Duty Cycle Maths

Learning Goals 5 min

1. Convert an analogWrite value (0–255) into its duty cycle as a percentage, and into the average voltage on the pin.
2. Calculate the period and frequency of a PWM signal — and explain why the Arduino's default 490 Hz looks steady to your eye but not to a fast camera.
3. Predict how bright an LED will look at a given duty cycle, knowing that the eye sees brightness on a curve, not a line.

Warm-Up 10 min

Yesterday you typed analogWrite(LED, 128) and called it "half brightness". Today we ask which half.

Pencil puzzle

Imagine a pin switching cleanly between 0 V and 5 V. In every 1 ms window, the pin spends:

• 0.25 ms HIGH (at 5 V),
• 0.75 ms LOW (at 0 V).

What's the average voltage across that 1 ms window?

New Concept · The duty-cycle formulas 20 min

Formula 1 — analogWrite value → duty percentage

The analogWrite function takes an 8-bit value. 0 is always off, 255 is always on. Anything in between is a proportional duty cycle:

duty% = (value / 255) × 100

Examples:

Formula 2 — duty percentage → average voltage

The pin only ever swings between 0 V and 5 V (or 3.3 V if you're on a 3.3 V board). The average voltage over many cycles is just the duty cycle times the supply:

V_avg = duty% × V_supply
      = (value / 255) × 5 V       on a 5 V UNO

Formula 3 — period and frequency

The PWM signal repeats. Period (T) is the length of one complete on-then-off cycle. Frequency (f) is how many cycles happen per second. They're reciprocals:

f = 1 / T          (Hz = cycles per second)
T = 1 / f          (seconds per cycle)

On most UNO pins the PWM frequency is about 490 Hz. So:

• One period T = 1 / 490 ≈ 2.04 ms.
• At 50% duty, the pin is HIGH for ~1.02 ms then LOW for ~1.02 ms.
• At 25% duty, the pin is HIGH for ~0.51 ms then LOW for ~1.53 ms.

On D5 and D6 the frequency is roughly 980 Hz — twice as fast, period ~1.02 ms. That's a quirk of the AVR chip's timers and won't matter for LEDs, but it can matter for motors and audio.

Formula 4 (sneaky) — perceived brightness ≠ duty cycle

Your eye is more sensitive at low light than at high light — a real-world "volume curve" for vision. So an LED at 50% duty doesn't look half as bright as one at 100% — it looks closer to 70%. The technical name is gamma correction, with γ ≈ 2.2 for most displays:

perceived = (duty)^(1/2.2)
           ≈ (duty)^0.45

So a duty cycle of 0.5 (50%) looks like roughly 0.5^0.45 ≈ 0.73 → about 73% perceived brightness. The full table:

The takeaway: when you fade an LED in even analogWrite steps (0, 32, 64, ...), the fade looks fast at the start, slow at the end. To get a fade that looks smooth, the steps need to be small near zero and large near 255. We'll exploit this in L02-04.

Worked Example · Calculate, then check on the board 20 min

Step 1 — pick a target voltage

Goal: drive the pin to an average of 2.0 V. What analogWrite value do you need on a 5 V UNO?

Rearrange Formula 2 for value:

V_avg = (value / 255) × 5 V
value  = (V_avg / 5) × 255
       = (2 / 5) × 255 = 0.4 × 255 = 102

So analogWrite(LED, 102) targets 2.0 V average. Round to the nearest whole number — 102.

Step 2 — write the sketch and check with a multimeter

// L02-03: target a specific average voltage
const int LED = 9;

void setup() {
  pinMode(LED, OUTPUT);
  analogWrite(LED, 102);   // ~2 V average on 5 V UNO
}

void loop() { }

Upload. Put your multimeter probes between pin 9 and GND on the breadboard, set to DC volts (200 V range or auto). The reading should be very close to 2.0 V — typically 1.95–2.05 V depending on the meter's averaging.

Step 3 — try the other classic targets

Real-world note

Your multimeter might be a tiny bit off (say, 0.95 V when you expected 1.00 V). That's usually the meter's input impedance or the PWM ripple averaging. Don't fight it — the maths is right.

Try It Yourself 20 min

Mini-Challenge · The phantom voltmeter 15 min

Without uploading or using a multimeter, predict the steady multimeter reading (DC volts, 5 V UNO) for the following sketch.

const int LED = 9;

void setup() {
  pinMode(LED, OUTPUT);
}

void loop() {
  analogWrite(LED, 85);
  delay(2000);
  analogWrite(LED, 170);
  delay(2000);
}

Predict:

1. What does the multimeter read during the first 2-second window?
2. What does it read during the second 2-second window?
3. If you set the meter to average over 4 seconds, what overall reading would you expect?

Recap 5 min

PWM is a percentage game. analogWrite value over 255 gives you the duty as a fraction; multiply by your supply voltage to get the average; multiply by 100 to read it as a percentage. The period is the inverse of the frequency, so 490 Hz means each cycle lasts roughly 2 ms. And your eye lies a bit — perceived brightness follows a γ ≈ 2.2 curve, which is why dim LEDs look brighter than the maths suggests. Tomorrow we put all this to work with a smooth LED fade.

Period (T)

The duration of one complete cycle of a repeating signal. Measured in seconds, milliseconds or microseconds.

Frequency (f)

How many cycles happen per second, measured in Hertz (Hz). f = 1 / T.

Average voltage

Duty fraction × supply voltage. For an UNO on 5 V: V_avg = (analogWrite_value / 255) × 5.

Gamma correction (γ ≈ 2.2)

The curve that maps linear duty cycle to perceived brightness. Half-duty looks like ~73% bright, not 50%.

Homework 5 min

Build a small reference card to keep next to your computer:

1. Draw a two-column table: column 1 = analogWrite value, column 2 = average voltage on 5 V UNO.
2. Fill in the rows for every multiple of 32 (0, 32, 64, 96, 128, 160, 192, 224, 255). Use the formula V_avg = (value / 255) × 5; round to 2 decimals.
3. Add a third column = perceived brightness as a percentage (use (duty)0.45 × 100, rounded to whole numbers).
4. Bring the card to L02-04 — we'll use it to plan a smooth fade tomorrow.

Bonus: on the back of the card, write down the formula for the time the pin spends HIGH in microseconds, given a PWM frequency f and an analogWrite value v. Hint: it's (v / 255) × (1,000,000 / f) µs.