Use of inductor for filtering
why they use specifically 4.7uH, any mathematical calculation is there


The main filtering is performed on the voltage across the motor with respect to the 48 volt power source (the input). Forget about it being a pi filter for now and ignore the capacitors at the power supply side on the left. The total capacitance across the motor is 445 uF and, with the 4.7 uH inductor, the resonant frequency can be calculated by: -

$$F = \dfrac{1}{2\pi\sqrt{LC}}$$

Plug in the numbers and it equals 3.48 kHz. 

The schematic also tells us what the saturation current is for the inductor. It says 78 amps so this circuit is intended for high current applications and an estimate of the equivalent load presented by the motor at full speed might be about 2 ohms. I'm basing this on a full-load current of ~26 amps i.e. one-third of the stated inductor saturation current. This is just an estimate of course.

And here's where I see a problem. If I calculated the quality factor of this circuit assuming a 2 ohm load I get a value of nearly 20. This is based on: -

$$Q = R\cdot\sqrt{\dfrac{C}{L}}$$

This level of Q worries me because it means that if a step input of 48 volts were applied to the input (left hand side) then the resulting transient response across the motor and capacitors could peak at 92.3 volts. See this simulation: -



Interactive filter link.

In other words this is too close the the voltage limits on the capacitors for my liking. I would want the peak voltage to be no more that 70% of the capacitor voltage rating. I am also concerned about the ripple current ratings for the capacitors too. 

These are not mentioned but ripple current can be estimated based on the light load scenario where a step input response could produce a long decaying sinewave across the capacitors of 96 volts p-p. This is an RMS voltage of 34 volts at a frequency of 3.48 kHz. At 3.48 kHz the overall capacitive reactance is 0.1 ohms. The capacitors will of course have an ESR of maybe 0.5 ohms so, if 0.5 ohms is used as the limiting factor on current, the RMS current could be 68 amps. This would concern me if the capacitors were chosen without respecting the potential ripple current.

As for the input capacitors across the 48 volt rail (forming the rest of the pi filter) I would regard these as noise suppressors. Any commutation noise that arises from the motor that isn't half beaten to death by the 445 uF would be further reduced by the input capacitors.

On the face of it I'd be concerned with this as a design (reading between the lines).