Two resistors in series provide a simple way to produce a fraction of an input voltage. The interesting question is what happens when something uses that output. A voltage divider is a circuit with assumptions, rather than a voltage setting that stays fixed under any conditions.

Start with the unloaded circuit

Place \(R_1\) between the input and output, and \(R_2\) between the output and ground. With no load connected, the same current flows through both resistors:

\[ I=\frac{V_{\mathrm{in}}}{R_1+R_2}. \]

The output is the voltage across \(R_2\). Using Ohm's law gives

\[ V_{\mathrm{out}}=IR_2 =V_{\mathrm{in}}\frac{R_2}{R_1+R_2}. \]

Equal resistances produce half the input voltage. Doubling both resistance values preserves that ratio while halving the current. These equations assume ideal resistors and an input source that maintains its stated voltage. Carnegie Mellon's circuit notes derive the divider relation from the series circuit.

A load changes the lower resistance

Connect a load \(R_L\) from the output to ground. It sits in parallel with \(R_2\), so the effective lower resistance becomes

\[ R_{\mathrm{lower}}=R_2\parallel R_L =\frac{R_2R_L}{R_2+R_L}. \]

Use this equivalent resistance in the divider formula. A finite load lowers the effective resistance, which lowers the output voltage for positive input voltage and positive resistances.

Here are calculated examples with a 5 V input and two 10 kΩ resistors:

Load Effective lower resistance Output voltage
No load 10 kΩ 2.500 V
100 kΩ 9.091 kΩ 2.381 V
10 kΩ 5 kΩ 1.667 V

These are predictions of the ideal circuit, not bench measurements. The interactive demonstration lets you vary the input, both resistors, and an optional resistive load. Compare the unloaded result with the loaded result before interpreting the current through \(R_1\).

INTERACTIVE EXPERIMENT

Explore a voltage divider

Ideal component model
Voltage divider circuitThe upper resistor connects input voltage to the output. The lower resistor connects the output to ground. An optional load runs in parallel with the lower resistor.VinR₁R₂VoutRLVin → R₁ → Vout → R₂ → ground
Output / input

The load, when connected, is in parallel with R₂.

Ideal unloaded output
Actual model output
Current through R₁
Resistor tolerance range

Calculated values, not measurements. The tolerance range varies R₁ and R₂; input voltage and any load remain fixed.

Nominal values have tolerances

A resistor marked 10 kΩ is a nominal value. A 5% tolerance allows its actual resistance to lie between 9.5 kΩ and 10.5 kΩ under the specified conditions.

For an unloaded divider with equal nominal resistances and both at 5% tolerance, a 5 V input can produce outputs from 2.375 V to 2.625 V. The lowest output pairs the highest \(R_1\) with the lowest \(R_2\); the highest output uses the opposite pair.

The tool evaluates those resistance extremes with the selected load held fixed. Its tolerance range is a worst-case interval, not a probability distribution or a statement that all values are equally likely.

Keep the model small and explicit

The loaded calculation is still only a few lines:

def divider(vin, r1, r2, load=None):
    lower = r2 if load is None else r2 * load / (r2 + load)
    current = vin / (r1 + lower)
    return current * lower, current

Use positive resistances in ohms and input voltage in volts. The model leaves out source resistance, temperature effects, resistor power limits, and loads that do not behave like a fixed resistor. A divider can help explain a sensor input or bias network; supplying a changing power load requires a fuller circuit analysis. Changing values in a browser reveals the consequences of these assumptions, but does not verify a physical circuit.

References

  1. Carnegie Mellon: Voltage divider