Rise Time - Bandwidth - Correlation

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A measurement 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

or

BW*tr≈0.35