation

Take a single supply voltage $V_{\text{in}}$ (from a battery or lab PSU) and connect two resistors $R_1$ and $R_2$ in series across it. Call the junction between them the "tap point". The output voltage $V_{\text{out}}$ is the pd between the tap and ground — i.e. the pd across $R_2$ .

flowchart LR
  VIN["V_in (supply +)"] --> R1["R_1"]
  R1 --> T((Tap point))
  T --> R2["R_2"]
  R2 --> GND["Ground (supply -)"]
  T --> VOUT["V_out (across R_2)"]

Assumption: no current is drawn from the tap point. This is the open-circuit assumption. Under it, the same current $I$ flows through both resistors (they are in series):

$I = \frac{V_{\text{in}}}{R_1 + R_2}$

The pd across $R_2$ is $V_{\text{out}} = IR_2$ , so:

$\boxed{\; V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2} \;}$

This is the potential divider equation. The factor $R_2/(R_1+R_2)$ is the dividing ratio; it is dimensionless and lies between $0$ and $1$ .

Some textbooks place $R_1$ across the output, giving $V_{\text{out}} = V_{\text{in}} \times R_1/(R_1+R_2)$ . Always identify which resistor is across the output before writing the equation.

Quick Sanity Checks

If $R_2 \gg R_1$ : $V_{\text{out}} \approx V_{\text{in}}$ .
If $R_2 \ll R_1$ : $V_{\text{out}} \approx 0$ .
If $R_1 = R_2$ : $V_{\text{out}} = V_{\text{in}} / 2$ (half-supply).

Worked Example 1 — Simple Ratio

A $12$ V supply feeds a divider of $1000$ $\Omega$ (top) and $2000$ $\Omega$ (bottom). What is $V_{\text{out}}$ across the $2000$ $\Omega$ ?

$V_{\text{out}} = 12 \times \frac{2000}{1000 + 2000} = 12 \times \frac{2}{3} = \mathbf{8.0 \text{ V}}$

And across the $1000$ $\Omega$ : $V_1 = 12 \times 1000/3000 = \mathbf{4.0}$ V. Check: $V_1 + V_{\text{out}} = 12$ V ✓ (Kirchhoff-2). The larger resistor drops the larger pd.

Why Use a Divider at All?

Fractional voltages on demand. A 12 V battery feeding multiple sub-circuits at different voltages.
Adjustable output. A potentiometer (mechanical divider) gives a continuously variable output.
Sensor readouts. A divider with a sensor resistor converts varying resistance into varying voltage that an ADC or comparator can read.
Simple and reliable. Two resistors. No transistors. No ICs.

In professional measurement systems, the divider is the front-end of almost every analogue sensor.

Worked Example 2 — Designing a 5 V Logic Rail

You have a 9 V battery and need to feed 5 V to a logic chip that draws negligible current.

$5 = 9 \times \frac{R_2}{R_1 + R_2} \Rightarrow \frac{R_2}{R_1 + R_2} = \frac{5}{9}$

Cross-multiplying: $9R_2 = 5(R_1 + R_2)$ , hence $\mathbf{R_2/R_1 = 5/4}$ . Pick $R_1 = 4$ k $\Omega$ , $R_2 = 5$ k $\Omega$ : $V_{\text{out}} = 9 \times 5/9 = 5.0$ V ✓.

Watch out: real logic chips draw input current that loads the divider; the output sags. We address loading next.

The Loading Effect (the Open-Circuit Caveat)

The formula assumes no current is drawn from the tap point. In reality, a load $R_L$ sits in parallel with $R_2$ , so the effective bottom resistance is:

$R_2 \| R_L = \frac{R_2 R_L}{R_2 + R_L}$

and the loaded output is:

$V_{\text{out, loaded}} = V_{\text{in}} \times \frac{R_2 \| R_L}{R_1 + R_2 \| R_L}$

The output falls under load. Loading is the single most common mark-scheme trap on this topic.

Worked Example 3 — Loading the 12 V/8 V Divider

Take the 12 V, $1000$ $\Omega$ / $2000$ $\Omega$ divider from Example 1. Unloaded $V_{\text{out}} = 8.0$ V. Connect a $2000$ $\Omega$ load:

$R_2 \| R_L = (2000 \times 2000)/4000 = 1000$ $\Omega$ .
$V_{\text{out, loaded}} = 12 \times 1000/(1000+1000) = \mathbf{6.0}$ V.

A 25% drop. Loading matters whenever $R_L$ is of the same order as $R_2$ .

Rule of Thumb

$R_L \geq 10\,R_2$ for the open-circuit formula to be accurate to $\sim 10\%$ . For precision, use a buffer amplifier (op-amp voltage-follower; input impedance $\sim 10^{12}$ $\Omega$ ) between divider and load.

The Potentiometer (Three-Terminal Variable Divider)

A potentiometer is a mechanical divider: a resistive track with a sliding wiper. Wiper at one end: $V_{\text{out}} = 0$ . Other end: $V_{\text{out}} = V_{\text{in}}$ . Linear-taper pot: linear in position.

A three-terminal potentiometer gives a continuously variable voltage. A two-terminal variable resistor — a rheostat — gives a variable resistance, cannot reach $0$ V output.

Exam Tip. OCR is strict:

Potential divider (potentiometer): 3 terminals, $0$ to $V_{\text{in}}$ .

Rheostat: 2 terminals, cannot output $0$ V.

Linear Pot — Voltage vs Wiper Position

Total track resistance $R$ , wiper at fraction $x$ from ground ( $0 \le x \le 1$ ):

$R_2 = xR$ (wiper-to-ground), $R_1 = (1-x)R$ (top-to-wiper).
$V_{\text{out}} = V_{\text{in}} \times xR/R = x V_{\text{in}}$ .

Linear in wiper position. This drives volume controls, joysticks, and resistive touchscreens.

Sensor Circuits — Resistance-to-Voltage Conversion

Replace one fixed resistor with a resistive sensor: the divider becomes a resistance-to-voltage converter.

Temperature: NTC Thermistor

An NTC thermistor has $R$ falling sharply as $T$ rises. Typical: $20$ k $\Omega$ at $20$ $^\circ$ C, $1$ k $\Omega$ at $60$ $^\circ$ C.

Thermistor as $R_2$ (bottom): $T$ rises $\Rightarrow$ $R_{\text{therm}}$ falls $\Rightarrow$ $V_{\text{out}}$ falls.
Thermistor as $R_1$ (top): $T$ rises $\Rightarrow$ $R_{\text{therm}}$ falls $\Rightarrow$ $V_{\text{out}}$ across fixed $R_2$ rises.

Choose orientation to match the application.

Worked Example 4 — Fire-Alarm Thermistor

$R_1 = 10$ k $\Omega$ fixed (top), $R_2 =$ NTC thermistor. $V_{\text{in}} = 9.0$ V. $R = 20$ k $\Omega$ at $20$ $^\circ$ C; $R = 1.0$ k $\Omega$ at $60$ $^\circ$ C.

At $20$ $^\circ$ C: $V_{\text{out}} = 9.0 \times 20/30 = \mathbf{6.0}$ V.

At $60$ $^\circ$ C: $V_{\text{out}} = 9.0 \times 1/11 = \mathbf{0.82}$ V.

A comparator triggers the alarm when $V_{\text{out}}$ falls below (say) $2$ V. This is exactly how domestic NTC fire detectors work.

Light: the LDR

A light-dependent resistor's $R$ falls with illumination. Typical: $\sim 100$ $\Omega$ in bright sunlight; $\sim 10$ k $\Omega$ in darkness.

flowchart LR
  V9["9 V"] --> RF["Fixed R_1"]
  RF --> T((Tap))
  T --> LDR["LDR (R_2)"]
  LDR --> GND["Ground"]
  T --> OUT["V_out → comparator"]

Worked Example 5 — Streetlamp Controller

$V_{\text{in}} = 9.0$ V, fixed $R_1 = 1.0$ k $\Omega$ , LDR as $R_2$ . Sunlight $R_{\text{LDR}} = 100$ $\Omega$ ; darkness $R_{\text{LDR}} = 10$ k $\Omega$ .

Sunlight: $V_{\text{out}} = 9.0 \times 100/1100 = \mathbf{0.82}$ V.

Darkness: $V_{\text{out}} = 9.0 \times 10000/11000 = \mathbf{8.18}$ V.

Controller switches the lamp on when $V_{\text{out}}$ rises above (say) $5$ V; off when it falls below.

Strain Gauge

A strain gauge is a fine wire whose resistance changes slightly when stretched ( $\Delta R / R_0 \propto$ strain). Typical: $R_0 = 120$ $\Omega$ , $\Delta R \sim \pm 1$ $\Omega$ at full-scale strain.

Worked Example 6 — Strain Gauge in a Divider

Strain gauge as $R_2 = 120$ $\Omega$ , fixed $R_1 = 120$ $\Omega$ , $V_{\text{in}} = 6$ V.

Unstrained: $V_{\text{out}} = 6 \times 120/240 = \mathbf{3.000}$ V.

Strained ( $R_2 = 121$ $\Omega$ ): $V_{\text{out}} = 6 \times 121/241 = \mathbf{3.012}$ V.

The change is $12$ mV — tiny. In practice, strain gauges are wired in a Wheatstone bridge (lesson 11) which subtracts the baseline, then an op-amp amplifies the millivolt signal. The underlying physics is still the divider equation.

Worked Example 7 — Why Headphones Break a Divider

A student builds a $9$ V, $4.7$ k $\Omega$ / $4.7$ k $\Omega$ divider (unloaded $V_{\text{out}} = 4.5$ V) and plugs in $32$ $\Omega$ headphones. Output collapses.

$R_2 \| R_L = (4700 \times 32)/(4700 + 32) = 31.8 \text{ }\Omega$

$V_{\text{out, loaded}} = 9 \times \frac{31.8}{4700 + 31.8} = \mathbf{0.060 \text{ V}} = 60 \text{ mV}$

A 75 $\times$ collapse. Lesson: the open-circuit equation only holds when $R_L \gg R_2$ . Low-impedance loads need a buffer amplifier, not a passive divider.

Worked Example 8 — Designing a Frost Alarm

You wish to build a frost alarm that triggers when the temperature falls below $2$ $^\circ$ C. The NTC thermistor has $R = 5$ k $\Omega$ at $2$ $^\circ$ C and $R = 15$ k $\Omega$ at $-10$ $^\circ$ C; $V_{\text{in}} = 5$ V. The comparator triggers when $V_{\text{out}}$ rises above $2.5$ V. Design $R_1$ to position the threshold correctly.

Wire the thermistor as $R_2$ (so $V_{\text{out}}$ rises as the thermistor's resistance rises, which happens as temperature falls). At the trigger point ( $T = 2$ $^\circ$ C, $R_{\text{therm}} = 5$ k $\Omega$ ), we want $V_{\text{out}} = 2.5$ V:

$2.5 = 5 \times \frac{5000}{R_1 + 5000} \Rightarrow \frac{R_1 + 5000}{5000} = 2 \Rightarrow R_1 = \mathbf{5000 \text{ }\Omega = 5 \text{ k}\Omega}$

Check at $T = -10$ $^\circ$ C, $R_{\text{therm}} = 15$ k $\Omega$ : $V_{\text{out}} = 5 \times 15/(5 + 15) = 3.75$ V — well above the threshold, alarm firmly triggered. At $T = +20$ $^\circ$ C the thermistor will be even smaller (say $\sim 2$ k $\Omega$ ), giving $V_{\text{out}} = 5 \times 2/(5+2) \approx 1.4$ V — comfortably below the threshold. Design works.

Practical detail: the comparator should have a small amount of hysteresis (a slightly higher trigger-on voltage than trigger-off voltage), otherwise the output oscillates near the threshold as ambient temperature drifts. Adding a small positive feedback resistor around the comparator implements hysteresis; it's standard in any practical sensor circuit.

Worked Example 9 — Comparing Series and Parallel Sensor Arrays

A scientist wants to monitor temperature at four points in an environmental chamber. Two architectures:

(a) Four independent dividers, each with $R_1 = 10$ k $\Omega$ fixed, $R_2 =$ thermistor. Each output goes to a separate ADC channel.

(b) Four thermistors in series, with $R_1 =$ four times the typical thermistor resistance. Single ADC reads the sum.

Architecture (a) gives four independent $V_{\text{out}}$ values, one per location — ideal for detecting hot/cold spots. Architecture (b) gives only the average temperature (sum of resistances), masking spatial variation but using fewer ADC channels. For most applications (a) is preferred; for a simple "average is the chamber too hot?" check, (b) is cheaper. The choice illustrates how the divider equation's linearity in $R_2$ (with $R_1$ fixed) gives different system-level architectures depending on what physical quantity needs measurement.

A practical compromise often used in industrial monitoring: a multiplexer chip selects one of $N$ sensor dividers in turn, routing each in sequence to a single high-precision ADC. This gives full per-location resolution (like architecture (a)) but at the cost of only a single ADC channel (cost of architecture (b)) and a small time-multiplexing delay. The same principle underlies modern temperature monitoring in data centres, where dozens of thermistor dividers per server rack feed back to a central controller via I²C or similar serial protocols.

Potential Dividers

Potential Dividers

The Basic Potential Divider — Derivation

Quick Sanity Checks

Worked Example 1 — Simple Ratio

Why Use a Divider at All?

Worked Example 2 — Designing a 5 V Logic Rail

The Loading Effect (the Open-Circuit Caveat)

Worked Example 3 — Loading the 12 V/8 V Divider

Rule of Thumb

The Potentiometer (Three-Terminal Variable Divider)

Linear Pot — Voltage vs Wiper Position

Sensor Circuits — Resistance-to-Voltage Conversion

Temperature: NTC Thermistor

Worked Example 4 — Fire-Alarm Thermistor

Light: the LDR

Worked Example 5 — Streetlamp Controller

Strain Gauge

Worked Example 6 — Strain Gauge in a Divider

Worked Example 7 — Why Headphones Break a Divider

Worked Example 8 — Designing a Frost Alarm

Worked Example 9 — Comparing Series and Parallel Sensor Arrays

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