Lowpass Butterworth

Low-pass Butterworth filter

Description

The Lowpass Butterworth component models an ${n}^{\mathrm{th}}$ order low-pass filter with Butterworth characteristics. Unlike the Critical Damping filter design, the Butterworth filter retains its shape, even at high order. The implemented filter is a series of second order filters and a first order filter. The transfer function component is

$\frac{y\left(s\right)}{u\left(s\right)}=\left\{\begin{array}{cc}{\left(\frac{\mathrm{\omega }}{{\left(\frac{s}{\mathrm{\omega }}\right)}^{2}+\frac{2}{\mathrm{\omega }}+1}\right)}^{n}& \mathrm{n even}\\ {\left(\frac{\mathrm{\omega }}{{\left(\frac{s}{\mathrm{\omega }}\right)}^{2}+\frac{2}{\mathrm{\omega }}+1}\right)}^{n-1}\left(\frac{\mathrm{\omega }}{\frac{s}{\mathrm{\omega }}+1}\right)& \mathrm{otherwise}\end{array}$

$\mathrm{\omega }=2\pi f$

 Name Description Modelica ID $u$ Real input signal u $y$ Real output signal y

Parameters

 Name Default Units Description Modelica ID n $2$ Order of filter n f $1$ $\mathrm{Hz}$ Cut-off frequency f Initial Values No initialization Type of initialization initType ${x}_{{1}_{0}}$ $\mathrm{zeros}\left(\mathrm{integer}\left(0.5n\right)\right)$ Initial or guess values of states 1 x1_start ${x}_{{2}_{0}}$ $\mathrm{zeros}\left(\mathrm{integer}\left(0.5n\right)\right)$ Initial or guess values of states 2 x2_start ${x}_{{r}_{0}}$ $0$ Initial or guess value of real pole for uneven order otherwise this value is null xr_start ${y}_{0}$ $0$ Initial value of output (states are initialized in steady state if possible) y_start

 Modelica Standard Library The component described in this topic is from the Modelica Standard Library. To view the original documentation, which includes author and copyright information, click here.