Summary

DescriptionButterworth poles 3.svg	English: pole location for a 3rd order Butterworth lowpass filter
Date	7 June 2012
Source	Own work
Author	Fcorthay
SVG development InfoField	The SVG code is valid.   This vector image was created with Inkscape, or with something else.

Source code

This graphics has been realized with the help of the following Octave script:

#-------------------------------------------------------------------------------
# Specifications
#
order = 3;
fc = 1;
pointNb = 100;
axesLength = 1.25;
textOffset = 0.05;
wLog = logspace(-2, 2, pointNb);
AdbMax = 10;
AdbMin = -60;

makeGraphics = 1;

figureIndex = 0;

#===============================================================================
# Butterworth filter
#
[zeroes, poles, gain] = butter(order, fc, 's');

[num, den] = butter(order, fc, 's');
while ( length(num) < length(den) )
  num = [0, num];
endwhile;

#-------------------------------------------------------------------------------
# Poles plot
#
figureIndex = figureIndex+1;
figure(figureIndex);
angle = linspace(pi/2, 3*pi/2, pointNb);
plot(cos(angle), sin(angle));

hold on;
plot([-axesLength, axesLength-1], [0, 0], 'k');
plot([0, 0], [-axesLength, axesLength], 'k');
plot(axesLength-1+textOffset*[0, 2, 0, 0], textOffset*3/4*[-1, 0, 1, -1], 'k');
plot(textOffset*3/4*[-1, 0, 1, -1], axesLength+textOffset*[0, 2, 0, 0], 'k');
plot(poles, 'xr');
hold off;

axis("equal", "off");
text(axesLength-1+textOffset, 1.5*textOffset, "re(s)");
text(textOffset, axesLength+1.5*textOffset, "im(s)");

for index = 1:order
  if imag(poles(index)) > 0
    text(real(poles(index))-textOffset, imag(poles(index))+textOffset, ...
    sprintf("p%i", index), "horizontalalignment", "right");
  elseif imag(poles(index)) == 0
    text(real(poles(index))-textOffset, imag(poles(index))+textOffset, ...
    "pr", "horizontalalignment", "right");
  endif
endfor

if (makeGraphics != 0)
  print -dsvg butterworth_poles_3.svg
endif

#-------------------------------------------------------------------------------
# Amplitude plot
#
Adb = 20*log10(abs(freqs(num, den, wLog)));

figureIndex = figureIndex+1;
figure(figureIndex);
semilogx(wLog, Adb);
hold on;
for index = 1:order
  if imag(poles(index)) > 0
    numP = [0, 0, 1];
    denP = [1, -2*real(poles(index)), abs(poles(index))^2];
    Adb = 20*log10(abs(freqs(numP, denP, wLog)));
    semilogx(wLog, Adb, 'g');
  elseif imag(poles(index)) == 0
    numP = [0, 1];
    denP = [1, -real(poles(index))];
    Adb = 20*log10(abs(freqs(numP, denP, wLog)));
    semilogx(wLog, Adb, 'g');
  endif
endfor
hold off;

axis([wLog(1), wLog(length(wLog)), AdbMin, AdbMax]);
xlabel('Frequency');
ylabel('Amplitude responses');
grid;

if (makeGraphics != 0)
  print -dsvg butterworth_amplitudes_3.svg
endif

#-------------------------------------------------------------------------------
# Phase plot
#
phase = arg(freqs(num, den, wLog));

figureIndex = figureIndex+1;
figure(figureIndex);
semilogx(wLog, phase);
hold on;
for index = 1:order
  if imag(poles(index)) > 0
    numP = [0, 0, 1];
    denP = [1, -2*real(poles(index)), abs(poles(index))^2];
    phase = arg(freqs(numP, denP, wLog));
    semilogx(wLog, phase, 'g');
  elseif imag(poles(index)) == 0
    numP = [0, 1];
    denP = [1, -real(poles(index))];
    phase = arg(freqs(numP, denP, wLog));
    semilogx(wLog, phase, 'g');
  endif
endfor
semilogx([wLog(1), wLog(length(wLog))], -pi/2*[1, 1], 'c');
semilogx([wLog(1), wLog(length(wLog))],   0  *[1, 1], 'c');
semilogx([wLog(1), wLog(length(wLog))],  pi/2*[1, 1], 'c');
hold off;

axis([wLog(1), wLog(length(wLog)), -pi, pi]);
xlabel('Frequency');
ylabel('Phase responses');

if (makeGraphics != 0)
  print -dsvg butterworth_phases_3.svg
endif

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:

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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

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