Robotics · Microcontrollers · Grade 8

PWM: brightness and sound

analogWrite(9, 64) — why does the LED glow at 25 % and what tune does the buzzer play?

The situation

The front light of a robot car should not be abrupt but fade smoothly on and off — and it should also sound a signal. But an Arduino pin knows only two states: 0 V or 5 V. How do we get “half” brightness? If you write analogWrite(9, 64), the LED glows at a quarter of its brightness — but on the oscilloscope there is no “quarter” voltage on the pin: only rapid switching on and off. Why does the eye take this for smooth light? Why does brightness that rises in equal steps from 0 to 255 not look even to the eye? And where does the 440 in tone(8, 440) come from?

Topic

PWM: analogWrite, brightness and sound. An LED fades smoothly, a buzzer plays a tune: duty cycle and frequency

Key ideas

Duty cycle
The eye and gamma
tone() — frequency

Formulas

D = \frac{v}{255}, T = \frac{1}{f}, t_{on} = D \cdot T

PWM (pulse-width modulation): `analogWrite(pin, v)` switches the pin on and off at frequency $f$ , HIGH for the fraction $D$ of the period. Uno: on pin 9 the frequency is 490 Hz → $T$ = 2.04 ms; $v$ = 64 → $D$ = 25 %, $t_{on}$ = 0.51 ms.

U_{avg} = D \cdot 5 V, I_{avg} = D \cdot I_{on}, I_{on} = \frac{5 - U_{L E D}}{R + R_{p i n}}

The average voltage and current are proportional to the duty cycle. The light of an LED is proportional to the average current — with 220 Ω, $I_{on}$ ≈ 12.7 mA.

L^{*} = 116 \sqrt[3]{Y} - 16

The eye (CIE 1976 $L^{*}$ ): perceived lightness $L^{*}$ (0–100) is close to the cube root of the light $Y$ (0–1). $Y$ = 10 % → $L^{*}$ ≈ 38, $Y$ = 50 % → $L^{*}$ ≈ 76: one tenth of the light looks to the eye like more than a third.

v_{k} = 255 {(\frac{k}{N})}^{γ}, γ \approx 2.2 \dots 3

Gamma correction: to make the steps look equal to the eye we raise $v$ to a power — small steps at the start, large ones at the end.

f_{n} = 440 \cdot 2^{n / 12}

Notes are frequencies: La (A4) = 440 Hz, each semitone is $\sqrt[12]{2}$ ≈ 1.0595 times higher. Do (C4) = 262, Re 294, Mi 330, Fa 349, Sol 392, La 440, Si 494, Do (C5) 523 Hz. `tone(pin, f)` gives a wave of that frequency with a 50 % duty cycle.

Theory

PWM: an “average” from two states

A microcontroller pin is digital: 0 V or 5 V, nothing in between. But `analogWrite(pin, v)` switches the pin on and off hundreds of times a second: $v$ = 255 — on all the time, 128 — half the time, 64 — a quarter of the time. This is PWM, pulse-width modulation; the fraction of time spent on, $D = v / 255$ , is called the duty cycle. For the circuit it is an “average” voltage of $5 D$ volts: the average current of the LED and its brightness are proportional to $D$ . The eye does not notice flicker at 490 Hz — it “collects” light over roughly 1/50 s: flicker faster than 50–60 Hz looks like steady light.

Which pins, which frequency

On the Uno, PWM exists only on the pins marked ~: 3, 5, 6, 9, 10, 11 — they are driven by three timers: Timer0 — 5 and 6 (≈ 980 Hz; this timer also counts `millis()`), Timer1 — 9 and 10 (490 Hz), Timer2 — 3 and 11 (490 Hz). The frequency does not affect the brightness — only $D$ matters; at 980 Hz the period is half as long. On other pins (13 too) `analogWrite` is a plain switch: $v < 128$ → LOW, $v \geq 128$ → HIGH. $v$ = 0 and $v$ = 255 on any pin give exactly LOW / HIGH — like `digitalWrite`.

The eye is not linear

In the Fade program $v$ grows evenly from 0 to 255, but the LED brightens very fast at the start and then hardly changes. The reason is the eye: perception is proportional not to the light itself but roughly to its cube root (the CIE 1976 formula $L^{*} = 116 \sqrt[3]{Y} - 16$ , on which colour science and screens are based). 10 % of the light looks to the eye like ≈ 38 %, 50 % like ≈ 76 %. To make the steps look equal we need gamma correction: $v_{k} = 255 {(k / N)}^{γ}$ , $γ$ ≈ 2.2–3 — small steps at the start, large ones at the end. Every screen and every LED lamp “dimmer” has such a table.

Reading the oscilloscope trace

The oscilloscope draws the pin voltage against time: 1 ms/div — 10 ms across the screen. At 490 Hz the period $T = 1 / 490$ ≈ 2.04 ms — about 5 periods on the screen; at 980 Hz about 10. The width of the on pulse is $t_{on} = D \cdot T$ : 0.51 ms at $v$ = 64, 1.53 ms at $v$ = 191. The amplitude is always 5 V — PWM changes not the voltage but the time. This is how PWM can be told apart from a true analog voltage (a DAC): the Uno has no DAC.

tone(): sound is PWM too

A passive piezo buzzer (a piezo disc) does not sound by itself — it has to be given vibrations: `tone(pin, f)` switches the pin on and off at frequency $f$ with a 50 % duty cycle (an active buzzer, in contrast, squeals by itself at one fixed tone when given 5 V). In music every note is a frequency: La (A4) 440 Hz, an octave — twice as high, a semitone — $\sqrt[12]{2}$ times. `tone(pin, f, ms)` stops by itself after the given time and does not hold up the program; `noTone(pin)` switches it off. On the Uno only one tone can play at a time, and it takes over Timer2 — PWM on pins 3 and 11 stops meanwhile. A piezo buzzer draws almost no current; a 16 Ω magnetic buzzer, however, cannot be connected straight to a pin: the current is $5 / (16 + 25)$ ≈ 120 mA — three times the 40 mA limit, and the chip burns out (a transistor or 100 Ω is needed).

About the model

The oscilloscope trace is drawn ideally from the duty cycle and the frequency (sharp edges; on a real pin the edges take a few nanoseconds because of ≈ 25 Ω and capacitance). The light of the LED is taken as proportional to the average current (which is true in the working range of an LED); the “seen by the eye” value is CIE $L^{*}$ : the eye is much more complicated (ambient light, adaptation), but the cube root is a good first approximation. The sound is a square wave from the computer speaker, at a low volume, and only if you switch it on; a real piezo buzzer sounds much louder, around 2–4 kHz (resonance). The heating of the chip is simplified as in `arduinoblink`.

Did you know?

The piezoelectric effect was discovered in 1880 by the brothers Jacques and Pierre Curie: if you squeeze a quartz crystal, a charge appears in it, and if you apply a voltage, it vibrates. The piezo disc in a buzzer, quartz watches and ultrasonic sensors all work on this effect.
The note A at 440 Hz is the international standard: ISO adopted it as a recommendation in 1955, and the current ISO 16 standard dates from 1975. Before that orchestras were tuned differently — 435 Hz in 19th-century Paris, in some places even higher than 450 Hz.

Inside the experiment

A real board in a hand-inked isometric drawing: turn it, place the parts, wire the circuit, write and upload the code — current and voltage are computed at once, and a wrong connection burns a part. Inside are missions, theory, a quiz and a guided tour.

More experiments in this section

Arduino Uno: the first program — Grade 7
Button and digital input — Grade 7
Analogue input and the Serial Monitor — Grade 7
millis() and multitasking — Grade 8