9.3, 10.3 - Potential dividers and sensors

Dividing a supply p.d.

Why can a divider give $8\,\mathrm{V}$ to a voltmeter but only $6\,\mathrm{V}$ after a load is connected? The voltage has not been used up. The load changes the circuit's resistance arrangement, so it changes the fraction of the supply appearing at the output.

A potential divider has two resistances in series across a supply. With no significant output current, the same current flows through both. Take $R_1$ as the upper resistor and $R_2$ as the lower resistor; measure the output from their junction to the lower supply terminal. The output is then the p.d. across $R_2$, rather than across the whole pair.

Using the series resistance and resistance definitions,

Combining them gives the useful fraction.

A real load bridges the same two output terminals as the lower resistor, so it is in parallel with that resistor. Replace the parallel pair before using the divider fraction. A high-resistance voltmeter has little effect; a low-resistance load can have a large one.

[DIAGRAM: asset_name: A divider with a connected load; asset_slug: circuits-loaded-divider; caption: The load is in parallel with the lower resistor, changing the divider fraction; description: A complete ideal 12-volt supply circuit has an upper 3.0-kilohm resistor and lower 6.0-kilohm resistor. A 6.0-kilohm load connects between the middle junction and the lower supply node, in parallel with the lower resistor. An ideal voltmeter bridges the same output nodes.]

Apply the fraction to a different pair.

Temperature-dependent voltage

A resistor whose resistance changes with temperature can make a divider's output respond to temperature. The thermistors in this course have a negative temperature coefficient: their resistance decreases as temperature rises.

This is different from the metallic filament you studied earlier, whose resistance increases as it becomes hotter. Do not transfer the filament explanation to a thermistor simply because both are components that warm up. The thermistor's semiconductor material gives a different resistance–temperature response.

[DIAGRAM: asset_name: Sensor position controls output direction; asset_slug: circuits-sensor-dividers; caption: A top thermistor can give an output that rises with temperature; a lower LDR can give an output that falls with light intensity; description: Two separate complete divider circuits are shown. Left: fixed supply, upper NTC thermistor and lower fixed resistor, with an ideal voltmeter across the lower resistor. Right: fixed supply, upper fixed resistor and lower LDR, with an ideal voltmeter across the LDR. Each output's two terminals are marked and the LDR's light arrows point towards it.]

Put a thermistor $R_T$ above a fixed resistor $R$, and measure the output across the fixed resistor. Heating reduces $R_T$, reducing the total series resistance. Current rises, so the p.d. $IR$ across the fixed resistor rises. Equivalently, $V_{\mathrm{out}}=V_sR/(R_T+R)$ has a decreasing denominator and a fixed numerator.