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In my opinion, speed means how long the opamp transient response to a step input impulse.
In frequency domain, wider gain bandwidth means faster speed in time domain.
the bandwidth is taken into consideration because it determines how fast the comparator/op amp can switch.... also the main idea is that opamp has more stages than the comparators normally used and hence has lesser bandwidth and also takes more space on the IC....
speed of the comparator depends on how much current you give the comparator so that the capacitors at the output of the opamp can charge/discharge as soon as possible, this is also a measure of the bandwidth of the opamp.
Speed for a comparator would mean its reponce time to change in input.
One has to take care of what responce time you need for what change in input step. The responce time would vary for a 1V step input and 5mV step input. One has to design depending on the step voltage input requirment.
Speed of opam would be time it take to give the desired ouput for a input change.Or if the oupput is disturbed , how much rime it take to restore to its original value. Limitied by 3dB bandwidth in closed loop.
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