Comressor engaging and disengaging?

I have a 2003 Ford Taurus with the DOHC engine. When the AC is turned on my compressor seems to repeatedly engage and disengage.

I recently installed a new compressor and installed a new coil pack to try to fix it but it still didn't help. The AC blows still blows cold air just fine. It's just noticeable when I'm idling in park or at a stoplight. I don't want it to seize and f my car up lol.
Any ideas what I should check?

Added (1). I recharged to AC system when I replaced the compressor.

It's supposed to cycle. Are you having A/C problems?

My 2004 Focus did that too a few years ago. The problem was that it was getting low on freon. It still worked but not to full strength. You likely need to get the coolant topped off.

As the AC achieves the temp you've set it to it will cut the compressor and whenit goes above that temp it will restart. Normal AC action. If it's cooling properly then it's just doing it's job. In the spring when outside temp isn't that high cycling will be more frequent. In July and August it may never shut off.

That is completely NORMAL.