Q) Why we only consider 3db  below the maximum gain in finding the bandwidth of a filter or amplifier? Why not 2db or 1 db?

Reply:-   

                         I'll start explanation with a term "Half power point".

Half Power point or cutoff freq. :-  A point in the frequency response of amplifier or filter where the power is half the maximum power...P(hp) = P(max)/2

 We can say that after crossing the half power point(should consider crossing from top to bottom since max power is on top) the response dies out or attenuate very rapidly and will not contain the useful information. So, the region of frequency response within half power points is important for us..and thus we find bandwidth at these points.

We know that Power (P) = i*i* R (or) V*V*R   Where i= Current , V= Voltage.

At half power points we already seen that P(hp) = Pmax/2 ---> (1)

So     i(hp) = imax/sqrt(2) = 0.707 imax ,  V(hp) = Vmax/ sqrt(2) = 0.707 Vmax since p is proportional to square of current or voltage.

Now consider the power diff between max power and half power point in db scale.

      P in db = 10 log( p(hp)/pmax ) = 10 log (0.5)  =  -3 db (approx)----->(2) from (1)

 So, the conclusion is that for any amplifier or filter we must first have an idea about its bandwidth (range of frequencies at which gain is constant),. To get bandwidth we need two extreme points(half power points) after which response attenuates rapidly. So indirectly to get bandwidth we need to know the half power points(also called cutoff frequencies). Calculation of half power points may not be easy all the time. So to find bandwidth easier we use the property of half power point that is shown in above equation (2) i.e the power at the half power point is 3db less than the max power. Since we use semilog graphs usually to draw freq. responses it will be easy to just mark the half power points 3 db down to the max value.So, this

is used by default in textbooks to find bandwidth. 3db is just convention which is best suited for many cases. There are few cases where the concept of 3db fails.

For example Butterworth filter  due to ripple in pass band).

This is why 3db came into picture...

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