A possible inrush current limiting circuit could be the one in the attached image.
But please note that I am a beginner in electronics so this circuit may not be fully correct or it may not be 100% safe. So use it at your own risk.
Maybe other, more experienced and more knowledgeable, members of the forum can shed a light on this.
R2 along with C2 sets the turn-on time. The D1 zener ensures that Vgs will not be lower than -12V (PMOS) so the MOSFET will not get damaged.
I believe that the zener should be selected based on your MOSFET specs. If a Vgs of -12V is too big for your selected MOSFET then a smaller value zener should be used to ensure that Vgs is kept within the specified limits of the MOSFET, but in the same time ensuring that Vgs will have a proper value to fully turn on the MOSFET.
I think that the turn-on time can also be affected by the MOSFET used, increasing or decreasing based on the amount of gate charge of the selected MOSFET.
Also the selected MOSFET should be able to handle the power dissipation, especially during the transition period (from off to on).
Note that in this setup if the supply voltage (Vs) is decreased the turn-on time will increase.
But again, I am a beginner so this my not be 100% correct. Please correct me if I am wrong.
Thank you
