- beta Bessel = 1/2*sqrt(3) ==> c2/c1 ratio = 1.333
- beta Butterworth 1/2*sqrt(2) ==> c2/c1 ratio = 2
- formula transmission & phase
- T = 1/sqrt(1+p*p*p*p/(w0*w0*w0*w0)+(4*beta*beta-2)*p*p/(w0*w0)) with p=jw and w0=1/r*sqrt(c1*c2)
- phi = atan (-2p*beta/w0)/(1+p*p/w0*w0) with p=jw and w0=1/r*sqrt(c1*c2)
- formula polar diagram
- real part : 2*beta*p/w0 / denominator
- imaginary part : 1-p*p/(w0*w0) / denominator
- denominator : (1+p*p*p*p/(w0*w0*w0*w0)+(4*beta*beta-2)*p*p/(w0*w0)) with p=jw and w0=1/r*sqrt(c1*c2)
- formula step response
- y(t) = 1 - 1/(sqrt(1-beta*beta))*exp(-beta*w0*t)*sin(wd*t+phi) with wd=w0*sqrt(1-beta*beta)
- y(t) = 1 + (tau2*exp(-t/tau2)-tau1*exp(-t/tau1))/(tau1-tau2)