Voltage Dividers Explained

Learning Goals 5 min

By the end of this lesson you will be able to:

1. Write down the voltage-divider formula — Vout = Vin × R2 / (R1 + R2) — and explain what each symbol means in plain English.
2. Calculate the voltage at the junction (and the predicted analogRead value) for any combination of Vin and two resistor values, without uploading.
3. Pick a sensible fixed resistor for a given sensor by aiming for the divider to sit near the middle of the 0–5 V range at "normal" conditions — the rule that makes 10 kΩ such a common default for hobby LDRs.

Warm-Up 10 min

You wired a voltage divider in L01-36 — LDR on top, 10 kΩ on the bottom, A0 in the middle — and trusted the readings. Today you open the hood and find out why the numbers come out the way they do. The same maths will let you pick the right "partner resistor" for any future sensor whose resistance changes — temperature probes, soil moisture, force, even your own fingertip's resistance.

Quick-fire puzzle

Imagine two identical 1 kΩ resistors connected end-to-end, with 5 V across the whole pair and your voltmeter touching the middle.

1. What voltage do you think the meter reads at the middle? Why?
2. Now replace the top resistor with a 9 kΩ. (Bottom is still 1 kΩ.) Does the middle voltage rise or fall? Roughly to what?
3. What would the middle read if the top resistor were replaced with a wire (zero ohms)?

New Concept — the divider formula 20 min

The setup — two resistors, one junction

Every voltage divider in this course has the same shape: a top resistor (R1) and a bottom resistor (R2) in series, with the supply voltage (Vin, usually 5 V on a Uno) across the pair and GND at the bottom. The output voltage (Vout) appears at the junction between them — exactly the point you tap with the A0 wire.

The formula

Here it is, the one equation you need:

Vout = Vin × R2 / (R1 + R2)

In words: the output voltage is the input voltage times R2 over the sum of both resistors. R2 is in the numerator because Vout is measured across R2 — the more of the total resistance R2 represents, the more of the total voltage drops across it.

Three sanity-check cases

Plug in extreme values and check the formula matches your intuition. These three checks confirm the formula is correct without any algebra:

• R1 = R2: Vout = 5 × R / (R + R) = 5 × 1/2 = 2.5 V. ✓ Matches the warm-up's "two equal resistors = halve the voltage".
• R1 = 0 (wire): Vout = 5 × R2 / (0 + R2) = 5 V. ✓ The full supply appears at the junction because R1 drops nothing.
• R2 = 0 (wire to GND): Vout = 5 × 0 / (R1 + 0) = 0 V. ✓ The junction is wired to GND through R2, so it sits at 0 V.

From V_out to analogRead

The Arduino's ADC turns 0–5 V into the integer 0–1023, linearly. So once you have Vout, you can predict the analogRead value:

analogRead value ≈ Vout × 1023 / 5

Or, combined with the divider formula, in one step:

analogRead value ≈ 1023 × R2 / (R1 + R2)

The 1023 replaces the 5 V because the maximum reading replaces the maximum voltage. Same shape, different units.

Why the partner resistor matters so much

For a varying-resistance sensor (LDR, thermistor, etc.) the goal is to pick the fixed resistor so the junction voltage swings across as much of the 0–5 V range as possible across the sensor's working range. If the fixed resistor is way too small or way too big, one of the two will always "win" and the junction will barely move.

• R2 way smaller than R1: the junction sits near 0 V at almost every R1 value. Tiny readings, tiny swings — you lose all the sensor's range to noise.
• R2 way bigger than R1: the junction sits near 5 V at almost every R1 value. Huge readings, tiny swings — same problem at the other end.
• R2 ≈ the "middle" value of R1: the junction swings fully as R1 changes — the sensor's range is mapped onto the ADC's range. Maximum sensitivity, easiest to set thresholds.

For a hobby LDR ranging from about 1 kΩ (bright) to about 100 kΩ (dark), a 10 kΩ fixed resistor sits in the geometric middle. That's why 10 kΩ is the universal default — it's not magic, it's just well-matched to the part.

Why it matters

Every analog sensor in the rest of this syllabus — thermistors, force-sensitive resistors, soil-moisture probes, even some MEMS gas sensors — is a "resistor that changes with something". They all need a partner resistor and they all give a junction voltage you read with analogRead. Master the formula once and you can wire and predict any of them. Skip it and you'll be re-soldering resistors all year trying to figure out why "the readings just don't change enough".

Worked Example — predict, then verify 20 min

Same L01-36 wiring: LDR on top (R1, variable), partner resistor on bottom (R2, fixed), junction to A0. The plan: calculate the expected reading for several scenarios on paper, then upload a tiny print sketch and check whether the real chip agrees.

Step 1 — The print-only sketch

You'll keep uploading this exact sketch — only the resistors on the breadboard change.

// LDR + partner — print live ADC reading
void setup() {
  Serial.begin(9600);
}

void loop() {
  Serial.println(analogRead(A0));
  delay(200);
}

Step 2 — Predict three readings on paper

Take a typical kit LDR with these approximate values:

• Bright room (lamp shining directly): R_LDR ≈ 2 kΩ
• Normal room light: R_LDR ≈ 10 kΩ
• Covered (fingertip pressed on top): R_LDR ≈ 100 kΩ

With R2 = 10 kΩ (today's partner), what does the formula predict?

So with a 10 kΩ partner, your readings should swing roughly from ~90 (covered) through ~510 (normal) up to ~850 (bright) — using about 75% of the ADC's full range. That's healthy.

Step 3 — Upload and confirm

1. Build the L01-36 wiring with a 10 kΩ partner. Upload the print sketch.
2. Open the Monitor. Note the reading at normal room light — should be somewhere near 500.
3. Cover the LDR with your fingertip — reading should drop toward 100.
4. Shine your phone torch directly at it — reading should climb toward 900.
5. Your real numbers will differ from the predictions (every LDR is slightly different and your room isn't exactly "normal"), but they should be in the same ballpark.

Step 4 — Now break it deliberately — swap to 1 kΩ

Replace the 10 kΩ with a 1 kΩ resistor (the same colour-banded part used as series resistance for LEDs in earlier lessons would also work — anything 220 Ω to 1 kΩ). Recalculate before testing:

Whole range squeezed into roughly 10–341 — only a third of the ADC. Upload and verify: cover the LDR and the reading barely changes from "covered" to "normal" — you've lost most of the sensor's resolution. The fixed resistor is so small that the LDR dominates at every brightness, and the junction stays near GND.

Step 5 — Try the other extreme — 100 kΩ partner

Now the readings are squeezed into ~510–1000: the whole sensor range is pinned to the top of the ADC and "bright" vs "normal" become almost indistinguishable. R2 too big this time.

The takeaway

10 kΩ swings the reading across most of 0–1023. 1 kΩ pins it to the bottom; 100 kΩ pins it to the top. The "best" R2 is approximately the geometric mean of the sensor's minimum and maximum resistance — for a 2 kΩ to 100 kΩ swing, that's √(2 × 100) ≈ 14 kΩ. The nearest standard part is 10 kΩ or 15 kΩ; both will work well. This is the rule behind every "use a 10 k" instruction in every Arduino tutorial.

Try It Yourself — three calculations and a swap 15 min

Mini-Challenge — design from scratch 10 min

"Choose the right resistor for a brand-new sensor"

You're given a hypothetical force-sensitive resistor (FSR) with these specs from its datasheet:

• No force applied: resistance is essentially infinite (open circuit).
• Light press (a fingertip resting): about 30 kΩ.
• Hard press (a thumb pushing): about 1 kΩ.

Your task: design a voltage divider for it.

1. Pick a partner resistor (R2) value from the E12 series (1 k, 2.2 k, 3.3 k, 4.7 k, 6.8 k, 10 k, 15 k, 22 k, 33 k, 47 k, 68 k, 100 k) to maximise sensitivity in the press range (1 kΩ to 30 kΩ).
2. Calculate the analogRead values you'd expect at no-force, light-press, and hard-press.
3. Write down a Serial-Monitor-style table of your predictions.
4. Explain in one sentence why you chose that R2 — what would have gone wrong with R2 = 1 kΩ, or with R2 = 100 kΩ?

It works if:

• Your R2 choice is a real E12 value.
• Your "light press" and "hard press" predictions are at least 200 ADC units apart (so you can reliably tell them apart in code).
• You can defend your R2 against both extremes (too small / too big) in one sentence each.

No force:    1023 × 4.7 / (∞ + 4.7)   = ~0   (FSR open → junction at GND)
Light press: 1023 × 4.7 / (30 + 4.7)   = ~139
Hard press:  1023 × 4.7 / (1  + 4.7)   = ~843

Recap 5 min

Every analog sensor in this course follows the same circuit: a variable resistor on top (R1), a fixed partner resistor on bottom (R2), the supply (V_in) across the pair, the junction (V_out) tapped by the analog pin. The voltage divider formula Vout = Vin × R2 / (R1 + R2) tells you what voltage to expect; multiplying through by 1023/5 tells you what analogRead will return. Picking R2 around the geometric mean of the sensor's resistance range gives the biggest swing in the ADC and the most sensitive readings. 10 kΩ is the universal default because most hobby sensors are roughly in its working range — but now you know why, and how to pick something better when the default doesn't fit.

Voltage divider

Two resistors in series between V_in and GND, with the analog pin tapping the junction. The shape of every variable-resistor sensor circuit in this course.

R1 (top resistor)

The resistor connected between V_in and the junction. In sensor circuits, this is usually the variable part (the LDR, thermistor, FSR, etc.).

R2 (bottom resistor)

The resistor connected between the junction and GND. Usually the fixed "partner" — picked by the designer to match the sensor's working range.

V_out (junction voltage)

The voltage at the tap point, measured against GND. Vout = Vin × R2 / (R1 + R2).

Geometric mean rule

For a sensor whose resistance ranges from R_min to R_max, the best fixed partner resistor is approximately √(R_min × R_max). This places the junction near 2.5 V at the middle of the sensor's range, swinging across most of the ADC.

E12 standard values

The 12 resistor values per decade you can actually buy: 1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2 — multiplied by 1, 10, 100, 1 k, etc. The most-stocked subset of these in hobby kits is 1 k, 2.2 k, 4.7 k, 10 k, 22 k, 47 k, 100 k.

ADC input impedance

The (very large) effective resistance the Arduino's analog pin presents to the junction. For R values under 100 kΩ, you can ignore it; above that, the ADC starts to pull current and skew readings.

Homework 5 min

Five divider calculations on paper. No Arduino needed. For each row of the table, calculate V_out and the predicted analogRead. Show working for at least three of the five.

Also: a design reflection on paper.

• Of rows 1–5 above, which ones put V_out in the middle of the 0–5 V range (between 2 V and 3 V)? Which put it near 0 V? Which near 5 V? ____
• A real soil-moisture sensor reads about 20 kΩ when soil is dry and 1 kΩ when soil is wet. What single E12 R2 would you pick to make the readings span most of the ADC? Justify in two sentences. ____
• You bought an LED-strip controller that has a 100 kΩ "ambient light" input pin labelled "0–5 V". You want to drive it from your existing LDR + 10 kΩ divider. Does that work as-is? ____ (Hint: the controller's input impedance might be lower than the ADC's; check the L01-38 watch-out about ADC impedance.)
• Why is the divider formula symmetric in shape but not in meaning — i.e. why does the choice of "which resistor is R1 and which is R2" matter? ____

Bring back next class:

• Your completed five-row table on a notebook page, with working shown for three of them.
• Your four written reflection answers.
• No .ino file this time — today's homework is pure paper-and-pencil thinking.

Heads up for next class: L01-39 "Tilt Switches" introduces your first motion-aware sensor — a tiny tube with a metal ball inside that closes a circuit when it tips. Not a resistor sensor, no divider needed — just a digital read with INPUT_PULLUP. Compared to today, it'll feel like a holiday.