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Init low pass report matlab part

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Apostolos Fanakis 6 years ago
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  1. 84
      report/1_low_pass/1_low_pass_design.pug
  2. 65
      report/1_low_pass/1_low_pass_transfer_function_matlab.pug
  3. 264
      report/1_low_pass/assets/diagrams/combined_transfer_functions_bode.svg
  4. 394
      report/1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_circuit_layout.svg
  5. 406
      report/1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_original_gain_attenuation.svg
  6. 406
      report/1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_total_attenuation.svg
  7. 406
      report/1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_total_transfer_function.svg
  8. 302
      report/1_low_pass/assets/diagrams/low_pass_notch_unit_1_transfer_function.svg
  9. 304
      report/1_low_pass/assets/diagrams/low_pass_notch_unit_2_transfer_function.svg
  10. 6
      report/report.scss

84
report/1_low_pass/1_low_pass_design.pug
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@ -30,7 +30,7 @@ p.latex-equation.
$$n = \left \lceil \frac{cos^{-1}\bigg(\sqrt{\frac{10^{\frac{a_{min}}{10}}-1}{10^{\frac{a_{max}}{10}}-1}}\bigg)}{cos^{-1}(\frac{1}{\Omega_p})} \right \rceil = \left \lceil \frac{cos^{-1}\bigg(\sqrt{\frac{10^{2.375}-1}{10^{0.035}-1}}\bigg)}{cos^{-1}(2.1)} \right \rceil = \left \lceil \frac{4.6642}{1.373} \right \rceil = \left \lceil 3.397 \right \rceil = 4$$ $$n = \left \lceil \frac{cos^{-1}\bigg(\sqrt{\frac{10^{\frac{a_{min}}{10}}-1}{10^{\frac{a_{max}}{10}}-1}}\bigg)}{cos^{-1}(\frac{1}{\Omega_p})} \right \rceil = \left \lceil \frac{cos^{-1}\bigg(\sqrt{\frac{10^{2.375}-1}{10^{0.035}-1}}\bigg)}{cos^{-1}(2.1)} \right \rceil = \left \lceil \frac{4.6642}{1.373} \right \rceil = \left \lceil 3.397 \right \rceil = 4$$
p. p.
Από τον παραπάνω τύπο φαίνεται ότι κατά τον υπολογισμό της τάξης του φίλτρου γίνεται στρογγυλοποίηση της τάξης προς τον επόμενο #[strong μεγαλύτερο] ακέραιο. Αυτό γίνεται επειδή δεν είναι δυνατή η υλοποίηση ενός φίλτρου ρητής τάξεως, έτσι είναι απαραίτητο η τάξη να στρογγυλοποιηθή. Η στρογγυλοποίηση είναι σημαντικό να γίνει προς τα επάνω (ceiling) ώστε να επιτευχθούν οι προδιαγραφές του φίλτρου. Μία πιθανή στρογγυλοποίηση προς τα κάτω θα είχε ως αποτέλεσμα την αποτυχία στη σχεδίαση. Από τον παραπάνω τύπο φαίνεται ότι κατά τον υπολογισμό της τάξης του φίλτρου γίνεται στρογγυλοποίηση της τάξης προς τον επόμενο #[strong μεγαλύτερο] ακέραιο. Αυτό γίνεται επειδή δεν είναι δυνατή η υλοποίηση ενός φίλτρου ρητής τάξεως, έτσι είναι απαραίτητο η τάξη να στρογγυλοποιηθεί. Η στρογγυλοποίηση είναι σημαντικό να γίνει προς τα επάνω (ceiling) ώστε να επιτευχθούν οι προδιαγραφές του φίλτρου. Μία πιθανή στρογγυλοποίηση προς τα κάτω θα είχε ως αποτέλεσμα την αποτυχία στη σχεδίαση.
p. p.
Λόγω της στρογγυλοποίησης αυτής, αναμένεται μάλιστα να υπάρχει υπερκάλυψη σε τουλάχιστον μία από τις προδιαγραφές. Στη συγκεκριμένη περίπτωση, λόγω της επιλογής κανονικοποίησης ως προς την ω#[sub s], αναμένεται να υπάρχει υπερκάλυψη της προδιαγραφής a#[sub max]. Λόγω της στρογγυλοποίησης αυτής, αναμένεται μάλιστα να υπάρχει υπερκάλυψη σε τουλάχιστον μία από τις προδιαγραφές. Στη συγκεκριμένη περίπτωση, λόγω της επιλογής κανονικοποίησης ως προς την ω#[sub s], αναμένεται να υπάρχει υπερκάλυψη της προδιαγραφής a#[sub max].
@ -55,7 +55,7 @@ p.latex-equation.
$$\omega_{hp} = \Omega_{hp} * \omega_s = 0.7196 * 72570.790 = 52222.58\frac{rad}{s}$$ $$\omega_{hp} = \Omega_{hp} * \omega_s = 0.7196 * 72570.790 = 52222.58\frac{rad}{s}$$
p. p.
Οι γωνίες Butterworth μπορούν να υπολογιστούν με βάση τον αλγόριθμο Guillemin ή να βρεθούν απευθείας από γνωστούς πίνακες γωνιών Butterworth. Για φίλτρο πέμπτης τάξης, οι γωνίες είναι: Οι γωνίες Butterworth μπορούν να υπολογιστούν με βάση τον αλγόριθμο Guillemin ή να βρεθούν απευθείας από γνωστούς πίνακες γωνιών Butterworth. Για φίλτρο τέταρτης τάξης, οι γωνίες είναι:
p.latex-equation. p.latex-equation.
$$\pm 22.5^\circ$$ και $$\pm 67.5^\circ$$ $$\pm 22.5^\circ$$ και $$\pm 67.5^\circ$$
@ -80,7 +80,7 @@ p.latex-equation.
$$\Omega_{0_k} = \sqrt{\sigma_k^2+\Omega_k^2}$$ $$\Omega_{0_k} = \sqrt{\sigma_k^2+\Omega_k^2}$$
$$Q_k = \frac{1}{2*\cos(\tan^{-1}(\frac{\Omega_k}{\sigma_k}))}$$ $$Q_k = \frac{1}{2*\cos(\tan^{-1}(\frac{\Omega_k}{\sigma_k}))}$$
p υπολογίζονται τα ω#[sub 0] και Q των πόλων αντίστοιχα: p υπολογίζονται τα Ω#[sub 0] και Q των πόλων αντίστοιχα:
p.latex-equation. p.latex-equation.
$$\text{Pole 1: } \Omega_0 = 1.038 \text{, }Q = 0.582$$ $$\text{Pole 1: } \Omega_0 = 1.038 \text{, }Q = 0.582$$
@ -117,7 +117,6 @@ p.
figure.block-center.width-19cm figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/inverse_chebyshev_zero_pole.svg").width-19cm img(src="1_low_pass/assets/diagrams/inverse_chebyshev_zero_pole.svg").width-19cm
//- include assets/diagrams/inverse_chebyshev_zero_pole.svg
figcaption figcaption
.reference #[span.plot-count] .reference #[span.plot-count]
.caption. .caption.
@ -145,7 +144,6 @@ figure.block-center
.row.top-5mm .row.top-5mm
p.center #[strong Unit 2] p.center #[strong Unit 2]
.two.wide.column .two.wide.column
//- include assets/diagrams/inverse_chebyshev_zero_pole.svg
figcaption figcaption
.reference #[span.plot-count] .reference #[span.plot-count]
.caption. .caption.
@ -154,11 +152,11 @@ figure.block-center
h4 Υλοποίηση συνάρτησης μεταφοράς h4 Υλοποίηση συνάρτησης μεταφοράς
p. p.
Από τον αριθμό ΑΕΜ (8261) υποδειηκύεται η χρήση των κυκλωμάτων low pass notch του κεφαλαίου 7, με χρήση του κυκλώματος του σχήματος 7.23. Από τον αριθμό ΑΕΜ (8261) υποδεικνύεται η χρήση των κυκλωμάτων low pass notch του κεφαλαίου 7, με χρήση του κυκλώματος του σχήματος 7.23.
h5 Μονάδα 1 h5 Μονάδα 1
p Η πρώτη μονάδα low pass notch πρέπει να υλοποιεί: p Η πρώτη μονάδα low pass notch, δεύτερης τάξης, πρέπει να υλοποιεί:
figure.block-center.width-15cm figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned table.ui.celled.table.teal.striped.center.aligned
@ -274,9 +272,20 @@ figure.block-center.width-15cm
.caption. .caption.
Τιμές των στοιχείων της πρώτης μονάδας και κέρδη στις υψηλές και χαμηλές συχνότητες Τιμές των στοιχείων της πρώτης 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.
$$\begin{align*}
T_{BE}(s) &= k_{high}\frac{s^2+\big[\frac{k_{high}-1}{k_{high}R_1C}+\frac{2}{R_2C}+\frac{2}{R_5C}\big]s+\big[\frac{1}{R_1R_5C^2}+\frac{1}{R_1R_2C^2}\big]}{s^2+\frac{2}{R_2C}s+\frac{1}{R_1R_2C^2}}= \\[3.5mm]
&=0.3492\frac{s^2+\big[\frac{0.3492-1}{0.3492*122.88*0.1*10^{-6}}+\frac{2}{166.59*0.1*10^{-6}}+\frac{2}{633*0.1*10^{-6}}\big]s+\big[\frac{1}{122.88*633*(0.1*10^{-6})^2}+\frac{1}{122.88*166.59*(0.1*10^{-6})^2}\big]}{s^2+\frac{2}{166.59*0.1*10^{-6}}s+\frac{1}{122.88*166.59*(0.1*10^{-6})^2}} = \\[3.5mm]
&=\frac{0.3492s^2+\cancelto{0}{(1.6228*10^{-12})}s+2.1547*10^9}{s^2+120055s+4.8846*10^9} = \\[3.5mm]
&=\frac{0.3492s^2+2.1547*10^9}{s^2+120055s+4.8846*10^9}
\end{align*}$$
h5 Μονάδα 2 h5 Μονάδα 2
p Η δεύτερη μονάδα low pass notch πρέπει να υλοποιεί: p Η δεύτερη μονάδα low pass notch, δεύτερης τάξης, πρέπει να υλοποιεί:
figure.block-center.width-15cm figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned table.ui.celled.table.teal.striped.center.aligned
@ -392,13 +401,23 @@ figure.block-center.width-15cm
.caption. .caption.
Τιμές των στοιχείων της πρώτης μονάδας και κέρδη στις υψηλές και χαμηλές συχνότητες Τιμές των στοιχείων της πρώτης 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.
$$\begin{align*}
T_{BE}(s) &= k_{high}\frac{s^2+\big[\frac{k_{high}-1}{k_{high}R_1C}+\frac{2}{R_2C}+\frac{2}{R_5C}\big]s+\big[\frac{1}{R_1R_5C^2}+\frac{1}{R_1R_2C^2}\big]}{s^2+\frac{2}{R_2C}s+\frac{1}{R_1R_2C^2}}= \\[3.5mm]
&=0.3492\frac{s^2+\big[\frac{0.3492-1}{0.3492*50.9*0.1*10^{-6}}+\frac{2}{665.9*0.1*10^{-6}}+\frac{2}{59.5*0.1*10^{-6}}\big]s+\big[\frac{1}{50.9*59.5*(0.1*10^{-6})^2}+\frac{1}{50.9*665.9*(0.1*10^{-6})^2}\big]}{s^2+\frac{2}{665.9*0.1*10^{-6}}s+\frac{1}{50.9*665.9*(0.1*10^{-6})^2}} = \\[3.5mm]
&=\frac{0.3492s^2+1.2559*10^{10}}{s^2+30031s+2.9499*10^9}
\end{align*}$$
h4 Ρύθμιση κέρδους h4 Ρύθμιση κέρδους
p. p.
Με βάση τον αριθμό ΑΕΜ (8261) πραγματοποιείται ρύθμιση κέρδους με στόχο την επίτευξη κέρδους 0 dB στις χαμηλές συχνότητες. Με βάση τον αριθμό ΑΕΜ (8261) πραγματοποιείται ρύθμιση κέρδους με στόχο την επίτευξη κέρδους 0 dB στις χαμηλές συχνότητες.
p. p.
Κατά την υλοποίηση των μονάδων Fried διαπιστώθηκε ότι κάθε μονάδα εισάγει μία απόσβεση. Κάτι το οποίο ήταν αναμενόμενο με βάση τη θεωρεία των μονάδω αυτών. Το συνολικό κέρδος που εισάγουν οι μονάδες είναι: Κατά την υλοποίηση των μονάδων Fried διαπιστώθηκε ότι κάθε μονάδα εισάγει μία απόσβεση. Κάτι το οποίο ήταν αναμενόμενο με βάση τη θεωρεία των μονάδων αυτών. Το συνολικό κέρδος που εισάγουν οι μονάδες είναι:
p.latex-equation. p.latex-equation.
$$k = k_{low}^1k_{low}^2 = 0.4411*4.2573 = 1.878$$ $$k = k_{low}^1k_{low}^2 = 0.4411*4.2573 = 1.878$$
@ -413,4 +432,49 @@ p.
Επιλέγεται η χρήση αντίστασης εισόδου ίσης με r#[sub 1]=10 kOhm. Έτσι η αντίσταση ανατροφοδότησης υπολογίζεται: Επιλέγεται η χρήση αντίστασης εισόδου ίσης με r#[sub 1]=10 kOhm. Έτσι η αντίσταση ανατροφοδότησης υπολογίζεται:
p.latex-equation. p.latex-equation.
$$r_2 = r_1{k}' = 10*10^3*0.5324 = 5324\text{ Ohm}$$ $$r_2 = r_1{k}' = 10*10^3*0.5324 = 5324\text{ Ohm}$$
h4 Συναρτήσεις μεταφοράς
p.
Στον παρακάτω πίνακα φαίνονται οι συναρτήσεις μεταφοράς των επιμέρους μονάδων που υπολογίστηκαν παραπάνω:
figure.block-center.width-15cm
table.ui.celled.table.teal.striped.center.aligned
thead
tr
th Μονάδα
th Συνάρτηση μεταφοράς
tbody
tr
td Πρώτη μονάδα (Unit 1)
td.
$$T_{BE}^1(s) = \frac{0.3492s^2+2.1547*10^9}{s^2+120055s+4.8846*10^9}$$
tr
td Δεύτερη μονάδα (Unit 2)
td.
$$T_{BE}^2(s) = \frac{0.3492s^2+1.2559*10^{10}}{s^2+30031s+2.9499*10^9}$$
figcaption
.reference #[span.table-count]
.caption.
Συναρτήσεις μεταφοράς των επιμέρους μονάδων
p.
Η συνολική συνάρτηση μεταφοράς του φίλτρου υπολογίζεται:
p.latex-equation.
$$\begin{align*}
T_{LP}(s) &= {k}'T_{BE}^1(s)T_{BE}^2(s) = \\[3.5mm]
&=\frac{0.064938s^4+\cancelto{0}{(1.4175*10^{-12})}s^3+(2.736*10^9)s^2+\cancelto{0}{0.050975}s+1.4409*10^{19}}{s^4+150088s^3+(1.144*10^{10})s^2+(5.0082*10^{14})s+1.4409*10^{19}} = \\[3.5mm]
&=\frac{0.064938s^4+(2.736*10^9)s^2+1.4409*10^{19}}{s^4+150088s^3+(1.144*10^{10})s^2+(5.0082*10^{14})s+1.4409*10^{19}}
\end{align*}$$
p.
Στο παρακάτω σχήμα φαίνεται το τελικό κύκλωμα του φίλτρου:
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_circuit_layout.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.
Κύκλωμα κατωδιαβατού φίλτρου

65
report/1_low_pass/1_low_pass_transfer_function_matlab.pug
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@ -1 +1,64 @@
h3 Μελέτη συνάρτησης μεταφοράς στο Matlab h3 Μελέτη συνάρτησης μεταφοράς στο Matlab
p.
Η σχεδίαση του φίλτρου έγινε στο λογισμικό Matlab. Σχεδιάστηκαν, με χρήση της συνάρτησης plot_transfer_function που δόθηκε καθώς και της ltiview που παρέχει το λογισμικό, οι αποκρίσεις πλάτους σε dB των επιμέρους μονάδων, καθώς και του συνολικού φίλτρου. Επίσης η συνολική απόκριση του φίλτρου ελέγχθηκαν σε σήμα εισόδου που καθόρισε η εκφώνηση.
p.
Παρακάτω φαίνονται οι αποκρίσεις όπως προέκυψαν στο Matlab από την plot_transfer_function, με ορίσματα κάθε φορά την συνάρτηση μεταφοράς των επί μέρους συστημάτων, καθώς και τυχόν κρίσιμες συχνότητες:
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_notch_unit_1_transfer_function.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Συνάρτηση μεταφοράς πρώτης μονάδας low pass notch (Fried).
.caption.
Παρατηρούμε ότι η μονάδα, ορθώς, έχει χαμηλότερη απόκριση στις υψηλές συχνότητες σε σχέση με τις χαμηλές. Στο γράφημα φαίνεται επίσης η απόκριση της συνάρτησης μεταφοράς στις χαμηλές συχνότητες, από όπου μπορούμε να επιβεβαιώσουμε ότι το κέρδος -7,1 dB που υπολογίστηκε νωρίτερα για την μονάδα είναι σωστό.
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_notch_unit_2_transfer_function.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Συνάρτηση μεταφοράς δεύτερης μονάδας low pass notch (Fried).
.caption.
Παρατηρούμε ότι η μονάδα, ορθώς, έχει χαμηλότερη απόκριση στις υψηλές συχνότητες σε σχέση με τις χαμηλές. Στο γράφημα φαίνεται επίσης η απόκριση της συνάρτησης μεταφοράς στις χαμηλές συχνότητες, από όπου μπορούμε να επιβεβαιώσουμε ότι το κέρδος 12,6 dB που υπολογίστηκε νωρίτερα για την μονάδα είναι σωστό.
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_total_transfer_function.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Συνολική συνάρτηση μεταφοράς φίλτρου.
.caption.
Παρατηρούμε ότι η συνάρτηση μεταφοράς του φίλτρου έχει το αναμενόμενο σχήμα ενός κατωδιαβατού αντίστροφου Chebyshev φίλτρου. Πιο συγκεκριμένα, στις χαμηλές συχνότητες η συνάρτηση μεταφοράς είναι επίπεδη, ενώ στην αποκοπή εμφανίζει ταλάντωση (ripples). Μάλιστα ο αριθμός των ripples (δύο) είναι σωστός με βάση τη τάξη του φίλτρου (τέταρτης τάξης) και τη θεωρία του αντίστροφου Chebyshev. Στο γράφημα φαίνεται επίσης η απόκριση της συνάρτησης μεταφοράς στις κρίσιμες συχνότητες: (τυχαία) χαμηλή συχνότητα=10 Hz, f#[sub p]=5500 Hz, f#[sub hp]=8313.16 Hz και f#[sub s]=11550 Hz.
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/combined_transfer_functions_bode.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Συνδυαστηκό γράφημα συναρτήσεων μεταφοράς επιμέρους μονάδων και συνολικού φίλτρου.
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_total_attenuation.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Απόσβεση φίλτρου.
.caption.
Στο γράφημα φαίνεται η απόσβεση του φίλτρου για τις κρίσιμες συχνότητες: (τυχαία) χαμηλή συχνότητα=10 Hz, f#[sub p]=5500 Hz, f#[sub hp]=8313.16 Hz και f#[sub s]=11550 Hz.
p.
Στα γραφήματα 2.2.3 και 2.2.5 έχουν σημειωθεί οι κρίσιμες συχνότητες οι οποίες καθορίζουν την ζώνη διόδου και αποκοπής, δηλαδή την f#[sub p]=5500 Hz και την f#[sub s]=11550 Hz, καθώς και οι αντίστοιχες αποσβέσεις. Παρατηρούμε ότι η απόκριση του φίλτρου στην ζώνη διόδου είναι 0.07 dB, ενώ στη ζώνη αποκοπής είναι 23.75 dB. Επομένως είναι φανερό ότι πληρείται ακριβώς η προδιαγραφή a#[sub min]=23.75 dB ενώ η προδιαγραφή a#[sub max]=0.35 dB υπερκαλύπτεται.
p.
Πριν τη ρύθμιση κέρδους η συνάρτηση απόσβεσης δίνεται από το παρακάτω διάγραμμα:
figure.block-center.width-19cm
img(src="1_low_pass/assets/diagrams/low_pass_inverse_chebyshev_original_gain_attenuation.svg").width-19cm
figcaption
.reference #[span.plot-count]
.caption.title.
Απόσβεση φίλτρου πριν τη ρύθμιση κέρδους.
.caption.
Στο γράφημα σημειώνονται οι ίδιες κρίσιμες συχνότητες: (τυχαία) χαμηλή συχνότητα=10 Hz, f#[sub p]=5500 Hz, f#[sub hp]=8313.16 Hz και f#[sub s]=11550 Hz.

264
report/1_low_pass/assets/diagrams/combined_transfer_functions_bode.svg
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@ -0,0 +1,264 @@
<?xml version="1.0"?>
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