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Work on band elimination report unit implementation part, Add half power freq transformation

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Apostolos Fanakis 6 years ago
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d79f7e4fd2
11 changed files with 1822 additions and 26 deletions
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  1. 28
      Band Elimination Chebyshev/band_elimination_design.m
  2. 568
      report/3_band_elimination/3_band_elimination_design.pug
  3. 125
      report/3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_1_zero_pole_grouping.svg
  4. 115
      report/3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_2_zero_pole_grouping.svg
  5. 115
      report/3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_3_zero_pole_grouping.svg
  6. 115
      report/3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_4_zero_pole_grouping.svg
  7. 25
      report/3_band_elimination/assets/diagrams/band_elimination_unit_1_block_diagram.svg
  8. 23
      report/3_band_elimination/assets/diagrams/band_elimination_unit_2_block_diagram.svg
  9. 23
      report/3_band_elimination/assets/diagrams/band_elimination_unit_3_block_diagram.svg
  10. 23
      report/3_band_elimination/assets/diagrams/band_elimination_unit_4_block_diagram.svg
  11. 688
      report/3_band_elimination/assets/diagrams/multisim_band_elimination_chebyshev_circuit_layout_only_filter.svg

28
Band Elimination Chebyshev/band_elimination_design.m
View File

@ -132,11 +132,19 @@ epsilon_parameter = sqrt(10^(specification_max_pass_attenuation/10)-1);
% Calculates alpha parameter using the eq. 9-92 % Calculates alpha parameter using the eq. 9-92
alpha_parameter = asinh(1/epsilon_parameter)/design_filter_order; alpha_parameter = asinh(1/epsilon_parameter)/design_filter_order;
% Calculates the frequency at which half power occurs using the eq. 9-80 % Calculates the frequency at which half power for the prototype low pass
% TODO: denormalize!! ====================%%%%%%%%%%%%%%%%%%%%%%%%============================ % filter occurs using the eq. 9-80
design_half_power_radial_frequency = cosh(acosh(( ... temp_low_pass_half_power_radial_frequency = cosh(acosh(1/epsilon_parameter)/ ...
10^(specification_max_pass_attenuation/10)-1)^(-1/2))/ ...
design_filter_order); % rad/s design_filter_order); % rad/s
% Calculates the frequency at which half power for the band pass filter
% occurs using the transformation eq. 13-4
design_half_power_radial_frequency = zeros([1 2]);
temp_polynomial = [1 ...
design_filter_bandwidth /temp_low_pass_half_power_radial_frequency ...
-design_geometric_central_radial_frequency^2];
temp_roots = roots(temp_polynomial);
design_half_power_radial_frequency(1,1) = abs(temp_roots(1));
design_half_power_radial_frequency(1,2) = abs(temp_roots(2));
% ----- % -----
% Calculates stable poles, zeros, angles and other characteristic sizes % Calculates stable poles, zeros, angles and other characteristic sizes
@ -238,12 +246,13 @@ fprintf(['\n' '===== PROTOTYPE LOW PASS DESIGN =====' '\n' ...
'Filter order ceiling = %d\n' ... 'Filter order ceiling = %d\n' ...
'Epsilon parameter = %.3f\n' ... 'Epsilon parameter = %.3f\n' ...
'Alpha parameter = %.3f\n' ... 'Alpha parameter = %.3f\n' ...
'Radial frequency at which half power occurs = %.3frad/s\n' ... 'Radial frequencies at which half power occurs = %.3frad/s, %.3frad/s\n' ...
'Butterworth angles are ' char(177) '%.2f' char(176) ' and ' ... 'Butterworth angles are ' char(177) '%.2f' char(176) ' and ' ...
char(177) '%.2f' char(176) '\n'], ... char(177) '%.2f' char(176) '\n'], ...
temp_filter_order, design_filter_order, ... temp_filter_order, design_filter_order, ...
epsilon_parameter, alpha_parameter, ... epsilon_parameter, alpha_parameter, ...
design_half_power_radial_frequency, design_butterworth_angles(1,1), ... design_half_power_radial_frequency(1,1), ...
design_half_power_radial_frequency(1,2), design_butterworth_angles(1,1), ...
design_butterworth_angles(1,2)); design_butterworth_angles(1,2));
fprintf('\nLow pass Chebyshev poles found:\n'); fprintf('\nLow pass Chebyshev poles found:\n');
@ -258,9 +267,10 @@ end
% Clears unneeded variables from workspace % Clears unneeded variables from workspace
clearVars = {'prototype_normalized_stop_radial_frequency', ... clearVars = {'prototype_normalized_stop_radial_frequency', ...
'epsilon_parameter', 'alpha_parameter', 'theta', 'temp_filter_order'}; 'epsilon_parameter', 'alpha_parameter', 'theta'};
clear(clearVars{:}) clear(clearVars{:})
clear clearVars clear clearVars
clear -regexp ^temp_
% ========== PROTOTYPE LOW PASS DESIGN END ========== % ========== PROTOTYPE LOW PASS DESIGN END ==========
@ -834,7 +844,7 @@ ltiview(unit_transfer_function(1,1), unit_transfer_function(1,2), ...
unit_transfer_function(1,3), unit_transfer_function(1,4), ... unit_transfer_function(1,3), unit_transfer_function(1,4), ...
total_transfer_function); total_transfer_function);
%} %}
%{
hold off hold off
sampling_time_seconds = 60; % s sampling_time_seconds = 60; % s
@ -884,7 +894,7 @@ Pyy = system_output_fft.*conj(system_output_fft)/sampling_length_L;
figure(3) figure(3)
semilogx(frequency_vector,Pyy(1:sampling_length_L/2+1)) semilogx(frequency_vector,Pyy(1:sampling_length_L/2+1))
grid on grid on
%}
% Clears unneeded variables from workspace % Clears unneeded variables from workspace
clearVars = {'temp', 'Pyy', 'frequency_vector', 'i' ... clearVars = {'temp', 'Pyy', 'frequency_vector', 'i' ...
'system_output', 'total_transfer_function', 'system_output_fft'}; 'system_output', 'total_transfer_function', 'system_output_fft'};

568
report/3_band_elimination/3_band_elimination_design.pug
View File

@ -229,23 +229,34 @@ p.
figure.block-center figure.block-center
.ui.grid .ui.grid
.two.wide.column .four.wide.column
.three.wide.column
.row .row
img(src="1_low_pass/assets/diagrams/inverse_chebyshev_unit_1_zero_pole_grouping.svg")/ img(src="3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_1_zero_pole_grouping.svg")/
.row.top-7mm .row.top-7mm
img(src="1_low_pass/assets/diagrams/low_pass_notch_unit_diagram.svg")/ img(src="3_band_elimination/assets/diagrams/band_elimination_unit_1_block_diagram.svg")/
.row.top-5mm .row.top-5mm
p.center #[strong Unit 1] p.center #[strong Unit 1]
.six.wide.column .four.wide.column
.three.wide.column
.row .row
img(src="1_low_pass/assets/diagrams/inverse_chebyshev_unit_2_zero_pole_grouping.svg")/ img(src="3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_2_zero_pole_grouping.svg")/
.row.top-7mm .row.top-7mm
img(src="1_low_pass/assets/diagrams/low_pass_notch_unit_diagram.svg")/ img(src="3_band_elimination/assets/diagrams/band_elimination_unit_2_block_diagram.svg")/
.row.top-5mm .row.top-5mm
p.center #[strong Unit 2] p.center #[strong Unit 2]
.two.wide.column .four.wide.column
.row
img(src="3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_3_zero_pole_grouping.svg")/
.row.top-7mm
img(src="3_band_elimination/assets/diagrams/band_elimination_unit_3_block_diagram.svg")/
.row.top-5mm
p.center #[strong Unit 3]
.four.wide.column
.row
img(src="3_band_elimination/assets/diagrams/band_elimination_chebyshev_unit_4_zero_pole_grouping.svg")/
.row.top-7mm
img(src="3_band_elimination/assets/diagrams/band_elimination_unit_4_block_diagram.svg")/
.row.top-5mm
p.center #[strong Unit 4]
figcaption figcaption
.reference #[span.plot-count] .reference #[span.plot-count]
.caption. .caption.
@ -256,6 +267,530 @@ h4 Υλοποίηση συνάρτησης μεταφοράς
p. p.
Από τον αριθμό ΑΕΜ (8261) υποδεικνύεται η χρήση των κυκλωμάτων high pass notch και low pass notch του κεφαλαίου 7, με χρήση των κυκλωμάτων των σχημάτων 7.21 και 7.23. Από τον αριθμό ΑΕΜ (8261) υποδεικνύεται η χρήση των κυκλωμάτων high pass notch και low pass notch του κεφαλαίου 7, με χρήση των κυκλωμάτων των σχημάτων 7.21 και 7.23.
h5 Μονάδα 1
p Η πρώτη μονάδα μονάδα του φίλτρου είναι το πρώτο από τα δύο high pass notch δεύτερης τάξης που πρέπει να υλοποιηθούν. Για τη μονάδα ισχύουν:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Προδιαγραφή
th Τιμή
tbody
tr
td ω#[sub 0]
td 21505.945
tr
td ω#[sub Z]
td 15707.963
tr
td ω#[sub Z]<ω#[sub 0]
td #[i.large.teal.checkmark.icon]
tr
td Q
td 1.751
figcaption
.reference #[span.table-count]
.caption.
Προδιαγραφές μονάδας 1 (πρώτη μονάδα high pass notch)
p.
Γίνεται κανονικοποίηση των συχνοτήτων ως προς το ω#[sub 0], ώστε Ω#[sub 0]=1:
p.latex-equation.
$$\Omega_Z = \frac{\omega_Z}{\omega_0} = \frac{15707.963}{21505.945} = 0.7304<1$$
p.
Υπολογίζονται τα κέρδη και οι τιμές των στοιχείων του κυκλώματος του φίλτρου, με χρήση της μεθοδολογίας που περιγράφεται στο κεφάλαιο 7.6-B (σελίδα 35) και των εξισώσεων #[span.course-notes-equation 7-135] εώς και #[span.course-notes-equation 7-140]:
p.latex-equation.
$$k_1=\frac{\Omega_0^2}{\Omega_z^2}-1 = \frac{1}{0.7304^2}-1 = 0.8745$$
p.latex-equation.
$$k_2=\frac{(2+k_1)Q^2}{(2+k_1)Q^2+1} = \frac{(2+0.8745)1.751^2}{(2+0.8745)1.751^2+1} = 0.8981$$
p.latex-equation.
$$k=k_2\frac{\Omega_0^2}{\Omega_z^2} = 0.8981\frac{1}{0.7304^2} = 1.6835\hspace{1cm}\text{(Gain at high frequencies)}$$
p.latex-equation.
$$R_1 = 1\text{ Ohm}$$
p.latex-equation.
$$R_2 = Q^2(k_1+2)^2 = 1.751^2(0.8745+2)^2 = 25.3394\text{ Ohm}$$
p.latex-equation.
$$R_3 = 1\text{ Ohm}$$
p.latex-equation.
$$R_4 = Q^2(k_1+2) = 1.751^2(0.8745+2) = 8.8154\text{ Ohm}$$
p.latex-equation.
$$C = \frac{1}{Q(2+k_1)} = \frac{1}{1.751(2+0.8745)} = 0.1987\text{ F}$$
p.latex-equation.
$$C_1 = k_1C = 0.8745*0.1987 = 0.1737\text{ F}$$
p #[strong Κλιμακοποίηση]
p.
Γίνεται κλιμακoποίηση των στοιχείων της μονάδας για να μεταφερθούν οι συχνότητες στις πραγματικές τιμές. Επιλέγεται:
p.latex-equation.
$$k_{f} = \omega_0 = 21505.945$$
p.
Με βάση τον αριθμό ΑΕΜ (8261) επιλέγεται κατάλληλος συντελεστής κλιμακοποίησης πλάτους ώστε να επιτευχθεί τιμή πυκνωτών ίση με 0.1μF, γίνεται χρήση του τύπου #[span.course-notes-equation 6-33]:
p.latex-equation.
$$k_{m} = \frac{C_{old}}{k_fC_{new}} = \frac{0.1987}{21505.945*0.1*10^{-6}} = 92.3725$$
p Οι τελικές τιμές των στοιχείων φαίνονται στον παρακάτω πίνακα:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Στοιχείο/Κέρδος
th(colspan="2") Τιμή
tbody
tr
td C
td(colspan="2") 0,1 μF
tr
td C#[sub 1]
td(colspan="2") 0.08745 μF
tr
td R#[sub 1]
td(colspan="2") 92.3725 Ohm
tr
td R#[sub 2]
td(colspan="2") 2340.7 Ohm
tr
td R#[sub 3]
td(colspan="2") 92.3725 Ohm
tr
td R#[sub 4]
td(colspan="2") 814.2974 Ohm
tr
td Κέρδος στις υψηλές συχνότητες
td 1.6835
td 4.52 dB
figcaption
.reference #[span.table-count]
.caption.
Οι τιμές των στοιχείων της πρώτης μονάδας και το κέρδος στις υψηλές συχνότητες
p.
Η συνάρτηση μεταφοράς της μονάδας υπολογίζεται χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-129], #[span.course-notes-equation 7-130] και #[span.course-notes-equation 7-131]:
p.latex-equation.
$$\begin{align*}
T_{BE}^1(s) &= k\frac{s^2+\frac{1}{(k_1+1)R_1R_2C^2}}{s^2+\frac{k_1+2}{R_2C}s+\frac{1}{R_1R_2C^2}} \\[3.5mm]
&=1.6835\frac{s^2+\frac{1}{(0.8745+1)*92.3725*2340.7*(0.1*10^{-6})^2}}{s^2+\frac{0.8745+2}{2340.7*0.1*10^{-6}}s+\frac{1}{92.6725*2340.7*(0.1*10^{-6})^2}} \\[3.5mm]
&=\frac{1.6835s^2+4.154*10^8}{s^2+12280.5s+4.625*10^8}
\end{align*}$$
h5 Μονάδα 2
p Η δεύτερη μονάδα μονάδα του φίλτρου είναι το πρώτο από τα δύο low pass notch δεύτερης τάξης που πρέπει να υλοποιηθούν. Για τη μονάδα ισχύουν:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Προδιαγραφή
th Τιμή
tbody
tr
td ω#[sub 0]
td 11473.112
tr
td ω#[sub Z]
td 15707.963
tr
td ω#[sub Z]>ω#[sub 0]
td #[i.large.teal.checkmark.icon]
tr
td Q
td 1.751
figcaption
.reference #[span.table-count]
.caption.
Προδιαγραφές μονάδας 2 (πρώτη μονάδα low pass notch)
p.
Γίνεται κανονικοποίηση των συχνοτήτων ως προς το ω#[sub 0], ώστε Ω#[sub 0]=1:
p.latex-equation.
$$\Omega_Z = \frac{\omega_Z}{\omega_0} = \frac{15707.963}{11473.112} = 1.3691>1$$
p.
Υπολογίζονται τα στοιχεία του κυκλώματος του φίλτρου, με χρήση της μεθοδολογίας που περιγράφεται στο κεφάλαιο 7.6-B (σελίδα 35) και των εξισώσεων #[span.course-notes-equation 7-150], #[span.course-notes-equation 7-152], #[span.course-notes-equation 7-155]:
p.latex-equation.
$$C = \frac{1}{2Q} = \frac{1}{2*1.751} = 0.2855\text{ F}$$
p.latex-equation.
$$R_2 = 4Q^2 = 4*1.751^2 = 12.2671\text{ Ohm}$$
p.latex-equation.
$$R_5 = \frac{4Q^2}{\Omega_Z^2-1} = \frac{4*1.751^2}{1.3691^2-1} = 14.0282\text{ Ohm}$$
p.latex-equation.
$$R_3 = \frac{\Omega_Z^2}{2Q^2} = \frac{1.3691^2}{2*1.751^2} = 0.3056\text{ Ohm}$$
p.latex-equation.
$$R_1 = R_4 = 1\text{ Ohm}$$
p.
Υπολογίζεται το κέρδος της μονάδας στις υψηλές συχνότητες, χρησιμοποιώντας την εξίσωση #[span.course-notes-equation 7-143]:
p.latex-equation.
$$k_{high} = \frac{R_4}{R_3+R_4} = \frac{1}{0.3056+1} = 0.7659$$
p.
Υπολογίζεται το κέρδος της μονάδας στις χαμηλές συχνότητες, χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-146], #[span.course-notes-equation 7-147] και #[span.course-notes-equation 7-148], θέτοντας s=0:
p.latex-equation.
$$k_{low} = k_{high}\Omega_Z^2 = 0.7659*1.3691^2 = 1.4357$$
p #[strong Κλιμακοποίηση]
p.
Γίνεται κλιμακoποίηση των στοιχείων της μονάδας για να μεταφερθούν οι συχνότητες στις πραγματικές τιμές. Επιλέγεται:
p.latex-equation.
$$k_{f} = \omega_0 = 11473.112$$
p.
Με βάση τον αριθμό ΑΕΜ (8261) επιλέγεται κατάλληλος συντελεστής κλιμακοποίησης πλάτους ώστε να επιτευχθεί τιμή πυκνωτών ίση με 0.1μF, γίνεται χρήση του τύπου #[span.course-notes-equation 6-33]:
p.latex-equation.
$$k_{m} = \frac{C_{old}}{k_fC_{new}} = \frac{0.2855}{11473.112*0.1*10^{-6}} = 248.8555$$
p Οι τελικές τιμές των στοιχείων φαίνονται στον παρακάτω πίνακα:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Στοιχείο/Κέρδος
th(colspan="2") Τιμή
tbody
tr
td C
td(colspan="2") 0,1 μF
tr
td R#[sub 1]
td(colspan="2") 248.8555 Ohm
tr
td R#[sub 2]
td(colspan="2") 3052.7 Ohm
tr
td R#[sub 3]
td(colspan="2") 76.0521 Ohm
tr
td R#[sub 4]
td(colspan="2") 248.8555 Ohm
tr
td R#[sub 5]
td(colspan="2") 3491 Ohm
tr
td Κέρδος στις υψηλές συχνότητες
td 0.7659
td -2.316 dB
tr
td Κέρδος στις χαμηλές συχνότητες
td 1.4357
td 3.14 dB
figcaption
.reference #[span.table-count]
.caption.
Τιμές των στοιχείων της δεύτερης μονάδας και κέρδη στις υψηλές και χαμηλές συχνότητες
p.
Η συνάρτηση μεταφοράς της μονάδας υπολογίζεται χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-146], #[span.course-notes-equation 7-147] και #[span.course-notes-equation 7-148]:
p.latex-equation.
$$\begin{align*}
T_{BE}^2(s) &= k_{high}\frac{s^2+\big[\frac{1}{R_1R_5C^2}+\frac{1}{R_1R_2C^2}\big]}{s^2+\frac{2}{R_2C}s+\frac{1}{R_1R_2C^2}}= \\[3.5mm]
&=0.7659\frac{s^2+\big[\frac{1}{248.8555*3491*(0.1*10^{-6})^2}+\frac{1}{248.8555*3052.7*(0.1*10^{-6})^2}\big]}{s^2+\frac{2}{3052.7*0.1*10^{-6}}s+\frac{1}{248.8555*3052.7*(0.1*10^{-6})^2}} = \\[3.5mm]
&=\frac{0.7659s^2+1.89*10^8}{s^2+6551s+1.316*10^8}
\end{align*}$$
h5 Μονάδα 3
p Η τρίτη μονάδα μονάδα του φίλτρου είναι το δεύτερο από τα δύο high pass notch δεύτερης τάξης που πρέπει να υλοποιηθούν. Για τη μονάδα ισχύουν:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Προδιαγραφή
th Τιμή
tbody
tr
td ω#[sub 0]
td 19957.291
tr
td ω#[sub Z]
td 15707.963
tr
td ω#[sub Z]<ω#[sub 0]
td #[i.large.teal.checkmark.icon]
tr
td Q
td 12.676
figcaption
.reference #[span.table-count]
.caption.
Προδιαγραφές μονάδας 3 (δεύτερη μονάδα high pass notch)
p.
Γίνεται κανονικοποίηση των συχνοτήτων ως προς το ω#[sub 0], ώστε Ω#[sub 0]=1:
p.latex-equation.
$$\Omega_Z = \frac{\omega_Z}{\omega_0} = \frac{15707.963}{19957.291} = 0.7871<1$$
p.
Υπολογίζονται τα κέρδη και οι τιμές των στοιχείων του κυκλώματος του φίλτρου, με χρήση της μεθοδολογίας που περιγράφεται στο κεφάλαιο 7.6-B (σελίδα 35) και των εξισώσεων #[span.course-notes-equation 7-135] εώς και #[span.course-notes-equation 7-140]:
p.latex-equation.
$$k_1=\frac{\Omega_0^2}{\Omega_z^2}-1 = \frac{1}{0.7871^2}-1 = 0.6142$$
p.latex-equation.
$$k_2=\frac{(2+k_1)Q^2}{(2+k_1)Q^2+1} = \frac{(2+0.6142)12.676^2}{(2+0.6142)12.676^2+1} = 0.9976$$
p.latex-equation.
$$k=k_2\frac{\Omega_0^2}{\Omega_z^2} = 0.9976\frac{1}{0.7871^2} = 1.6104\hspace{1cm}\text{(Gain at high frequencies)}$$
p.latex-equation.
$$R_1 = 1\text{ Ohm}$$
p.latex-equation.
$$R_2 = Q^2(k_1+2)^2 = 12.676^2(0.6142+2)^2 = 1098\text{ Ohm}$$
p.latex-equation.
$$R_3 = 1\text{ Ohm}$$
p.latex-equation.
$$R_4 = Q^2(k_1+2) = 12.676^2(0.6142+2) = 420.0291\text{ Ohm}$$
p.latex-equation.
$$C = \frac{1}{Q(2+k_1)} = \frac{1}{12.676(2+0.6142)} = 0.0302\text{ F}$$
p.latex-equation.
$$C_1 = k_1C = 0.6142*0.0302 = 0.0185\text{ F}$$
p #[strong Κλιμακοποίηση]
p.
Γίνεται κλιμακoποίηση των στοιχείων της μονάδας για να μεταφερθούν οι συχνότητες στις πραγματικές τιμές. Επιλέγεται:
p.latex-equation.
$$k_{f} = \omega_0 = 19957.291$$
p.
Με βάση τον αριθμό ΑΕΜ (8261) επιλέγεται κατάλληλος συντελεστής κλιμακοποίησης πλάτους ώστε να επιτευχθεί τιμή πυκνωτών ίση με 0.1μF, γίνεται χρήση του τύπου #[span.course-notes-equation 6-33]:
p.latex-equation.
$$k_{m} = \frac{C_{old}}{k_fC_{new}} = \frac{0.0302}{19957.291*0.1*10^{-6}} = 15.1212$$
p Οι τελικές τιμές των στοιχείων φαίνονται στον παρακάτω πίνακα:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Στοιχείο/Κέρδος
th(colspan="2") Τιμή
tbody
tr
td C
td(colspan="2") 0,1 μF
tr
td C#[sub 1]
td(colspan="2") 0.06142 μF
tr
td R#[sub 1]
td(colspan="2") 15.1212 Ohm
tr
td R#[sub 2]
td(colspan="2") 16604 Ohm
tr
td R#[sub 3]
td(colspan="2") 15.1212 Ohm
tr
td R#[sub 4]
td(colspan="2") 6351.4 Ohm
tr
td Κέρδος στις υψηλές συχνότητες
td 1.6104
td 4.14 dB
figcaption
.reference #[span.table-count]
.caption.
Οι τιμές των στοιχείων της τρίτης μονάδας και το κέρδος στις υψηλές συχνότητες
p.
Η συνάρτηση μεταφοράς της μονάδας υπολογίζεται χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-129], #[span.course-notes-equation 7-130] και #[span.course-notes-equation 7-131]:
p.latex-equation.
$$\begin{align*}
T_{BE}^3(s) &= k\frac{s^2+\frac{1}{(k_1+1)R_1R_2C^2}}{s^2+\frac{k_1+2}{R_2C}s+\frac{1}{R_1R_2C^2}} \\[3.5mm]
&=1.6104\frac{s^2+\frac{1}{(0.6142+1)*15.1212*16604*(0.1*10^{-6})^2}}{s^2+\frac{0.6142+2}{16604*0.1*10^{-6}}s+\frac{1}{15.1212*16604*(0.1*10^{-6})^2}} \\[3.5mm]
&=\frac{1.6104s^2+3.973*10^8}{s^2+1574.4s+3.983*10^8}
\end{align*}$$
h5 Μονάδα 4
p Η τέταρτη μονάδα μονάδα του φίλτρου είναι το δεύτερο από τα δύο low pass notch δεύτερης τάξης που πρέπει να υλοποιηθούν. Για τη μονάδα ισχύουν:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Προδιαγραφή
th Τιμή
tbody
tr
td ω#[sub 0]
td 12363.407
tr
td ω#[sub Z]
td 15707.963
tr
td ω#[sub Z]>ω#[sub 0]
td #[i.large.teal.checkmark.icon]
tr
td Q
td 12.676
figcaption
.reference #[span.table-count]
.caption.
Προδιαγραφές μονάδας 4 (δεύτερη μονάδα low pass notch)
p.
Γίνεται κανονικοποίηση των συχνοτήτων ως προς το ω#[sub 0], ώστε Ω#[sub 0]=1:
p.latex-equation.
$$\Omega_Z = \frac{\omega_Z}{\omega_0} = \frac{15707.963}{ 12363.407} = 1.2705>1$$
p.
Υπολογίζονται τα στοιχεία του κυκλώματος του φίλτρου, με χρήση της μεθοδολογίας που περιγράφεται στο κεφάλαιο 7.6-B (σελίδα 35) και των εξισώσεων #[span.course-notes-equation 7-150], #[span.course-notes-equation 7-152], #[span.course-notes-equation 7-155]:
p.latex-equation.
$$C = \frac{1}{2Q} = \frac{1}{2* 12.676} = 0.0394\text{ F}$$
p.latex-equation.
$$R_2 = 4Q^2 = 4* 12.676^2 = 642.7\text{ Ohm}$$
p.latex-equation.
$$R_5 = \frac{4Q^2}{\Omega_Z^2-1} = \frac{4* 12.676^2}{1.2705^2-1} = 1046.3\text{ Ohm}$$
p.latex-equation.
$$R_3 = \frac{\Omega_Z^2}{2Q^2} = \frac{1.2705^2}{2* 12.676^2} = 0.0050\text{ Ohm}$$
p.latex-equation.
$$R_1 = R_4 = 1\text{ Ohm}$$
p.
Υπολογίζεται το κέρδος της μονάδας στις υψηλές συχνότητες, χρησιμοποιώντας την εξίσωση #[span.course-notes-equation 7-143]:
p.latex-equation.
$$k_{high} = \frac{R_4}{R_3+R_4} = \frac{1}{0.0050+1} = 0.995$$
p.
Υπολογίζεται το κέρδος της μονάδας στις χαμηλές συχνότητες, χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-146], #[span.course-notes-equation 7-147] και #[span.course-notes-equation 7-148], θέτοντας s=0:
p.latex-equation.
$$k_{low} = k_{high}\Omega_Z^2 = 0.995*1.2705^2 = 1.6062$$
p #[strong Κλιμακοποίηση]
p.
Γίνεται κλιμακoποίηση των στοιχείων της μονάδας για να μεταφερθούν οι συχνότητες στις πραγματικές τιμές. Επιλέγεται:
p.latex-equation.
$$k_{f} = \omega_0 = 12363.407$$
p.
Με βάση τον αριθμό ΑΕΜ (8261) επιλέγεται κατάλληλος συντελεστής κλιμακοποίησης πλάτους ώστε να επιτευχθεί τιμή πυκνωτών ίση με 0.1μF, γίνεται χρήση του τύπου #[span.course-notes-equation 6-33]:
p.latex-equation.
$$k_{m} = \frac{C_{old}}{k_fC_{new}} = \frac{0.0394}{ 12363.407*0.1*10^{-6}} = 31.9053$$
p Οι τελικές τιμές των στοιχείων φαίνονται στον παρακάτω πίνακα:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Στοιχείο/Κέρδος
th(colspan="2") Τιμή
tbody
tr
td C
td(colspan="2") 0,1 μF
tr
td R#[sub 1]
td(colspan="2") 31.9053 Ohm
tr
td R#[sub 2]
td(colspan="2") 20505 Ohm
tr
td R#[sub 3]
td(colspan="2") 0.1603 Ohm
tr
td R#[sub 4]
td(colspan="2") 31.9053 Ohm
tr
td R#[sub 5]
td(colspan="2") 33384 Ohm
tr
td Κέρδος στις υψηλές συχνότητες
td 0.995
td -0.044 dB
tr
td Κέρδος στις χαμηλές συχνότητες
td 1.6062
td 4.116 dB
figcaption
.reference #[span.table-count]
.caption.
Τιμές των στοιχείων της τέταρτης μονάδας και κέρδη στις υψηλές και χαμηλές συχνότητες
p.
Η συνάρτηση μεταφοράς της μονάδας υπολογίζεται χρησιμοποιώντας τις εξισώσεις #[span.course-notes-equation 7-146], #[span.course-notes-equation 7-147] και #[span.course-notes-equation 7-148]:
p.latex-equation.
$$\begin{align*}
T_{BE}^4(s) &= k_{high}\frac{s^2+\big[\frac{1}{R_1R_5C^2}+\frac{1}{R_1R_2C^2}\big]}{s^2+\frac{2}{R_2C}s+\frac{1}{R_1R_2C^2}}= \\[3.5mm]
&=0.995\frac{s^2+\big[\frac{1}{31.9053*33384*(0.1*10^{-6})^2}+\frac{1}{31.9053*20505*(0.1*10^{-6})^2}\big]}{s^2+\frac{2}{20505*0.1*10^{-6}}s+\frac{1}{31.9053*20505*(0.1*10^{-6})^2}} = \\[3.5mm]
&=\frac{0.995s^2+2.455*10^8}{s^2+975.4s+1.529*10^8}
\end{align*}$$
h4 Ρύθμιση κέρδους
p.
Με βάση τον αριθμό ΑΕΜ (8261) πραγματοποιείται ρύθμιση κέρδους με στόχο την επίτευξη κέρδους 0 dB στη ζώνη διόδου.
p.
Κατά την υλοποίηση των μονάδων HPN και LPN διαπιστώθηκε ότι κάθε μονάδα εισάγει ένα κέρδος. Το συνολικό κέρδος που εισάγουν οι μονάδες είναι:
p.latex-equation.
$$k = k_{high}^1k_{high}^2k_{high}^3k_{high}^4 = 1.6835*0.7659*1.6104*0.995 = 2.066$$
p.
Για τη ρύθμιση του κέρδους χρησιμοποιείται μία αναστρέφουσα συνδεσμολογία με κέρδος:
p.latex-equation.
$${k}' = \frac{1}{k} = \frac{1}{2.066} = 0.484$$
p.
Επιλέγεται η χρήση αντίστασης εισόδου ίσης με r#[sub 1]=10 kOhm. Έτσι η αντίσταση ανατροφοδότησης υπολογίζεται:
p.latex-equation.
$$r_2 = r_1{k}' = 10*10^3*0.484 = 4840\text{ Ohm}$$
h4 Συναρτήσεις μεταφοράς h4 Συναρτήσεις μεταφοράς
p. p.
@ -271,19 +806,19 @@ figure.block-center.width-15cm
tr tr
td Πρώτη μονάδα (Unit 1) td Πρώτη μονάδα (Unit 1)
td. td.
$$T_{BP(s)}^1 =-\frac{795.697s}{s^2+542.1426s+35270884.89}$$ $$T_{BE}^1(s) = \frac{1.6835s^2+4.154*10^8}{s^2+12280.5s+4.625*10^8}$$
tr tr
td Δεύτερη μονάδα (Unit 2) td Δεύτερη μονάδα (Unit 2)
td. td.
$$T_{BP(s)}^2 =-\frac{721.397s}{s^2+491.5187s+28991510.41}$$ $$T_{BE}^2(s) = \frac{0.7659s^2+1.89*10^8}{s^2+6551s+1.316*10^8}$$
tr tr
td Τρίτη μονάδα (Unit 3) td Τρίτη μονάδα (Unit 3)
td. td.
$$T_{BP(s)}^3 =-\frac{1524.1s}{s^2+239.223s+40486530.84}$$ $$T_{BE}^3(s) = \frac{1.6104s^2+3.973*10^8}{s^2+1574.4s+3.983*10^8}$$
tr tr
td Τέταρτη μονάδα (Unit 4) td Τέταρτη μονάδα (Unit 4)
td. td.
$$T_{BP(s)}^4 =-\frac{1203.7s}{s^2+188.933s+25255181.75}$$ $$T_{BE}^4(s) = \frac{0.995s^2+2.455*10^8}{s^2+975.4s+1.529*10^8}$$
figcaption figcaption
.reference #[span.table-count] .reference #[span.table-count]
.caption. .caption.
@ -293,15 +828,14 @@ p.
Η συνολική συνάρτηση μεταφοράς του φίλτρου υπολογίζεται: Η συνολική συνάρτηση μεταφοράς του φίλτρου υπολογίζεται:
p.latex-equation. p.latex-equation.
$$\begin{align*} $$\begin{align*} T_{BP(s)} &= {k}'T_{BP(s)}^1T_{BP(s)}^2T_{BP(s)}^3T_{BP(s)}^4 \\[3.5mm] &=\frac{s^8+9.8696*10^8s^6+3.6528*10^{17}s^4+6.0087*10^{25}s^2+3.7065*10^{33}}{s^8+21382*s^7+1.2753*10^{9}*s^6+1.7404*10^{13}*s^5+5.1829*10^{17}*s^4+} \hspace{3mm}\text{...} \\[3.5mm] &\text{...}\hspace{3mm}\frac{s^8+9.8696*10^8s^6+3.6528*10^{17}s^4+6.0087*10^{25}s^2+3.7065*10^{33}}{+4.2943*10^{21}*s^3+7.7641*10^{25}*s^2+3.2119*10^{29}*s+3.7065*10^{33}} \end{align*}$$
T_{BP(s)} &= T_{BP(s)}^1T_{BP(s)}^2T_{BP(s)}^3T_{BP(s)}^4 \\[3.5mm] &=\frac{1.053*10^{12}*s^4}{s^8+1461.8*s^7+1.3076*10^{8}*s^6+1.4238*10^{11}*s^5+6.3188*10^{15}*s^4+} \hspace{3mm}\text{...} \\[3.5mm] &\text{...}\hspace{3mm}\frac{1.053*10^{12}*s^4}{+4.5529*10^{18}*s^3+1.3371*10^{23}*s^2+4.78*10^{25}*s+1.04556*10^{30}} \end{align*}$$
p. p.
Στο παρακάτω σχήμα φαίνεται το τελικό κύκλωμα του φίλτρου: Στο παρακάτω σχήμα φαίνεται το τελικό κύκλωμα του φίλτρου:
figure.block-center.width-19cm figure.block-center.width-19cm
img(src="2_band_pass/assets/diagrams/multisim_band_pass_chebyshev_circuit_layout_only_filter.svg").width-19cm img(src="3_band_elimination/assets/diagrams/multisim_band_elimination_chebyshev_circuit_layout_only_filter.svg").width-19cm
figcaption figcaption
.reference #[span.plot-count] .reference #[span.plot-count]
.caption. .caption.
Κύκλωμα κατωδιαβατού φίλτρου Κύκλωμα ζωνοφρακτικού φίλτρου

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