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This is how it works: the circuit is assumed or approximated to behave like a simple RC integrator.
The time constant will then be: τ=RC
The bandwidth is then BW=1/(2*π*RC)=1/(2*π*τ)
On the other hand, the rise time for this simple RC circuit is calculated starting from the equation of the voltage across the cap when a step impulse is applied at the input:
u(t)=U*(1-exp(-t/τ))
For u(t)=0.1U you get 0.1U=U*(1-exp(-t10/τ)), where t10 is the time when the voltage reaches 10% of the final value.
After simplification
0.9=exp(-t10/τ)
Applying the natural log to the equation you get:
ln0.9=-t10/τ or t10=-τ*ln0.9
Similarly, for u(t)=0.9U, we have t90=-τ*ln0.1
The rise time is then:
tr=t90-t10=τ*(ln0.9-ln0.1)≈2.2*τ or τ=tr/2.2
Substitute this value of τ into the expression for bandwidth:
BW=1/(2*π*tr/(2.2))=2.2/(2*π*tr)≈0.35/tr
One additional comment:
The mentioned formula can be used with good accuracy also for higher order systems when there is a dominant pole which is far below the other poles.
This, for example, is true for universal-compensated opamps.
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