I'm new user and hope that someone know the answer,

I have installed a 250 gal. water storage vessel and a booster set up including a 25 gal. pressure tank and a pump which has the max. capacity of 40 gpm and 60 psi. the problem is that when I set the pressure tank to 15/30 psi cut-in/cut-out the pump starts and the pressure rises quickly to 30 psi so that pressure switch turn it off then suddenly pressure falls to 15 psi although there is no water consumption in the house. Therefore, pump turn on and off quickly. I changed the set point of the pressure switch to 15/45 psi and then pump rises the pressure to 45psig but when it's turned off the pressure falls 15 psi.
I don't know why the pressure falls after turning off. One possibility could be the foot valve and the other could be the pressure tank. I checked the pressure tank and it has air so its bladder isn't damaged.
I would be glad to have your ideas.

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I don't know why the pressure falls after turning off. One possibility could be the foot valve

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BINGO! The check/foot valve would be the first place I would check, If the bladder were ruptured in your tank the pump would pump to pressure and cut off. However, the pump would kick on every time you drew a glass of water or
Flushed a toilet. Good luck, Tom

I checked the check valve it is not a spring type and installed at the pump suction, so, I want to install a spring type check valve at the outlet of the pump to act as quickly as possible. Please let me know your idea about that.

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so, I want to install a spring type check valve at the outlet of the pump to act as quickly as possible.

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No! If you install the check valve on the outlet (discharge side) of the pump you would have to prime it every time the pump shut down. The check valve installs on the suction line not the discharge line. Good luck, Tom