As stated before, a Pulse Width Modulation (PWM) signal is a square signal with fixed frequency but variable width. On the image we see an example. On each row there is one of such signals with a different duty cycle
the period portion in which the signal is High). Taking the mean signal over a period, we get the red line that correspond to the duty cycle with respect to the signal maximum.
Electronically "taking the mean of a signal" can be translated to "passing it to a low-pass filter", as seen on the preceding step.
How does the Arduino generate a PWM signal? There is a really good tutorial about PWM here:http://arduino.cc/en/Tutorial/SecretsOfArduinoPWM
We will see just the points that are needed for this project.
In the ATMega328P there are three timers that can be used to generate PWM signals, each one of those has different characteristics that you can use. For each timer correspond two registers called Output Compare Registers A/B
(OCRnx) that are used to set the signal duty cycle.
As for the ADC there is a prescaler (see image), that slows down the main clock to have a precise control of the PWM frequency. The slowed down clock is fed to a counter that increments a Timer/Counter Register
(TCNTn). This register is continuously compared to the OCRnx, when they are equal a signal is sent to a Waveform Generator that generate a pulse on the output pin. So the trick is setting the OCRnx register to some value to change the mean value of the signal.
If we want a 5 V signal (maximum) we must set a 100% duty cycle or a 255 in the OCRnx (maximum for a 8-bit number), while if we want a 0.5 V signal we must set a 10% duty cycle or a 25 in the OCRnx.
Since the clock has to fill the TCNTn register before starting from the beginning for a new pulse the output frequency of the PWM is:
f = (Main clock) / prescaler / (TCNTn maximum)
for the Timer 0 and 2 (8-bit) with no prescaler it will be: 16 MHz / 256 = 62.5 KHz while for Timer 1 (16-bit) it will be 16 MHz / 65536 = 244 Hz.
I decided to use the Timer number 2 because
Timer 0 is used internally by the Arduino IDE for functions such as millis();
Timer 1 has an output frequency too slow because it is a 16-bit timer.
In the ATMega328P there are different kinds of operation mode of the timers, but what I wanted was the Fast PWM
one with no prescaling to get the maximum possible output frequency.