Voltage and current dividers
Dividers split a voltage or current in predictable proportions. They appear everywhere, from sensor interfaces to biasing transistors.
The voltage divider
Two resistors in series across a voltage \(V_{in}\) divide it in proportion to their resistances. The output across \(R_2\) is \(V_{out} = V_{in}\frac{R_2}{R_1 + R_2}\). With \(R_1 = R_2\), the output is half the input. Voltage dividers set reference voltages, scale signals for measurement, and form the basis of potentiometers.
Potentiometers
A potentiometer is a resistor with a sliding contact, or wiper, forming an adjustable voltage divider. Turning the knob changes the ratio of the two resistances, so the wiper voltage varies smoothly from zero to the full input. Volume controls, joysticks, and position sensors use potentiometers; used with only two terminals, it becomes a variable resistor (rheostat).
Loading effects
The divider formula assumes nothing draws current from the output. Connecting a load \(R_L\) puts it in parallel with \(R_2\), lowering the effective resistance and the output voltage. To keep the error small, the load should be much larger than \(R_2\), typically at least ten times. Equivalently, the divider's Thevenin resistance, \(R_1 \parallel R_2\), should be small compared with the load.
The current divider
Current entering two parallel resistors divides inversely with resistance: \(I_1 = I_{in}\frac{R_2}{R_1 + R_2}\), so the smaller resistor carries more current. In terms of conductance, each branch takes a share proportional to its conductance: \(I_k = I_{in}\frac{G_k}{G_{total}}\). Ammeter shunts use this principle to measure large currents.
Sensors and dividers
Many sensors change resistance with a physical quantity: light-dependent resistors with light, thermistors with temperature, strain gauges with stretch. Placing such a sensor in a voltage divider converts the resistance change into a voltage that a microcontroller's analogue-to-digital converter can read.
The Wheatstone bridge
The Wheatstone bridge, popularised by Charles Wheatstone in 1843, places two voltage dividers side by side and measures the voltage between their midpoints. When the ratios match, R1/R2 = R3/R4, the bridge is balanced and that voltage is zero. Small changes in one resistor produce a measurable imbalance, making bridges highly sensitive for strain gauges and precision measurement.
Design considerations
Resistor values in dividers trade power consumption against loading: large values waste less current but are more sensitive to loading and noise; small values are stiff but draw more current. Tolerances matter too: with 5% resistors, a nominally 2.5 V divider output could be off by around a tenth of a volt, so precision applications use 1% or better.
Worked example
A 5 V microcontroller must read a 12 V battery, so its input must stay below 5 V. Choose \(R_1 = 20\) k\(\Omega\) and \(R_2 = 10\) k\(\Omega\): \(V_{out} = 12 \times \frac{10}{30} = 4\) V. The divider draws \(\frac{12}{30{,}000} = 0.4\) mA. If the input has an equivalent resistance of 1 M\(\Omega\), it lowers \(R_2\) to about 9.9 k\(\Omega\), changing the reading by under 1%, which is acceptable.
Key takeaways
A voltage divider gives V_out = V_in R2 / (R1 + R2); potentiometers make it adjustable. Loads in parallel with R2 lower the output, so keep loads much larger than R2, or the divider's Thevenin resistance small. A current divider splits current inversely with resistance: I1 = I_in R2 / (R1 + R2), proportional to conductance. Resistive sensors in dividers produce readable voltages. The Wheatstone bridge (1843) balances when R1/R2 = R3/R4 and detects tiny changes. Divider design trades power against loading, noise, and tolerance.
Flashcard
V_out = V_in R2 / (R1 + R2)
Check your understanding
A voltage divider has R1 = 30 kilohms and R2 = 10 kilohms across 8 V. What is the unloaded output across R2?