There is something I can't really get about PWM. Hope someone can help clarify my mind.
Look at the picture attached.

I know that regardless of whether the supply for my FPGA/MCU is 5 V or 3.3 V, I want the low state to turn into 0 V across my fan, and the high state to turn into 12 V across it.

This is the question: If I have a duty cycle of 50%, is the voltage at the base of the transistor 2.5 (if 5V) and the voltage across the fan 6V?

 

Hello,

The fan has (roughly) 0v across it and then 12v across it. Since the wave is 50 percent duty cycle, that means the average voltage could be called 6v across the fan. But that's an average and that is provided the back emf does not last very long.

Average is often called the 'mean' and so the definition is:
(1/T)*integral(y(t),t,0,T)

where t is time and T is the period of the wave.

 

There is something I can't really get about PWM. Hope someone can help clarify my mind.
Look at the picture attached.

I know that regardless of whether the supply for my FPGA/MCU is 5 V or 3.3 V, I want the low state to turn into 0 V across my fan, and the high state to turn into 12 V across it.

This is the question: If I have a duty cycle of 50%, is the voltage at the base of the transistor 2.5 (if 5V) and the voltage across the fan 6V?

The output voltage from the MCU us either Hi or Low (no matter what the duty cycle may be). A 50% duty cycle means the output is Hi for half of the total on/off cycle of the PWM frequency.

 

This is the question: If I have a duty cycle of 50%, is the voltage at the base of the transistor 2.5 (if 5V) and the voltage across the fan 6V?

The thing to keep in mind is the average Current level changes across the coil. at a 50% voltage pulse, with an inductive device the current waveform is quite different from the voltage.
It appears as a mean level sawtooth.
Max.

 

PWM is the approx to the average over a time period. So, over a time period, a 50% PWM signal with yield an average voltage of 50% the source.

 

... I want the low state to turn into 0 V across my fan, and the high state to turn into 12 V across it.

It is doing exactly that.

This is the question: If I have a duty cycle of 50%, is the voltage at the base of the transistor 2.5 (if 5V) and the voltage across the fan 6V?

If by "voltage" you mean time-weighted-averaged (as @MrAl described), then yes. But if you watched the voltage on an oscilloscope, you'd see it does not linger at the average, it's either high or low. Changing the duty cycle changes the time-weighting of on versus off, but the on or off voltages are unchanged.