The Voltage Divider, and What Loading Does to It

You want a microcontroller to measure a 12 V battery, but its input only reads 0 to 3.3 V. A microcontroller is a small computer on one chip, like the one on an Arduino board. Its ADC (analog-to-digital converter) is the part that measures a voltage and turns it into a number.

Two resistors solve this. Their formula has one catch: it only holds while nothing takes current from the point between them.

The ideal divider

A voltage divider is two resistors in series across a supply. R₁ runs from the supply to the middle point, and R₂ from the middle point to ground. The output is the middle point, measured from ground. V_in is the voltage you feed in, and V_out is the voltage at the output:

V_out = V_in × R₂ / (R₁ + R₂)

Working it out takes one line. The resistors are in series, so the same current flows through both (Kirchhoff's current law, which the next lesson covers in full):

I = V_in / (R₁ + R₂)

That current through R₂ makes a voltage across it, and that voltage is the output:

V_out = I × R₂ = V_in × R₂ / (R₁ + R₂)

So the output is the input scaled by R₂'s share of the total resistance. Equal resistors give half the input, whatever their value: 1 kΩ and 1 kΩ divide by two, and so do 1 MΩ and 1 MΩ.

A potentiometer is a divider you can adjust: a knob slides the middle point along one resistor. The volume knob on an electric guitar is one.

What happens when you connect something

A load is whatever you connect to the output to use its voltage: an ADC input, a sensor board, a small motor. With nothing connected, the divider is unloaded. In the instrument below, the Load switch starts off and reads "None", and the formula holds. Switch it on and watch the output drop, and the formula at the top of the instrument change to the loaded form.

With the starting values (9 V, two 10 kΩ resistors and a 10 kΩ load), the output drops from 4.5 V to 3 V. Nothing is broken. The load is another resistor, and connecting it across R₂ puts it in parallel with R₂. The lower half of the divider now has less resistance than the R₂ you drew.

R_load is the load's resistance, and ∥ means "in parallel with", so R₂ ∥ R_load is (R₂ × R_load) / (R₂ + R_load). The instrument writes it as R2||RL. The loaded form is:

V_out = V_in × (R₂ ∥ R_load) / (R₁ + (R₂ ∥ R_load))

Now try other values with the same ratio. A 10 kΩ / 10 kΩ divider and a 100 Ω / 100 Ω divider both give 4.5 V unloaded. Under a 10 kΩ load, the first drops to 3 V, a third lower. The second only drops to 4.48 V.

The ratio of R₁ to R₂ sets the voltage. Their size sets how much it drops under a load: a divider of smaller resistors drops less. A divider that holds its voltage well under a load is called stiff.

At 9 V, the "Wasted in the divider" readout shows 4.05 mW for the 10 kΩ pair and 405 mW for the 100 Ω pair.

Output impedance: one number for the drop

The instrument calls the number that sets the drop output impedance. Impedance is the general word for how much something opposes current. With DC (direct current, which flows one way and holds steady, as from a battery) and only resistors, impedance is just resistance. In this lesson you can read it as resistance.

Output impedance acts like a resistor hidden inside the output. For a divider it is R₁ in parallel with R₂, because seen from the output, each one leads to a fixed voltage (the supply or ground). So a 10 kΩ / 10 kΩ divider has 5 kΩ. The loaded output is the unloaded output times R_load / (Z_out + R_load), where Z_out is the output impedance. A 10 kΩ load gives 4.5 V × 10 / 15 = 3 V, as the instrument showed. A 5 kΩ load halves the output, and a 500 kΩ load lowers it by about 1 %.

Every input has an input impedance: the resistance it presents to whatever feeds it. A high input impedance takes little current. One circuit can feed another well when the second one's input impedance is much larger than the first one's output impedance, by the same ten-times rule. A multimeter's 10 MΩ is two thousand times a 5 kΩ output, so it barely changes the reading. A guitar's pickup is the part under the strings that turns their movement into a voltage. It has a high output impedance, so guitar amplifiers typically have an input impedance of about 1 MΩ.

Where dividers are the right answer

Reading a voltage that is too high for your microcontroller. A car's 12 V battery is not always at 12 V. It reads about 11 V when it is nearly flat and powering a load, and up to about 14.4 V while the engine charges it (typical figures). The car's own electronics usually read that voltage through a divider like this one. To try it yourself, feed the divider from a bench supply (a power supply with a knob that sets its output voltage) set anywhere from 11 V to 14.4 V, not from a car battery. The divider must keep even 14.4 V under 3.3 V. Two standard E12 values, R₁ = 10 kΩ and R₂ = 2.7 kΩ, do it:

• At 14.4 V: 14.4 V × 2.7 / 12.7 = 3.06 V
• At 11 V: 11 V × 2.7 / 12.7 = 2.34 V
• The output stays under 3.3 V up to 3.3 V × 12.7 / 2.7 = 15.5 V.

The ADC takes almost no current, but it needs a low output impedance behind it. Each reading, it fills a tiny capacitor inside it (a part that stores a little charge), and a high impedance fills it too slowly. The datasheet for the ATmega328P, the microcontroller on an Arduino Uno, asks for 10 kΩ or less. (A datasheet is the maker's document that lists what a part does and its limits.) The Uno runs this chip at 5 V, so its input reads up to 5 V. The 10 kΩ limit is the same when the chip runs at 3.3 V. This divider has 10 kΩ ∥ 2.7 kΩ = 2.13 kΩ, well under that.

Setting a reference or a threshold. A reference is a fixed voltage that another circuit compares against, and a threshold is the voltage at which something switches. A battery-low light can use a comparator, a chip that tells you which of two voltages is higher, to compare a divided-down battery voltage with a reference. A comparator's inputs take very little current, so they do not load the divider.

Reading a resistive sensor. A resistive sensor changes its resistance with what it measures. A thermistor is a resistor whose value changes with temperature, as in many digital thermostats. With a thermistor as R₁ and a fixed resistor as R₂, the output follows the temperature. Module 10 starts here.

Where a divider is the wrong answer

Powering anything. A common beginner mistake: a divider from 12 V to 5 V to run a small board, such as a Wi-Fi board. Say it takes 100 mA. It cannot work. To hold up under 100 mA, the resistors would have to be around 10 Ω. They would turn several watts into heat all the time, and the output would still move whenever the board's current draw (the current it takes) changes. That is the job of a regulator, a part that holds its output voltage steady whatever the load takes. Module 05 covers regulators. A car's USB charger makes 5 V from 12 V with a regulator, not a divider.

Anything whose current changes. Even at low currents, if the load's current draw changes, your reference voltage moves with it.

Try to predict before you drag

Before you touch the controls above, decide what the output will be, then check. A 9 V supply, R₁ = 10 kΩ, R₂ = 4.7 kΩ, with a 10 kΩ load.

Unloaded, the output would be 9 V × 4.7 / 14.7 = 2.88 V. The 10 kΩ load in parallel with 4.7 kΩ gives 3.20 kΩ, so the real output is 9 V × 3.20 / 13.20 = 2.18 V. The Error readout shows it is 24.2 % low. If that divider fed an ADC reading a battery, the reading would be about a quarter low.

Next: Kirchhoff's laws, for circuits that the series and parallel rules cannot reduce.