4 changed files with 388 additions and 6 deletions
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%% AUTHOR : Apostolos Fanakis |
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%% $DATE : 16-Aug-2018 19:23:40 $ |
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%% $Revision : 1.00 $ |
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%% DEVELOPED : 9.0.0.341360 (R2016a) |
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%% FILENAME : band_pass_design.m |
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%% AEM : 8261 |
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%% |
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%% ========== DESIGN SPECIFICATIONS START ========== |
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% Figures out design specifications according to my AEM number |
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AEM = [8 2 6 1]; |
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if AEM(4)<4 |
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design_param_m = 1; |
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elseif AEM(3)<7 |
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design_param_m = 0; |
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else |
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design_param_m = 3; |
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end |
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|
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specification_pass_frequency = (4+design_param_m)*1000; % Hz |
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specification_pass_radial_frequency = ... |
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specification_pass_frequency*(2*pi); % rad/s |
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specification_stop_frequency = specification_pass_frequency/2.6; % Hz |
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specification_stop_radial_frequency = ... |
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specification_stop_frequency*(2*pi); % rad/s |
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specification_min_stop_attenuation = 24+AEM(4)*(6/9); % dB |
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specification_max_pass_attenuation = 0.5+AEM(3)/36; % dB |
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|
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clear design_param_m |
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|
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%{ |
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specification_pass_radial_frequency = 15707.96; % rad/s |
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specification_stop_radial_frequency = 6283.2; % rad/s |
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specification_min_stop_attenuation = 25; % dB |
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specification_max_pass_attenuation = 0.5; % dB |
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%} |
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% ========== DESIGN SPECIFICATIONS END ========== |
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|
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%% ========== PROTOTYPE LOW PASS DESIGN SPECIFICATIONS START ========== |
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% Calculates the specifications for the low pass design that will later be |
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% converted to the desired high pass filter |
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|
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% Calculates the specifications using the eq. 12-4 |
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% prototype_normalized_pass_radial_frequency = 1; % rad/s |
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prototype_normalized_stop_radial_frequency = ... |
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specification_pass_radial_frequency/ ... |
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specification_stop_radial_frequency; % rad/s |
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% Min and max attenuations remain the same |
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% ========== PROTOTYPE LOW PASS DESIGN SPECIFICATIONS END ========== |
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%% ========== PROTOTYPE LOW PASS DESIGN START ========== |
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% Designs the prototype normalized filter. |
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% Calculates the filter's order using the eq. 9-52 |
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design_filter_order = ceil(log10(((10^ ... |
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(specification_min_stop_attenuation/10)-1)/(10^ ... |
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(specification_max_pass_attenuation/10)-1)))/ ... |
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(2*log10(prototype_normalized_stop_radial_frequency))); |
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% Calculates the frequency at which half power of the low pass prototype |
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% occurs using the eq. 9-48 |
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low_pass_prototype_half_power_radial_frequency = 1/ ... |
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(10^(specification_max_pass_attenuation/10)-1)^ ... |
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(1/(2*design_filter_order)); % rad/s |
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|
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% Transforms the result to get the corresponding frequency for the high |
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% pass using the eq. 12-12 |
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design_half_power_radial_frequency = specification_pass_radial_frequency/ ... |
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low_pass_prototype_half_power_radial_frequency; % rad/s |
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% ----- |
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% Calculates stable poles, zeros, angles and other characteristic sizes |
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% using the Guillemin algorithm for a normalized low pass Butterworth |
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% filter. |
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% |
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% So for the time being we assume that the pass radial frequency is equal |
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% to unity (1). |
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% ----- |
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% Initializes necessary variables |
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low_pass_prototype_number_of_poles = idivide(design_filter_order, ... |
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int32(2),'ceil'); |
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% Creates five vector arrays of dimensions [1 * number_of_poles] filled |
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% with zeros to store: |
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% - the Butterworth angles with reference to the negative horizontal axes, |
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% - the real parts of the poles, |
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% - the imaginary parts of the poles, |
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% - the radial frequencies (Omega0) of the poles and |
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% - the Q's of the poles |
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design_butterworth_angles = zeros([1 low_pass_prototype_number_of_poles]); |
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low_pass_prototype_poles_real_parts = ... |
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zeros([1 low_pass_prototype_number_of_poles]); |
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low_pass_prototype_poles_imaginary_parts = ... |
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zeros([1 low_pass_prototype_number_of_poles]); |
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low_pass_prototype_poles_radial_frequencies = ... |
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zeros([1 low_pass_prototype_number_of_poles]); |
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low_pass_prototype_poles_Q = zeros([1 low_pass_prototype_number_of_poles]); |
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% Calculates the Butterworth angles using the method suggested in chapter |
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% 9 (page 10) of the course notes and then uses them to calculate the |
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% Chebyshev poles |
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if mod(design_filter_order,2)~=0 % Odd number of poles |
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% First pole has a zero angle |
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design_butterworth_angles(1,1)=0; |
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% The rest of the poles are scattered in the left half pane with |
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% equal angles |
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% Theta is a helper parameter |
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theta=180/design_filter_order; |
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% Calculates the first pole's real part using the eq. 9-31 |
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low_pass_prototype_poles_real_parts(1,1) = ... |
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-cosd(design_butterworth_angles(1,1)); |
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% Calculates the first pole's imaginary part using the eq. 9-31 |
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low_pass_prototype_poles_imaginary_parts(1,1) = ... |
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sind(design_butterworth_angles(1,1)); |
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low_pass_prototype_poles_radial_frequencies(1,1) = 1; |
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% Calculates the first pole's Q using the eq. 9-38 |
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low_pass_prototype_poles_Q(1,1) = 1/ ... |
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(2*cosd(design_butterworth_angles(1,1))); |
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% Calculates the rest of the poles in the same way |
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for i=2:low_pass_prototype_number_of_poles |
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design_butterworth_angles(1,i) = double((i-1)*theta); |
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% Pole's real part, eq. 9-31 |
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low_pass_prototype_poles_real_parts(1,i) = ... |
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-cosd(design_butterworth_angles(1,i)); |
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% Pole's imaginary part, eq. 9-31 |
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low_pass_prototype_poles_imaginary_parts(1,i) = ... |
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sind(design_butterworth_angles(1,i)); |
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low_pass_prototype_poles_radial_frequencies(1,i) = 1; |
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% Pole's Q, eq. 9-38 |
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low_pass_prototype_poles_Q(1,i) = 1/ ... |
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(2*cosd(design_butterworth_angles(1,i))); |
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end |
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else % Even number of poles |
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% Theta is a helper parameter |
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theta=90/low_pass_prototype_number_of_poles; |
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for i=1:low_pass_prototype_number_of_poles |
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design_butterworth_angles(1,i) = double(90)/ ... |
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double(design_filter_order)+double((i-1)*theta); |
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% Pole's real part, eq. 9-31 |
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low_pass_prototype_poles_real_parts(1,i) = ... |
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-cosd(design_butterworth_angles(1,i)); |
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% Pole's imaginary part, eq. 9-31 |
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low_pass_prototype_poles_imaginary_parts(1,i) = ... |
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sind(design_butterworth_angles(1,i)); |
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low_pass_prototype_poles_radial_frequencies(1,i) = 1; |
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% Pole's Q, eq. 9-106 |
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low_pass_prototype_poles_Q(1,i) = 1/ ... |
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(2*cosd(design_butterworth_angles(1,i))); |
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end |
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end |
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% Clears unneeded variables from workspace |
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% |
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clearVars = {'i', 'prototype_normalized_stop_radial_frequency', ... |
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'low_pass_prototype_half_power_radial_frequency', 'theta'}; |
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clear(clearVars{:}) |
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clear clearVars |
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% |
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% ========== PROTOTYPE LOW PASS DESIGN END ========== |
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%% ========== POLES TRANSFORMATION START ========== |
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% Transforms the prototype's poles |
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% Initializes necessary variables |
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% Calculates the number of poles that will occur after the transformation |
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high_pass_number_of_poles = low_pass_prototype_number_of_poles; |
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% According to the course notes (chapter 12, end of page 5) the |
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% transformation leaves the poles unchanged. |
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high_pass_poles_real_parts = low_pass_prototype_poles_real_parts; |
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high_pass_poles_imaginary_parts = ... |
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low_pass_prototype_poles_imaginary_parts; |
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high_pass_poles_radial_frequencies = ... |
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low_pass_prototype_poles_radial_frequencies; |
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high_pass_poles_Q = low_pass_prototype_poles_Q; |
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% The transormation also produces a number of zeros at (0,0) equal to the |
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% filter order |
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high_pass_transfer_function_zeros = zeros([1 design_filter_order]); |
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% Clears unneeded variables from workspace |
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clear low_pass_prototype_number_of_poles |
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clear -regexp ^low_pass_prototype_ |
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% ========== POLES TRANSFORMATION END ========== |
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%% ========== POLES DE-NORMALIZATION START ========== |
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% The high pass filter poles calculated above are those of a normalized |
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% filter. De-normalization is needed to get the actual poles for the |
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% desired high pass filter. |
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for i=1:high_pass_number_of_poles |
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high_pass_poles_real_parts(1,i) = high_pass_poles_real_parts(1,i)* ... |
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design_half_power_radial_frequency; |
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high_pass_poles_imaginary_parts(1,i) = ... |
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high_pass_poles_imaginary_parts(1,i)* ... |
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design_half_power_radial_frequency; |
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high_pass_poles_radial_frequencies(1,i) = ... |
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design_half_power_radial_frequency; |
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end |
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% Clears unneeded variables from workspace |
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clear i |
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clear -regexp ^geffe_ |
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clear -regexp ^transformation_ |
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% ========== POLES DE-NORMALIZATION END ========== |
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%% ========== UNITS IMPLEMENTATION START ========== |
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% AEM(2) = 2, so the first design strategy is going to be used in the |
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% Sallen-Key high pass circuits. |
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% ------------------------------------------------------------------------- |
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% Unit 1 has a pole pair with Q equal to 0.5412 and lies on a circle with a |
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% radius equal to 25097.78. |
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% ------------------------------------------------------------------------- |
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% Unit 1 has a pole pair with Q equal to 1.3066 and lies on a circle with a |
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% radius equal to 25097.78. |
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% ------------------------------------------------------------------------- |
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% Initializes necessary arrays, each array is 1X2, the first element (1,1) |
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% corresponds to the first unit and the second element (1,2) to second |
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% unit. |
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units_R = ones([1 high_pass_number_of_poles]); |
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units_C = ones([1 high_pass_number_of_poles]); |
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units_r1 = ones([1 high_pass_number_of_poles]); |
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units_r2 = zeros([1 high_pass_number_of_poles]); |
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units_k = zeros([1 high_pass_number_of_poles]); |
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units_frequency_scale_factors = zeros([1 2]); |
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units_amplitude_scale_factors = zeros([1 2]); |
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units_transfer_functions = [tf(1) tf(1)]; |
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for i=1:high_pass_number_of_poles |
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% Calculates k and r2 using the eq. 6-75 |
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units_r2(1,i) = 2-1/high_pass_poles_Q(1,i); |
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units_k(1,i) = 3-1/high_pass_poles_Q(1,i); |
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% Selects the appropriate frequency scale factor to transfer the |
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% normalized radial frequency back to the original |
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units_frequency_scale_factors(1,i) = ... |
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high_pass_poles_radial_frequencies(1,i); |
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% AEM(3) = 6, so the magnitude scaling will be performed to achieve a |
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% capacitor value of 0.1uF using the eq. 6-33 |
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units_amplitude_scale_factors(1,i) = ... |
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units_C(1,i)/ ... |
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(units_frequency_scale_factors(1,i)*0.1*10^(-6)); |
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% Performs scaling |
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units_R(1,i) = units_R(1,i)* ... |
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units_amplitude_scale_factors(1,i); |
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units_C(1,i) = 0.1*10^(-6); |
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units_r1(1,i) = units_r1(1,i)* ... |
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units_amplitude_scale_factors(1,i); |
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units_r2(1,i) = units_r2(1,i)* ... |
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units_amplitude_scale_factors(1,i); |
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% Builds unit's transfer function |
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% Builds numerator and denominator of the transfer function using the |
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% eq. 6-68 |
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G = (units_R(1,i)+units_r2(1,i))/units_R(1,i); |
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unit_numerator = [G ... |
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0 ... |
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0]; |
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unit_denominator = [1 ... |
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2/(units_C(1,i)*units_R(1,i))+(1-G)/(units_C(1,i)*units_R(1,i)) ... |
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1/(units_C(1,i)^2*units_R(1,i)^2)]; |
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units_transfer_functions(1,i) = ... |
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tf(unit_numerator, unit_denominator); |
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end |
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% Clears unneeded variables from workspace |
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clearVars = {''}; |
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clear(clearVars{:}) |
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clear clearVars |
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clear -regexp _transfer_function$ |
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% ========== UNITS IMPLEMENTATION END ========== |
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%% ========== GAIN ADJUSTMENT START ========== |
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% |
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total_gain_high = units_k(1,1)*units_k(1,2); |
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unit_adjustment_gain = 1/total_gain_high; |
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% We arbitrarily choose to use a 10KOhm series resistor in the adjustment |
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% unit |
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unit_adjustment_feedback_resistor = 10*10^3*unit_adjustment_gain; |
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% |
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total_transfer_function = series(units_transfer_functions(1,1), ... |
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units_transfer_functions(1,2)); |
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total_transfer_function = total_transfer_function*unit_adjustment_gain; |
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%{ |
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ltiview(units_transfer_functions(1,1), ... |
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units_transfer_functions(1,2)); |
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%} |
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%{ |
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ltiview(units_transfer_functions(1,1), ... |
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units_transfer_functions(1,2), ... |
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total_transfer_function); |
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%} |
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%ltiview(total_transfer_function); |
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% |
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plot_transfer_function(total_transfer_function, ... |
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[design_half_power_radial_frequency/(2*pi) ... |
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specification_stop_frequency ... |
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specification_pass_frequency ... |
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15000]); |
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% |
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% Clears unneeded variable from workspace |
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clearVars = {'total_transfer_function'}; |
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clear(clearVars{:}) |
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clear clearVars |
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clear -regexp _transfer_functions$ |
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% ========== GAIN ADJUSTMENT END ========== |
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@ -0,0 +1,44 @@ |
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function plot_transfer_function( tf, frequency_markers ) |
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%PLOT_TRANSFER_FUNCTION Plots bode of a transfer function with markers |
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% |
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% tf - The transfer function (created using tf) |
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% frequency_markers - A matrix of frequencies in Hz |
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% |
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% Example: |
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% plot_transfer_function( tf([1000], [1 1000]), [10 1000 10000] ); |
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figure; |
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x_space = logspace(1,5,1000); % 1000 points between 10^1 and 10^5 |
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x_space = 2 * pi * x_space; % to rad / sec |
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[mag,~,wout] = bode(tf,x_space); |
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mag = squeeze(mag); |
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wout = squeeze(wout); |
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mag = 20*log10(mag); |
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wout = wout/2/pi; |
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semilogx(wout,mag,'-b'); |
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axis([min(wout) max(wout) min(mag)-10 max(mag)+10]); |
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[num,den] = tfdata(tf); |
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syms s; |
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d1 = digits(5); |
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ltx = latex(vpa(poly2sym(cell2mat(num),s)/poly2sym(cell2mat(den),s))); |
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digits(d1); |
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title(strcat('$',ltx,'$'), 'Interpreter','latex', 'FontSize', 24); |
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xlabel('Frequency (Hz)', 'FontSize', 18); |
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ylabel('Magnitude (dB)', 'FontSize', 18); |
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grid on; |
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hold on; |
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[dbMarks,~,frequency_markers] = bode(tf,2 * pi * frequency_markers); |
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dbMarks = squeeze(dbMarks); |
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frequency_markers = squeeze(frequency_markers); |
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dbMarks = 20*log10(dbMarks); |
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frequency_markers = frequency_markers/2/pi; |
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Aw = cell(size(frequency_markers, 1) + 1, 1); |
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Aw{1} = 'Transfer function'; |
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for i = 1 : size(frequency_markers, 1) |
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semilogx(frequency_markers(i),dbMarks(i),'o'); |
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Aw{i+1} = sprintf('Attenuation at %.2f Hz is %.2f dB', ... |
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frequency_markers(i), dbMarks(i)); |
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end |
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legend(Aw,'Location','best','FontSize',12); |
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set(gca,'FontSize',14); |
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end |
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