Please explain the relation between bandwidth and gain of op-amp.

I have learned that negative feedback could reduce gain and increase bandwidth of an amplifier. Since gain= Vout/Vin, I understand that less gain means higher Vin. The larger Vin, the greater bandwidth. (I made my own assumption here. Please correct me if I am wrong!)

However, this seemed contradictory with what I knew about the properties of an ideal operational amplifier (op-amp). The ideal op-amp should has infinite open-loop voltage gain and infinite bandwidth. How can infinite open-loop voltage gain has infinite bandwidth? Or this is why it is an ideal op-amp (impossible to be made).

This confuses me a lot. Thank you for your help!

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    $\begingroup$ Those two concepts are not a priori related. Their connection arrises from a particular circuit. Also, the equation you gave is to be understood as: given a Vin, what is the ratio Vout/Vin. $\endgroup$ – G. Bergeron Dec 11 '16 at 8:06

Less gain doesn't mean higher Vin. In a constant gain the ratio Vout/Vin remains the same. You are right an ideal op-amp should have an infinite gain and an infinite bandwidth but this does not happen in a real op-amp due to the output going down with increasing frequency. You can imagine the output going through an RC low-pass filter within the op-amp so the output starts to fall off at some stage as the frequency is increased. The gain thus decreases with frequency. This means if the op-amp gain is reduces as in negative feedback, you can have a large range of frequency where the gain remains constant, that is larger bandwidth.

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