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% AUTHOR : Apostolos Fanakis |
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% $DATE : 02-May-2018 17:21:18 $ |
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% $Revision : 1.00 $ |
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% DEVELOPED : 9.0.0.341360 (R2016a) |
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% FILENAME : low_pass_design_single.m |
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% AEM : 8261 |
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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(3)<3 |
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design_param_m = 1; |
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elseif AEM(3)<7 |
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design_param_m = 2; |
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else |
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design_param_m = 3; |
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end |
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specification_pass_frequency = (1.1*(3+design_param_m))*1000; % Hz |
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specification_pass_radial_frequency = specification_pass_frequency*(2*pi); % rad/s |
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specification_stop_frequency = 2.1*specification_pass_frequency; % Hz |
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specification_stop_radial_frequency = specification_stop_frequency*(2*pi); % rad/s |
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specification_min_stop_attenuation = 23+(max(1,AEM(3))-5)*(3/4); % dB |
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specification_max_pass_attenuation = 0.6+((max(1,AEM(4))-5)/16); % dB |
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clear design_param_m |
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specification_pass_radial_frequency = 1000; % rad/s |
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specification_stop_radial_frequency = 1400; % rad/s |
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specification_min_stop_attenuation = 18; % dB |
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specification_max_pass_attenuation = 0.25; % dB |
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% ========== DESIGN SPECIFICATIONS END ========== |
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% ========== NORMALIZED DESIGN START ========== |
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% Calculates normalized design specifications and designs a normalized |
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% filter using them |
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normalized_pass_radial_frequency = specification_pass_radial_frequency/ ... |
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specification_stop_radial_frequency; % rad/s |
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% normalized_stop_radial_frequency = 1; % Hz (stop_frequency/stop_frequency) |
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% Calculates the filter's order using the eq. 9-137 |
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design_filter_order = ceil( ... |
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acosh(((10^(specification_min_stop_attenuation/10)-1)/ ... |
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(10^(specification_max_pass_attenuation/10)-1))^(1/2)) ... |
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/acosh(1/normalized_pass_radial_frequency)); |
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% Calculates epsilon parameter using the eq. 9-123 |
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epsilon_parameter = 1/(10^(specification_min_stop_attenuation/10)-1)^(1/2); |
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% Calculates alpha using the eq. ?? wtf is this? =========^^^^^^^^^^^^^^^^^^&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&========== |
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alpha_parameter = asinh(1/epsilon_parameter)/design_filter_order; |
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% Calculates the frequency at which half power occurs using the eq. 9-139 |
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design_half_power_radial_frequency = specification_stop_radial_frequency/ ... |
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(cosh(acosh(1/epsilon_parameter)/design_filter_order)); % 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 |
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% ----- |
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% Initializes necessary variables |
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design_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 Q's of the poles |
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design_butterworth_angles = zeros([1 design_number_of_poles]); |
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poles_real_parts = zeros([1 design_number_of_poles]); |
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poles_imaginary_parts = zeros([1 design_number_of_poles]); |
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poles_radial_frequencies = zeros([1 design_number_of_poles]); |
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inverse_poles_Q = zeros([1 design_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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% inverse 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. 9-102 |
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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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poles_imaginary_parts(1,1) = cosh(alpha_parameter)* ... |
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sind(design_butterworth_angles(1,1)); |
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% Calculates the first pole's radial frequency using the eq. 9-150 |
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poles_radial_frequencies(1,1) = (poles_real_parts(1,1)^2+ ... |
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poles_imaginary_parts(1,1)^2)^(1/2); |
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% Calculates the first pole's Q using the eq. 9-151 |
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inverse_poles_Q(1,1) = 1/ ... |
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(2*cos(atan(poles_imaginary_parts(1,1)/poles_real_parts(1,1)))); |
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% Calculates the rest of the poles in the same way |
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for i=2:design_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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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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poles_imaginary_parts(1,i) = cosh(alpha_parameter)* ... |
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sind(design_butterworth_angles(1,i)); |
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% Pole's radial frequency, eq. 9-150 |
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poles_radial_frequencies(1,i) = (poles_real_parts(1,i)^2+ ... |
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poles_imaginary_parts(1,i)^2)^(1/2); |
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% Pole's Q, eq. 9-151 |
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inverse_poles_Q(1,i) = 1/ ... |
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(2*cos(atan(poles_imaginary_parts(1,i)/poles_real_parts(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/design_number_of_poles; |
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for i=1:design_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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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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poles_imaginary_parts(1,i) = cosh(alpha_parameter)* ... |
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sind(design_butterworth_angles(1,i)); |
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% Pole's radial frequency, eq. 9-150 |
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poles_radial_frequencies(1,i) = (poles_real_parts(1,i)^2+ ... |
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poles_imaginary_parts(1,i)^2)^(1/2); |
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% Pole's Q, eq. 9-151 |
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inverse_poles_Q(1,i) = 1/ ... |
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(2*cos(atan(poles_imaginary_parts(1,i)/poles_real_parts(1,i)))); |
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end |
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end |
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% Initializes array to hold the inversed poles |
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inverse_poles_radial_frequencies = zeros([1 design_number_of_poles]); |
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% Calculates inverse Chebyshev poles by inversing the Chebyshev poles |
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% using the eq. 9-146 |
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for i=1:design_number_of_poles |
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inverse_poles_radial_frequencies(1,i) = 1/poles_radial_frequencies(1,i); |
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end |
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% Initializes array to hold the transfer function zeros and a temporary |
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% helper variable |
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inverse_transfer_function_zeros = zeros([1 length(design_butterworth_angles)]); |
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temp_index = 1; |
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% Calculates the transfer function's zeros using the eq. 9-143 |
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for i=1:2:design_filter_order |
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inverse_transfer_function_zeros(1,temp_index) = sec((i*pi)/ ... |
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(2*design_filter_order)); |
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temp_index = temp_index + 1; |
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end |
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% Clears unneeded variable from workspace |
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clearVars = {'theta', 'i', 'temp_index', 'alpha_parameter' ... |
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'epsilon_parameter', 'normalized_pass_radial_frequency', ... |
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'poles_real_parts', 'poles_imaginary_parts', 'poles_radial_frequencies'}; |
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clear(clearVars{:}) |
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clear clearVars |
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% ========== NORMALIZED DESIGN END ========== |
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% ========== ZEROS-POLES GROUPING START ========== |
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% Grouping is done "by hand". The first pole, that has a radial frequency |
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% equal to 0.9631, is grouped with the first zero (1.0824) and the second |
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% pole, that has a radial frequency equal to 0.7484, is grouped with the |
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% second zero (2.6131). |
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% Two low pass notch units are required to implement the desired low pass filter. |
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% ========== ZEROS-POLES GROUPING END ========== |
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% ========== UNITS IMPLEMENTATION START ========== |
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% AEM(3) = 6, so the circuit shown in 7.23 is going to be used for the low |
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% pass notch units. |
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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 unit. |
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unit_low_pass_notch_resistors_1 = zeros([1 2]); |
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unit_low_pass_notch_resistors_2 = zeros([1 2]); |
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unit_low_pass_notch_resistors_3 = zeros([1 2]); |
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unit_low_pass_notch_resistors_4 = zeros([1 2]); |
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unit_low_pass_notch_resistors_5 = zeros([1 2]); |
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unit_low_pass_notch_capacitors = zeros([1 2]); |
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unit_low_pass_notch_gains_high = zeros([1 2]); |
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unit_low_pass_notch_gains_low = zeros([1 2]); |
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% The specifications for the first unit are: |
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% - inverse pole radial frequency = 0.9631 |
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% - zero at 1.0824 (zero > inverse pole radial frequency) |
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% - pole angle = 0.5822 |
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% normalized_inverse_pole_radial_frequency = 1; |
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normalized_transfer_function_zero = inverse_transfer_function_zeros(1,1)/ ... |
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inverse_poles_radial_frequencies(1,3); |
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% According to the design method outlined in 7.6-B, at page 35 |
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unit_low_pass_notch_resistors_1(1,1) = 1; % Ohm |
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% Calculates the capacity of the normalized circuit capacitors using the |
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% eq. 7-150 |
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unit_low_pass_notch_capacitors(1,1) = 1/(2*inverse_poles_Q(1,3)); % Farad |
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% Calculates the resistance of R2 using the same equations (7-150) |
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unit_low_pass_notch_resistors_2(1,1) = 4*inverse_poles_Q(1,3)^2; % Ohm |
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% Calculates the resistance of R5 using the eq. 7-152 |
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unit_low_pass_notch_resistors_5(1,1) = (4*inverse_poles_Q(1,3)^2)/ ... |
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(normalized_transfer_function_zero^2-1); % Ohm |
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unit_low_pass_notch_resistors_4(1,1) = 1; % Ohm |
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% Calculates the resistance of R3 using the eq. 7-155 |
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unit_low_pass_notch_resistors_3(1,1) = (normalized_transfer_function_zero^2)/ ... |
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(2*inverse_poles_Q(1,3)^2); % Ohm |
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% Calculates the gain of this unit in high frequencies using the eq. 7-143 |
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unit_low_pass_notch_gains_high(1,1) = (unit_low_pass_notch_resistors_4(1,1))/ ... |
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(unit_low_pass_notch_resistors_3(1,1)+unit_low_pass_notch_resistors_4(1,1)); |
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% Calculates the gain of this unit in low frequencies using the |
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% eq. 7-146, 7-147, 7-148, setting s = 0 |
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unit_low_pass_notch_gains_low(1,1) = unit_low_pass_notch_gains_high(1,1)* ... |
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normalized_transfer_function_zero^2; |
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% The specifications for the first unit are: |
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% - inverse pole radial frequency = $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
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% - zero at $$$$$$$$$$$$$$$$$$$$ (zero > inverse pole radial frequency) |
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% - pole angle = $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
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normalized_transfer_function_zero = inverse_transfer_function_zeros(1,2)/ ... |
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inverse_poles_radial_frequencies(1,2); |
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% According to the design method outlined in 7.6-B, at page 35 |
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unit_low_pass_notch_resistors_1(1,2) = 1; % Ohm |
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% Calculates the capacity of the normalized circuit capacitors using the |
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% eq. 7-150 |
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unit_low_pass_notch_capacitors(1,2) = 1/(2*inverse_poles_Q(1,2)); % Farad |
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% Calculates the resistance of R2 using the same equations (7-150) |
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unit_low_pass_notch_resistors_2(1,2) = 4*inverse_poles_Q(1,2)^2; % Ohm |
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% Calculates the resistance of R5 using the eq. 7-152 |
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unit_low_pass_notch_resistors_5(1,2) = (4*inverse_poles_Q(1,2)^2)/ ... |
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(normalized_transfer_function_zero^2-1); % Ohm |
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unit_low_pass_notch_resistors_4(1,2) = 1; % Ohm |
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% Calculates the resistance of R3 using the eq. 7-155 |
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unit_low_pass_notch_resistors_3(1,2) = (normalized_transfer_function_zero^2)/ ... |
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(2*inverse_poles_Q(1,2)^2); % Ohm |
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% Calculates the gain of this unit in high frequencies using the eq. 7-143 |
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unit_low_pass_notch_gains_high(1,2) = (unit_low_pass_notch_resistors_4(1,2))/ ... |
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(unit_low_pass_notch_resistors_3(1,2)+unit_low_pass_notch_resistors_4(1,2)); |
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% Calculates the gain of this unit in low frequencies using the |
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% eq. 7-146, 7-147, 7-148, setting s = 0 |
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unit_low_pass_notch_gains_low(1,2) = unit_low_pass_notch_gains_high(1,2)* ... |
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normalized_transfer_function_zero^2; |
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% Clears unneeded variable from workspace |
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clear normalized_transfer_function_zero |
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% ========== UNITS IMPLEMENTATION END ========== |
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% ========== DENORMALIZATION START ========== |
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% Unit sizes rescale |
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unit_frequency_scale_factors = zeros([1 2]); |
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unit_amplitude_scale_factors = zeros([1 2]); |
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% For unit 1 |
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unit_frequency_scale_factors(1,1) = specification_stop_radial_frequency* ... |
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inverse_poles_radial_frequencies(1,3); |
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% AEM(4) = 1, so the scaling will be performed to achieve a capacitor value |
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% of 0.1uF using the eq. 6-33 |
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unit_amplitude_scale_factors(1,1) = unit_low_pass_notch_capacitors(1,1)/ ... |
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(unit_frequency_scale_factors(1,1)*0.1*10^(-6)); |
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unit_low_pass_notch_resistors_1(1,1) = unit_low_pass_notch_resistors_1(1,1)* ... |
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unit_amplitude_scale_factors(1,1); % Ohm |
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unit_low_pass_notch_resistors_2(1,1) = unit_low_pass_notch_resistors_2(1,1)* ... |
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unit_amplitude_scale_factors(1,1); % Ohm |
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unit_low_pass_notch_resistors_3(1,1) = unit_low_pass_notch_resistors_3(1,1)* ... |
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unit_amplitude_scale_factors(1,1); % Ohm |
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unit_low_pass_notch_resistors_4(1,1) = unit_low_pass_notch_resistors_4(1,1)* ... |
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unit_amplitude_scale_factors(1,1); % Ohm |
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unit_low_pass_notch_resistors_5(1,1) = unit_low_pass_notch_resistors_5(1,1)* ... |
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unit_amplitude_scale_factors(1,1); % Ohm |
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unit_low_pass_notch_capacitors(1,1) = 0.1*10^(-6); % Farad |
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% For unit 2 |
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unit_frequency_scale_factors(1,2) = specification_stop_radial_frequency* ... |
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inverse_poles_radial_frequencies(1,2); |
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% AEM(4) = 1, so the scaling will be performed to achieve a capacitor value |
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% of 0.1uF using the eq. 6-33 |
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unit_amplitude_scale_factors(1,2) = unit_low_pass_notch_capacitors(1,2)/ ... |
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(unit_frequency_scale_factors(1,2)*0.1*10^(-6)); |
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unit_low_pass_notch_resistors_1(1,2) = unit_low_pass_notch_resistors_1(1,2)* ... |
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unit_amplitude_scale_factors(1,2); % Ohm |
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unit_low_pass_notch_resistors_2(1,2) = unit_low_pass_notch_resistors_2(1,2)* ... |
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unit_amplitude_scale_factors(1,2); % Ohm |
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unit_low_pass_notch_resistors_3(1,2) = unit_low_pass_notch_resistors_3(1,2)* ... |
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unit_amplitude_scale_factors(1,2); % Ohm |
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unit_low_pass_notch_resistors_4(1,2) = unit_low_pass_notch_resistors_4(1,2)* ... |
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unit_amplitude_scale_factors(1,2); % Ohm |
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unit_low_pass_notch_resistors_5(1,2) = unit_low_pass_notch_resistors_5(1,2)* ... |
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unit_amplitude_scale_factors(1,2); % Ohm |
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unit_low_pass_notch_capacitors(1,2) = 0.1*10^(-6); % Farad |
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% ========== DENORMALIZATION END ========== |
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% ========== GAIN ADJUSTMENT START ========== |
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total_fried_units_attenuation = unit_low_pass_notch_gains_low(1,1)*unit_low_pass_notch_gains_low(1,2); |
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unit_adjustment_gain = 1/total_fried_units_attenuation; |
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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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% ========== GAIN ADJUSTMENT END ========== |
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% ========== TRANSFER FUNCTIONS START ========== |
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% Builds numerator and denominator of the transfer function of each unit |
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% using the eq. 7-146, 7-147 & 7-148 |
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unit_1_numerator = [1 ... |
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(((unit_low_pass_notch_gains_high(1,1)-1)/ ... |
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(unit_low_pass_notch_gains_high(1,1)*unit_low_pass_notch_resistors_1(1,1)* ... |
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unit_low_pass_notch_capacitors(1,1)))+ ... |
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(2/(unit_low_pass_notch_resistors_2(1,1)* ... |
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unit_low_pass_notch_capacitors(1,1)))+ ... |
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(2/(unit_low_pass_notch_resistors_5(1,1)* ... |
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unit_low_pass_notch_capacitors(1,1)))) ... |
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(1/(unit_low_pass_notch_resistors_1(1,1)* ... |
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unit_low_pass_notch_resistors_5(1,1)*unit_low_pass_notch_capacitors(1,1)^2)+ ... |
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1/(unit_low_pass_notch_resistors_1(1,1)* ... |
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unit_low_pass_notch_resistors_2(1,1)*unit_low_pass_notch_capacitors(1,1)^2))]; |
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unit_1_denominator = [1 ... |
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2/(unit_low_pass_notch_resistors_2(1,1)* ... |
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unit_low_pass_notch_capacitors(1,1)) ... |
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1/(unit_low_pass_notch_resistors_1(1,1)* ... |
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unit_low_pass_notch_resistors_2(1,1)*unit_low_pass_notch_capacitors(1,1)^2)]; |
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unit_1_transfer_function = tf(unit_1_numerator, unit_1_denominator); |
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unit_1_transfer_function = unit_1_transfer_function*unit_low_pass_notch_gains_high(1,1); |
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|
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unit_2_numerator = [1 ... |
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((unit_low_pass_notch_gains_high(1,2)-1)/ ... |
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(unit_low_pass_notch_gains_high(1,2)*unit_low_pass_notch_resistors_1(1,2)* ... |
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unit_low_pass_notch_capacitors(1,2))+ ... |
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2/(unit_low_pass_notch_resistors_2(1,2)* ... |
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unit_low_pass_notch_capacitors(1,2))+ ... |
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2/(unit_low_pass_notch_resistors_5(1,2)* ... |
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unit_low_pass_notch_capacitors(1,2))) ... |
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(1/(unit_low_pass_notch_resistors_1(1,2)* ... |
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unit_low_pass_notch_resistors_5(1,2)*unit_low_pass_notch_capacitors(1,2)^2)+ ... |
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1/(unit_low_pass_notch_resistors_1(1,2)* ... |
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unit_low_pass_notch_resistors_2(1,2)*unit_low_pass_notch_capacitors(1,2)^2))]; |
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unit_2_denominator = [1 ... |
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2/(unit_low_pass_notch_resistors_2(1,2)* ... |
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unit_low_pass_notch_capacitors(1,2)) ... |
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1/(unit_low_pass_notch_resistors_1(1,2)* ... |
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unit_low_pass_notch_resistors_2(1,2)*unit_low_pass_notch_capacitors(1,2)^2)]; |
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|
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unit_2_transfer_function = tf(unit_2_numerator, unit_2_denominator); |
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unit_2_transfer_function = unit_2_transfer_function*unit_low_pass_notch_gains_high(1,2); |
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|
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tmpHpRC = double(1)/double(0.1*10^(-6)*4.14*10^3); |
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unit_3_transfer_function = tf([0 tmpHpRC], [1 tmpHpRC]); |
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|
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total_transfer_function = series(series(unit_1_transfer_function,unit_2_transfer_function), unit_3_transfer_function); |
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total_transfer_function = total_transfer_function*unit_adjustment_gain; |
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%plot_transfer_function(unit_1_transfer_function, [1 10]); |
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%plot_transfer_function(unit_2_transfer_function, [1 192.82 222.817 234 238.73]); |
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%plot_transfer_function(0.656*series(unit_1_transfer_function,unit_2_transfer_function), ... |
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% [1 159.155 192.82 222.817 234 238.73]); |
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%{ |
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plot_transfer_function(total_transfer_function, ... |
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[1 ... |
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specification_pass_radial_frequency/(2*pi) ... |
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design_half_power_radial_frequency/(2*pi) ... |
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specification_stop_radial_frequency/(2*pi)]); |
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%} |
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|
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%ltiview(unit_1_transfer_function); |
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%ltiview(unit_2_transfer_function); |
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%ltiview(unit_3_transfer_function); |
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%ltiview(total_transfer_function); |
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%ltiview(unit_1_transfer_function, unit_2_transfer_function, ... |
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% unit_3_transfer_function, total_transfer_function); |
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|
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% Clears unneeded variable from workspace |
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clear tmpHpRC |
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clear -regexp _numerator$ |
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clear -regexp _denominator$ |
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clear -regexp _transfer_function$ |
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|
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% ========== TRANSFER FUNCTIONS 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) |
|||
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)); |
|||
end |
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legend(Aw,'Location','best','FontSize',12); |
|||
set(gca,'FontSize',14); |
|||
end |
|||
Loading…
Reference in new issue