How to control this system by using a single Opamp circuit?
1) You mean: one opamp-timer for all 20 outputs. Sure that is possible but then you would need extra circuitry to detect which switch was pressed and only enable the corresponding output. In the end that would probably be as complex as using 20 opamps.

2) The TIP120 is a darlington transistor so pushing 25 mA into its base is overkill ! The minimum Hfe is 1000 so if you would switch 10 A then only 10 mA would suffice. I would double R11 to 1 kΩ making life a bit easier for the LM324.

But if you want something more simple why not replace the opamp timers with a (I hate to say this, me being a very analog type of person :-) ) microcontroller ?

Or replace the TIP122 with a power NMOSFET and make something very simple like this:

I see your supply for the magnets is 48 V, this might be too much for the gate-source of the MOSFET. If so you could derive a lower voltage from that 48 V and use that for the timer part. The NMOSFET usually only needs 5 - 10 V to fully conduct. To adjust the 10 seconds there's a trimmer. If your 1 uF capacitors are accurate enough you might need only a singe resistor.







simulate this circuit &ndash; Schematic created using CircuitLab