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%% AUTHOR : Apostolos Fanakis |
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%% $DATE : 03-Aug-2018 12:32:39 $ |
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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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|
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AEM = [8 2 6 1]; |
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|
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% Specification f0 |
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specification_central_frequency = 0.9*1000; % Hz |
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specification_central_radial_frequency = 2*pi* ... |
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specification_central_frequency; % rad/s |
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|
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% Specification f1 |
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specification_low_pass_frequency = 650+25*AEM(3); % Hz |
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specification_low_pass_radial_frequency = 2*pi* ... |
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specification_low_pass_frequency; % rad/s |
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|
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% Specification f2 |
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specification_high_pass_frequency = specification_central_frequency^2/ ... |
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specification_low_pass_frequency; % Hz |
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specification_high_pass_radial_frequency = 2*pi* ... |
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specification_high_pass_frequency; % rad/s |
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|
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% Specification D |
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design_param_D = 2.2*((specification_central_frequency^2- ... |
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specification_low_pass_frequency^2)/specification_low_pass_frequency); |
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|
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% Specification f3 |
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specification_low_stop_frequency = (-design_param_D+ ... |
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sqrt(design_param_D^2+4*specification_central_frequency^2))/2; % Hz |
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specification_low_stop_radial_frequency = 2*pi* ... |
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specification_low_stop_frequency; % rad/s |
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|
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% Specification f4 |
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specification_high_stop_frequency = specification_central_frequency^2/ ... |
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specification_low_stop_frequency; % Hz |
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specification_high_stop_radial_frequency = 2*pi* ... |
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specification_high_stop_frequency; % rad/s |
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|
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specification_min_stop_attenuation = 28+AEM(4)*5/9; % 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_D |
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|
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%{ |
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specification_low_pass_radial_frequency = 500; % rad/s |
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specification_high_pass_radial_frequency = 800; % rad/s |
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specification_low_stop_radial_frequency = 400; % rad/s |
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specification_high_stop_radial_frequency = 1000; % rad/s |
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specification_min_stop_attenuation = 18; % 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 band pass filter |
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|
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% Calculates the specifications using the eq. 11-56 |
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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_high_stop_radial_frequency- ... |
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specification_low_stop_radial_frequency)/ ... |
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(specification_high_pass_radial_frequency- ... |
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specification_low_pass_radial_frequency); % rad/s |
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|
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% Min and max attenuations remain the same |
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|
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% Calculates the geometric middle radial frequency using the eq. 11-2 |
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design_geometric_central_radial_frequency = ... |
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sqrt(specification_low_pass_radial_frequency* ... |
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specification_high_pass_radial_frequency); % rad/s |
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|
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% Calculates the pass bandwidth using the eq. 11-52 |
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design_filter_bandwidth = specification_high_pass_radial_frequency- ... |
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specification_low_pass_radial_frequency; % rad/s |
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|
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% ========== PROTOTYPE LOW PASS DESIGN SPECIFICATIONS END ========== |
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|
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%% ========== PROTOTYPE LOW PASS DESIGN START ========== |
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% The calculated low pass design specifications have a form fit for a low |
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% pass Chebyshev filter design (the pass radial frequency is normalized to |
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% one). |
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% Designs the prototype normalized filter. |
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|
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% Calculates the filter's order using the eq. 9-83 |
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design_filter_order = ceil(acosh(((10^ ... |
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(specification_min_stop_attenuation/10)-1)/(10^ ... |
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(specification_max_pass_attenuation/10)-1))^(1/2))/ ... |
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acosh(prototype_normalized_stop_radial_frequency)); |
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|
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% Calculates epsilon parameter using the eq. 9-76 |
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epsilon_parameter = sqrt(10^(specification_max_pass_attenuation/10)-1); |
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% Calculates alpha parameter using the eq. 9-92 |
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alpha_parameter = asinh(1/epsilon_parameter)/design_filter_order; |
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|
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% Calculates the frequency at which half power occurs using the eq. 9-80 |
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% TODO: denormalize!! ====================%%%%%%%%%%%%%%%%%%%%%%%%============================ |
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design_half_power_radial_frequency = cosh(acosh(( ... |
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10^(specification_max_pass_attenuation/10-1))^(-1/2))/design_filter_order); % rad/s |
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|
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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 |
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% ----- |
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|
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% Initializes necessary variables |
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prototype_number_of_poles = idivide(design_filter_order,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 angles (Qk) of the poles |
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design_butterworth_angles = zeros([1 prototype_number_of_poles]); |
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prototype_poles_real_parts = zeros([1 prototype_number_of_poles]); |
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prototype_poles_imaginary_parts = zeros([1 prototype_number_of_poles]); |
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|
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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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|
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% Calculates the first pole's real part using the eq. 9-102 |
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prototype_poles_real_parts(1,1) = -sinh(alpha_parameter)* ... |
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cosd(design_butterworth_angles(1,1)); |
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% Calculates the first pole's imaginary part using the eq. 9-103 |
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prototype_poles_imaginary_parts(1,1) = cosh(alpha_parameter)* ... |
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sind(design_butterworth_angles(1,1)); |
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|
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% Calculates the rest of the poles in the same way |
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for i=2: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. 9-102 |
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prototype_poles_real_parts(1,i) = -sinh(alpha_parameter)* ... |
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cosd(design_butterworth_angles(1,i)); |
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% Pole's imaginary part, eq. 9-103 |
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prototype_poles_imaginary_parts(1,i) = cosh(alpha_parameter)* ... |
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sind(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/prototype_number_of_poles; |
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|
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for i=1: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. 9-102 |
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prototype_poles_real_parts(1,i) = -sinh(alpha_parameter)* ... |
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cosd(design_butterworth_angles(1,i)); |
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% Pole's imaginary part, eq. 9-103 |
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prototype_poles_imaginary_parts(1,i) = cosh(alpha_parameter)* ... |
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sind(design_butterworth_angles(1,i)); |
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end |
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end |
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|
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% Clears unneeded variables from workspace |
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clearVars = {'prototype_normalized_stop_radial_frequency', ... |
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'epsilon_parameter', 'alpha_parameter', '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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|
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%% ========== POLES TRANSFORMATION START ========== |
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% Transforms the prototype's poles according to the Geffe algorithm |
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|
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% Initializes necessary variables |
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% Calculates the parameter qc, required for the transformation of the |
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% poles, using the eq. 11-6 |
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transformation_qc_parameter = design_geometric_central_radial_frequency/ ... |
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design_filter_bandwidth; |
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% Calculates the number of poles that will occur after the transformation |
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if mod(design_filter_order,2)~=0 |
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band_pass_number_of_poles = 2*prototype_number_of_poles-1; |
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else |
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band_pass_number_of_poles = 2*prototype_number_of_poles; |
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end |
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% Creates four vector arrays of dimensions [1 * 4] filled with zeros to |
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% store: |
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% - the Q's of the transformed poles |
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% - the angles of the transformed poles |
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% - the radial frequencies of the transformed poles |
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% - the tranfer function zeros |
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band_pass_poles_Q = zeros([1 band_pass_number_of_poles]); |
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band_pass_poles_angle = zeros([1 band_pass_number_of_poles]); |
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band_pass_poles_radial_frequencies = zeros([1 band_pass_number_of_poles]); |
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% Every pole transormation produces one transfer function zero at (0,0) for |
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% every new pole |
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band_pass_transfer_function_zeros = zeros([1 band_pass_number_of_poles]); |
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% temp_index is a helper variable |
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temp_index = 1; |
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|
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for i=1:prototype_number_of_poles |
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if prototype_poles_imaginary_parts(1,i)==0 % Real pole |
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transformation_sigma_1 = -prototype_poles_real_parts(1,i); |
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|
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% Calculates the transformed pole's Q using the eq. 11-11 |
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band_pass_poles_Q(1,temp_index) = transformation_qc_parameter/ ... |
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transformation_sigma_1; |
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% Calculates the transformed pole's angle using the eq. 11-12 |
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band_pass_poles_angle(1,temp_index) = acosd(1/(2*band_pass_poles_Q(1,i))); |
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band_pass_poles_radial_frequencies(1,temp_index) = ... |
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design_geometric_central_radial_frequency; |
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temp_index = temp_index+1; |
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else % Complex pole |
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geffe_sigma_2 = -prototype_poles_real_parts(1,i); |
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geffe_Omega_2 = prototype_poles_imaginary_parts(1,i); |
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|
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% Calculates the parameter C using the eq. 11-28 |
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geffe_C = geffe_sigma_2^2+geffe_Omega_2^2; |
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% Calculates the parameter D using the eq. 11-29 |
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geffe_D = (2*geffe_sigma_2)/transformation_qc_parameter; |
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% Calculates the parameter E using the eq. 11-30 |
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geffe_E = 4+geffe_C/transformation_qc_parameter^2; |
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% Calculates the parameter G using the eq. 11-31 |
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geffe_G = sqrt(geffe_E^2-4*geffe_D^2); |
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|
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% Calculates the parameter Q using the eq. 11-32 |
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geffe_Q = sqrt((geffe_E+geffe_G)/2)/geffe_D; |
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% Calculates the parameter k using the eq. 11-33 |
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geffe_k = (geffe_sigma_2*geffe_Q)/transformation_qc_parameter; |
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% Calculates the parameter W using the eq. 11-34 |
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geffe_W = geffe_k+sqrt(geffe_k^2-1); |
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|
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% Calculates the radius of the circles upon which the two poles |
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% reside using the eq. 11-15 |
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geffe_Omega_0_1 = design_geometric_central_radial_frequency* ... |
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geffe_W; |
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geffe_Omega_0_2 = design_geometric_central_radial_frequency/ ... |
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geffe_W; |
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|
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% The two poles have the same Q |
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band_pass_poles_Q(1,temp_index) = geffe_Q; |
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band_pass_poles_Q(1,temp_index+1) = geffe_Q; |
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% Calculates the transformed pole's angle using the eq. 11-37b |
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band_pass_poles_angle(1,temp_index) = acosd(1/ ... |
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(2*band_pass_poles_Q(1,temp_index))); |
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band_pass_poles_angle(1,temp_index+1) = ... |
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band_pass_poles_angle(1,temp_index); |
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|
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band_pass_poles_radial_frequencies(1,temp_index) = geffe_Omega_0_1; |
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band_pass_poles_radial_frequencies(1,temp_index+1) = geffe_Omega_0_2; |
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temp_index = temp_index+2; |
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end |
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end |
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|
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% Clears unneeded variables from workspace |
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clearVars = {'prototype_number_of_poles', 'i', 'temp_index', ... |
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'prototype_poles_imaginary_parts', 'prototype_poles_real_parts'}; |
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clear(clearVars{:}) |
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clear clearVars |
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clear -regexp ^geffe_ |
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clear -regexp ^transformation_ |
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|
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% ========== POLES TRANSFORMATION END ========== |
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|
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%% ========== UNITS IMPLEMENTATION START ========== |
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% AEM(3) = 6, so the first design strategy is going to be used in the |
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% Delyiannis-Fried circuits. |
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|
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% ------------------------------------------------------------------------- |
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% Unit 1 has a pole pair with ±87.38 degrees of angle, Q equal to 10.954 |
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% and lies on a circle with a radius equal to 5938.94. |
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% ------------------------------------------------------------------------- |
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% Unit 2 has a pole pair with ±87.38 degrees of angle, Q equal to 10.954 |
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% and lies on a circle with a radius equal to 5384.37. |
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% ------------------------------------------------------------------------- |
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% Unit 3 has a pole pair with ±88.92 degrees of angle, Q equal to 26.599 |
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% and lies on a circle with a radius equal to 6363.12. |
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% ------------------------------------------------------------------------- |
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% Unit 4 has a pole pair with ±88.92 degrees of angle, Q equal to 26.599 |
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% and lies on a circle with a radius equal to 5025.44. |
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% ------------------------------------------------------------------------- |
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|
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% Initializes necessary arrays, each array is 1X4, the first element (1,1) |
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% corresponds to the first unit and the second element (1,2) to second unit |
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% etc. |
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units_BW = zeros([1 band_pass_number_of_poles]); |
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units_C21 = zeros([1 band_pass_number_of_poles]); |
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units_C22 = zeros([1 band_pass_number_of_poles]); |
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units_R21 = zeros([1 band_pass_number_of_poles]); |
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units_R22 = zeros([1 band_pass_number_of_poles]); |
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units_frequency_scale_factors = zeros([1 band_pass_number_of_poles]); |
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units_amplitude_scale_factors = zeros([1 band_pass_number_of_poles]); |
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units_central_frequency_gain = zeros([1 band_pass_number_of_poles]); |
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units_alpha = zeros([1 band_pass_number_of_poles]); |
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units_Z22 = zeros([1 band_pass_number_of_poles]); |
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units_Z23 = zeros([1 band_pass_number_of_poles]); |
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unit_transfer_function = [tf(1) tf(1) tf(1) tf(1)]; |
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|
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for i=1:band_pass_number_of_poles |
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% Calculates BW using the eq. 7-62 |
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units_BW(1,i) = band_pass_poles_radial_frequencies(1,i)/ ... |
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band_pass_poles_Q(1,i); |
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% Calculates C21 (=C22=C) using the eq. 7-87 |
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units_C21(1,i) = 1/(2*band_pass_poles_Q(1,i)); |
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units_C22(1,i) = units_C21(1,i); |
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% Using the eq. 7-86 |
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units_R21(1,i) = 1; |
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% Calculates R12 using the eq. 7-87 |
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units_R22(1,i) = 4*band_pass_poles_Q(1,i)^2; |
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|
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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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band_pass_poles_radial_frequencies(1,i); |
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% AEM(4) = 1, 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) = units_C21(1,i)/ ... |
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(units_frequency_scale_factors(1,i)*0.1*10^(-6)); |
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|
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% Scales the circuit elements |
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units_C21(1,i) = 0.1*10^(-6); |
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units_C22(1,i) = 0.1*10^(-6); |
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units_R21(1,i) = units_amplitude_scale_factors(1,i); |
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units_R22(1,i) = units_R22(1,i)*units_amplitude_scale_factors(1,i); |
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|
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% Calculates the gain at the central radial frequency and the alpha |
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% parameter using the eq. 7-89 |
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units_central_frequency_gain(1,i) = 2*band_pass_poles_Q(1,i)^2; |
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units_alpha(1,i) = 1/units_central_frequency_gain(1,i); |
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|
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% Calculates the values of the resistors used to diminish the entry |
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% using the eq. 7-90 (to include the scaling already done the equations |
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% are used in the form presented at example 7.2) |
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units_Z22(1,i) = units_R21(1,i)/units_alpha(1,i); |
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units_Z23(1,i) = units_R21(1,i)/(1-units_alpha(1,i)); |
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|
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%TODO: build the tfs |
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% |
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unit_numerator = [-1/(units_R21(1,i)*units_C21(1,i)) ... |
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0]; |
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unit_denominator = [1 ... |
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(2/units_C21(1,i))/units_R22(1,i) ... |
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1/(units_R22(1,i)*units_R21(1,i)*units_C21(1,i)^2)]; |
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% |
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%{ |
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unit_numerator = [-2*band_pass_poles_Q(1,i)^2*units_BW(1,i) ... |
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0]; |
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unit_denominator = [1 ... |
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units_BW(1,i) ... |
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band_pass_poles_radial_frequencies(1,i)^2]; |
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%} |
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unit_transfer_function(i) = tf(unit_numerator, unit_denominator); |
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end |
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|
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total_transfer_function = series(series(series(unit_transfer_function(1), ... |
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unit_transfer_function(2)), unit_transfer_function(3)), ... |
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unit_transfer_function(4)); |
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%total_transfer_function = total_transfer_function*38.2; |
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|
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%{ |
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plot_transfer_function(total_transfer_function, ... |
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[1 ... |
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specification_low_stop_frequency ... |
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specification_low_pass_frequency ... |
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900 ... |
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specification_high_pass_frequency ... |
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specification_high_stop_frequency]); |
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%} |
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%plot_transfer_function(unit_transfer_function(2), [1 10]); |
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%ltiview(unit_transfer_function(1), unit_transfer_function(2), ... |
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% unit_transfer_function(3), unit_transfer_function(4)); |
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|
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%ltiview(total_transfer_function); |
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|
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% Clears unneeded variables from workspace |
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clearVars = {'units_central_frequency_gain', 'i', 'units_alpha', ... |
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'unit_denominator', 'unit_numerator', ... |
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'units_amplitude_scale_factors', 'units_frequency_scale_factors'}; |
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clear(clearVars{:}) |
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clear clearVars |
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clear -regexp _transfer_function$ |
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|
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% ========== UNITS IMPLEMENTATION 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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|
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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); |
|||
end |
|||
Loading…
Reference in new issue