EGR 232, Section A, Circuit Theory II

Circuits 2 Project: Filter Circuits

Prepared for: Dr. Butler

Michael Sandy

David Sandy

Marcus Duringer

College of EngineeringCalifornia Baptist University

4/20/16

Table of Contents

Introduction…………………………………………………………………2Development……………………………………………………………......2 Filter Design Calculation……………………………………………..2 Simulation…………………………………………………………….2 Equipment/Part List………………………………………………......9 Implementation………………………………………………….......10Discussion………………………………………………………………....20Conclusion………………………………………………………………...20References…………………………………………………………………20Appendix A………………………………………………………………..21

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IntroductionThe purpose of this project was to design crossover active filter circuits, in order to drive

music through three different types of speakers. So, high frequencies would be sent through a Tweeter speaker, low frequencies would be sent through a Woofer speaker, and middle frequencies would be sent through a Midbass driver speaker. Three circuits were created to drive these speakers. Multisim, MATLAB, and Excel, were all used in the design process in order to create the filter circuits correctly.

DevelopmentThe three filter circuits are all Butterworth filter designs. A 2nd order Sallen-Key high

pass filter was designed first, because the gain of the filter was set at a fixed value. Then a 1st order low pass filter was designed to have a total gain, including the Woofer gain, that was the same as the high pass filter and Tweeter gain. The same consideration was made when designing the Band pass filter. MATLAB was used in order to verify the circuit designs by using the transfer function of each filter, and plotting their bode plots. Multisim was also used to verify the actual circuit designs, using the AC Analysis feature to plot the bode plots. Then excel was used to compare the Multisim and MATLAB bode plots. Finally, the three filter circuits were built on breadboards, and an oscilloscope was used to manually recreate the bode plots for each circuit. These bode plots were compared with the data from MATLAB and Multisim. The circuits were also tested by driving music through the speakers.

Filter Design CalculationThe filter design calculations consisted of doing hand calculations in order to design the

three filter circuits. These hand calculations can be seen in the end of this report in the Appendix B. The main results for each filter design can be seen below:

Low Pass Filter: R1 = 44203.2Ω, R2 = 55648.6Ω, C = .022µF

Band Pass Filter: R1 = 55648.6Ω, R2 = 206695.19Ω, C1 = .022µF, C2 = 220pF

High Pass Filter: R = 96750.4Ω, C = 470pF

SimulationIn the project Multisim and MATLAB were used to simulate the filter circuits, in order to

design them correctly. The MATLAB code can be seen in Appendix A. Furthermore, the three Multisim schematics for the filters can be seen below. Figure 1 below shows the bode plot of the low pass filter that was simulated in Multisim. Figure 2 below shows the bode plot of the high pass filter that was simulated in Multisim. Figure 3 below shows the bode plot of the band pass filter that was simulated in Multisim. Figures 4, 5, and 6 show the bode plot from the MATLAB simulation for the low pass filter, high pass filter, and band pass filter respectively. Figures 7, 8, and 9 show the bode plots using MATLAB that include the gain of each speaker with the gain of

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the filter, for the low pass filter, high pass filter, and band pass filter respectively. Figures 10, 11, and 12 show the Multisim schematics of each of the filters.

Figure 1: Multisim Simulated Bode Plot Low Pass Filter

Figure 2: Multisim Simulated Bode Plot High Pass Filter

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Figure 3: Multisim Simulated Bode Plot Band Pass Filter

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B) System: BWLPF_1

Frequency (rad/s): 15Magnitude (dB): 2

System: BWLPF_1Frequency (rad/s): 817Magnitude (dB): -1.01

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Bode Diagram

Frequency (rad/s)

Figure 4: MATLAB Bode Plot Low Pass Filter

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System: SallenkeyBWHPF_2Frequency (rad/s): 2.2e+04Magnitude (dB): 0.981

System: SallenkeyBWHPF_2Frequency (rad/s): 8.15e+05Magnitude (dB): 4

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Bode Diagram

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Figure 5: MATLAB Bode Plot High Pass Filter

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System: BWBPF_2Frequency (rad/s): 816Magnitude (dB): 8.37

System: BWBPF_2Frequency (rad/s): 2.18e+04Magnitude (dB): 8.41

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Bode Diagram

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Figure 6: MATLAB Bode Plot Band Pass Filter

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Figure 7: MATLAB Bode Plot Low Pass Filter with Speaker

Gain

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Figure 8: MATLAB Bode Plot High Pass Filter with Speaker Gain

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System: BW_LPFFrequency (rad/s): 22.3Magnitude (dB): 93

System: BW_LPFFrequency (rad/s): 817Magnitude (dB): 90

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Bode Diagram

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System: BW_HPFFrequency (rad/s): 8.04e+05Magnitude (dB): 93

System: BW_HPFFrequency (rad/s): 2.2e+04Magnitude (dB): 90

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Bode Diagram

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System: BW_BPFFrequency (rad/s): 816Magnitude (dB): 89.4

System: BW_BPFFrequency (rad/s): 2.2e+04Magnitude (dB): 89.4

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Bode Diagram

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Figure 9: MATLAB Bode Plot Band Pass Filter with Speaker Gain

Figure 10: Multisim Schematic Low Pass Filter

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Figure 11: Multisim Schematic High Pass Filter

Figure 12: Multisim Schematic Band Pass Filter

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Equipment/Part List Power Supply – TENMA 72-7700 (1) Function generator – TENMA 72-7650 (1) Tektronix TDS 2024B Oscilloscope (1) Breadboard (3) MATLAB software computation tool (1)

Low Pass Filter: 741 Operational Amplifier (1) 3.9kΩ resistor (5% tolerance) (1) 22kΩ resistor (5% tolerance) (3) 33kΩ resistor (5% tolerance) (1) 200Ω resistor (5% tolerance) (1) 3.3Ω resistor (5% tolerance) (1) .022uF (CM472) capacitor (1)

High Pass Filter: 741 Operational Amplifier (1) 47kΩ resistor (5% tolerance) (6) 2.7kΩ resistor (5% tolerance) (3) 47Ω resistor (5% tolerance) (2) 4.7kΩ resistor (5% tolerance) (1) 150Ω resistor (5% tolerance) (1) 33kΩ resistor (5% tolerance) (1) 1.5kΩ resistor (5% tolerance) (1) 22kΩ resistor (5% tolerance) (1) 470pF (CM472) capacitor (2)

Band Pass Filter: 741 Operational Amplifier (1) 51kΩ resistor (5% tolerance) (1) 3.3Ω resistor (5% tolerance) (1) 200kΩ resistor (5% tolerance) (1) 6.2kΩ resistor (5% tolerance) (1) 430Ω resistor (5% tolerance) (1) 620Ω resistor (5% tolerance) (1) .022uF (CM472) capacitor (1) 220pF (CM472) capacitor (1)

Implementation

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To start off, the three filter circuits were designed by hand. The high pass filter was designed first, because the gain of the filter was fixed at a certain value. This gain was 4dB,

because Q= 13−A =

1√2

→ A = 1.58579 and A = k, where k is the gain of the filter. The k in

decibels is 20 log10 1.58579=4 dB . Next, the low pass filter was designed, followed by the band pass filter. The low pass filter was designed with a gain of 2dB, whereas the band pass filter had a gain of 12dB. The data tables from Matlab, Multisim, and the measured data are limited to 30 data points in order to limit the length of the lab report. Tables 1, 4, and 7 show the Multisim data for the low pass filter, high pass filter, and band pass filter respectively. Tables 2, 5, and 8 show the Matlab data for the low pass filter, high pass filter, and band pass filter respectively. Tables 3, 5, and 8 show the measured data for the low pass filter, high pass filter, and band pass filter respectively. Figures 13, 14, and 15 show the excel graphs comparing the simulated data and the measured data of the bode plots for the low pass filter, high pass filter, and band pass filter respectively. The data with the formulas used in excel to plot the bode plots can be seen attached to the report in Appendix B.

Once each filter was designed, the students tested each using Multisim to verify the results using the AC Analysis feature to create a bode plot. The plots verified the data as correct. The results are shown in Tables 1, 4, and 7 and Figures 1, 2, and 3. Next, the students used Matlab to confirm the designs, which was also confirmed. Lastly, the Multisim and Matlab results were compared using Excel, which proved the two findings. The cutoff frequency for the low pass filter was determined to be 130Hz, which is approximately 816.8rad/s. The cutoff frequency for the high pass filter was 3500Hz, which is approximately 21,991rad/s. The two cutoff frequencies for the band pass filter were the cutoffs for the low pass and high pass filters. By determining the gain for each to be approximately 93dB, the students calculated the appropriate gain for each filter and added that value to the gain of the speaker connected to each circuit. The calculated gains were 4dB, 2dB, and 12dB for the high pass, low pass, and band pass, respectively. The tweeter, woofer, and midbass gains were 89dB, 91dB, and 81dB, respectively. After testing the circuits by connecting them to the speakers and finding that the filters were successful, the students tested the filters again using Excel to verify the data. Figures 13, 14, and 15 show the comparison between the Multisim and Matlab data, along with the measured data points. The blue line shows the Multisim data converted to Excel, the orange is the Matlab data, and the grey points are the measured data points.

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Multisim X--Trace 1::[V(vo)] Multisim Y--Trace 1::[V(vo)] MultisimVo (dB)0.1 1.258911075 1.999901086

0.100023029 1.258911075 1.9999010850.100046062 1.258911075 1.9999010840.100069101 1.258911075 1.9999010830.100092146 1.258911075 1.9999010820.100115196 1.258911075 1.999901080.100138251 1.258911074 1.9999010790.100161311 1.258911074 1.9999010780.100184377 1.258911074 1.9999010770.100207448 1.258911074 1.9999010760.100230524 1.258911074 1.9999010740.100253605 1.258911074 1.9999010730.100276692 1.258911073 1.9999010720.100299785 1.258911073 1.9999010710.100322882 1.258911073 1.999901070.100345985 1.258911073 1.9999010690.100369093 1.258911073 1.9999010670.100392207 1.258911072 1.9999010660.100415325 1.258911072 1.9999010650.10043845 1.258911072 1.999901064

0.100461579 1.258911072 1.9999010630.100484714 1.258911072 1.9999010610.100507854 1.258911072 1.999901060.100530999 1.258911071 1.9999010590.10055415 1.258911071 1.999901058

0.100577306 1.258911071 1.9999010570.100600468 1.258911071 1.9999010550.100623635 1.258911071 1.9999010540.100646807 1.258911071 1.999901053

Table 1: Multisim Low Pass Filter Data

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Matlab Frequency (rad/s) Matlab Magnitude (dB) Matlab Frequency (Hz)1.63363 2 0.2600002916.3363 1.99828 2.60000289719.1035 1.99764 3.04041645522.3394 1.99677 3.55542593626.1234 1.99558 4.1576682430.5484 1.99395 4.86192886435.7229 1.99172 5.68547611741.774 1.98867 6.64853859348.85 1.98451 7.77471897

57.1247 1.97883 9.09167837866.8009 1.97107 10.6316934478.1162 1.96048 12.4325793791.3482 1.94604 14.53851757106.822 1.92637 17.00124933124.916 1.89962 19.88099887146.075 1.8633 23.24855831170.819 1.81412 27.18668822199.753 1.74776 31.79167735233.589 1.65862 37.176844273.157 1.53962 43.47428679319.426 1.38199 50.83822685373.533 1.1753 59.44962336436.805 0.907689 69.51967492510.795 0.566548 81.29554916597.318 0.139593 95.0661123698.497 -0.38365 111.1692503816.814 -1.01028 129.9999857955.173 -1.74283 152.02050451116.97 -2.57876 177.77129681306.17 -3.51098 207.883412

Table 2: Matlab Low Pass Filter Data

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Measured Frequency (Hz) Measured Linear Max. Voltage Measured Magnitude (dB)10 1.3 2.27886704625 1.28 2.14419939350 1.22 1.727196613

100 1.04 0.340666786120 0.98 -0.175478486130 0.94 -0.537442928140 0.9 -0.915149811150 0.88 -1.110346557200 0.74 -2.615365605300 0.56 -5.03623946400 0.44 -7.13094647500 0.38 -8.404328068750 0.26 -11.70053304

1000 0.22 -13.15154638Table 3: Measured Low Pass Filter Data

0.1 1 10 100 1000 10000 100000 1000000-60

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Low Pass Filter Vo (dB)

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Figure 13: Excel Graph Low Pass Filter

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Multisim X--Trace 1::[V(vo)] Multisim Y--Trace 1::[V(vo)] Multisim Vo (dB)0.1 1.29457E-09 -177.757483

0.100230524 1.30055E-09 -177.7174830.100461579 1.30655E-09 -177.6774830.100693167 1.31258E-09 -177.6374830.100925289 1.31864E-09 -177.5974830.101157945 1.32473E-09 -177.5574830.101391139 1.33084E-09 -177.5174830.101624869 1.33698E-09 -177.4774830.101859139 1.34315E-09 -177.4374830.102093948 1.34935E-09 -177.3974830.102329299 1.35558E-09 -177.3574830.102565193 1.36184E-09 -177.3174830.10280163 1.36813E-09 -177.277483

0.103038612 1.37444E-09 -177.2374830.103276141 1.38078E-09 -177.1974830.103514217 1.38716E-09 -177.1574830.103752842 1.39356E-09 -177.1174830.103992017 1.39999E-09 -177.0774830.104231743 1.40646E-09 -177.0374830.104472022 1.41295E-09 -176.9974830.104712855 1.41947E-09 -176.9574830.104954243 1.42602E-09 -176.91748310.105196187 1.4326E-09 -176.87748310.10543869 1.43922E-09 -176.8374831

0.105681751 1.44586E-09 -176.79748310.105925373 1.45253E-09 -176.75748310.106169556 1.45924E-09 -176.71748310.106414302 1.46597E-09 -176.67748310.106659612 1.47274E-09 -176.6374831

Table 4: Multisim High Pass Filter Data

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Matlab Frequency (rad/s) Matlab Magnitude (dB) Matlab Frequency (Hz)0.439822 -183.954 0.06999984543.9822 -103.954 6.999984538439.822 -63.9539 69.99984538514.323 -61.2355 81.8570478601.443 -58.5172 95.72262644703.321 -55.7988 111.9370137822.455 -53.0805 130.8977787961.769 -50.3621 153.07029051124.68 -47.6438 178.99838141315.19 -44.9255 209.31898961537.97 -42.2072 244.77552781798.48 -39.4889 286.23698212103.12 -36.7707 334.72194392459.37 -34.0527 391.42089242875.96 -31.3349 457.72325013363.11 -28.6176 535.25558073932.78 -25.9013 625.92137714598.95 -23.1868 731.94562555377.96 -20.4756 855.92891786288.93 -17.7706 1000.9142967354.2 -15.0772 1170.457282

8599.91 -12.4052 1368.71818710056.6 -9.77222 1600.55760111760.1 -7.2092 1871.67804613752.1 -4.76728 2188.71469316081.6 -2.52302 2559.46613318805.6 -0.57261 2993.00419821991.1 0.99593 3499.99226925716.1 2.14574 4092.84443230072.2 2.91368 4786.13928

Table 5: Matlab High Pass Filter Data

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Measured Frequency (Hz) Measured Linear Max. Voltage Measured Magnitude (dB)1000 0.16 -15.917600351250 0.24 -12.395775171500 0.32 -9.8970004342000 0.52 -5.6799331272500 0.76 -2.3837281543000 0.96 -0.3545753393500 1.14 1.1380970274000 1.26 2.0074109024500 1.34 2.5420959675000 1.4 2.9225607145500 1.44 3.1672498426000 1.48 3.405234308

10000 1.54 3.75041441750000 1.58 3.973141739

100000 1.64 4.296876961Table 6: Measured High Pass Filter Data

0.05 0.5 5 50 500 5000 50000-190

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Figure 14: Excel Graph High Pass Filter

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Multisim X--Trace 1::[V(vo)] Multisim Y--Trace 1::[V(vo)] Multisim Vo (dB)0.1 0.002857017 -50.88174312

0.100023029 0.002857675 -50.879743120.100046062 0.002858333 -50.877743120.100069101 0.002858991 -50.875743120.100092146 0.00285965 -50.873743120.100115196 0.002860308 -50.871743120.100138251 0.002860967 -50.869743120.100161311 0.002861626 -50.867743130.100184377 0.002862285 -50.865743130.100207448 0.002862944 -50.863743130.100230524 0.002863603 -50.861743130.100253605 0.002864263 -50.859743130.100276692 0.002864922 -50.857743130.100299785 0.002865582 -50.855743130.100322882 0.002866242 -50.853743130.100345985 0.002866902 -50.851743140.100369093 0.002867562 -50.849743140.100392207 0.002868223 -50.847743140.100415325 0.002868883 -50.845743140.10043845 0.002869544 -50.84374314

0.100461579 0.002870205 -50.841743140.100484714 0.002870865 -50.839743140.100507854 0.002871527 -50.837743140.100530999 0.002872188 -50.835743150.10055415 0.002872849 -50.83374315

0.100577306 0.002873511 -50.831743150.100600468 0.002874173 -50.829743150.100623635 0.002874834 -50.827743150.100646807 0.002875496 -50.82574315

Table 7: Multisim Band Pass Filter Data

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Matlab Frequency (rad/s) Matlab Magnitude (dB) Matlab Frequency (Hz)1.63337 -42.582 0.25995890916.3337 -22.5837 2.59958909419.0609 -21.2432 3.03363645522.2433 -19.9029 3.54013114625.9572 -18.5629 4.13121668930.2911 -17.2233 4.82097829735.3486 -15.8844 5.62590442141.2506 -14.5461 6.56523689648.1379 -13.209 7.66138473556.1752 -11.8733 8.94056075965.5544 -10.5396 10.433306876.4997 -9.20846 12.175305489.2724 -7.88098 14.20814374104.178 -6.55837 16.58044366121.572 -5.24234 19.34878474141.87 -3.93517 22.57931178

165.557 -2.63987 26.34921491193.199 -1.3604 30.74857585225.456 -0.101856 35.88243685263.099 1.12926 41.87350637307.027 2.3249 48.86486471358.289 3.4754 57.02346541418.111 4.56965 66.54443241487.92 5.59559 77.65487983

569.385 6.54115 90.62043727664.452 7.3955 105.7508202775.392 8.15056 123.4074696904.854 8.80225 144.01198691055.93 9.35116 168.05647911232.24 9.80243 196.1170871

Table 8: Matlab Band Pass Filter Data

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Measured Frequency (Hz) Measured Linear Max. Voltage Measured Magnitude (dB)5 0.124 -18.1315663

10 0.316 -10.0062583550 1.28 2.144199393

100 2.12 6.526717219130 2.56 8.164799306150 2.76 8.818181641200 3.12 9.88309188500 3.72 11.4108588

1000 3.72 11.41085882000 3.36 10.526785553000 2.92 9.3076570293250 2.8 8.9431606273500 2.72 8.6913780813750 2.6 8.2994669594000 2.52 8.0280108165000 2.2 6.848453616

10000 1.28 2.144199393Table 9: Measured Band Pass Filter Data

0.015 0.15 1.5 15 150 1500 15000 150000-60

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Figure 15: Excel Graph Band Pass Filter

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Discussiona. The purpose of this project was to construct active crossover circuits and be able to

pass music through each circuit and produce the desired output for each filter. A 2nd order Sallen-key Butterworth high pass filter, 1st order Butterworth low pass filter, and a 2nd order Butterworth band pass filter were the desired filters to be created. Using Multisim, MATLAB, and Excel, the students compared the results to each other and found that the data was correct and each plot resulted in the same plot. For example, the Excel graph of the band pass filter shown above in Figure 15 shows that the MATLAB data (orange) rests right on top of the Excel data (blue) found using Multisim, which was expected.

b. The only discrepancies encountered were the resistance of the resistors chosen for the desired resistor values and their respective tolerance. These were overcome by swapping out a couple resistors for either the R1 or R2 value and leaving the other as is.

Conclusionsa. In conclusion, this lab proved successful. The purpose of this project was to design

crossover active filter circuits, in order to drive music through three different types of speakers. High frequencies were sent through a Tweeter speaker, low frequencies were sent through a Woofer speaker, and middle frequencies were sent through a Midbass driver speaker. Three circuits were created to drive these speakers. Multisim, MATLAB, and Excel, were all used in the design process in order to create and test the filter circuits correctly. The results for each test were a match the values recorded were close to the desired results. Using standard filter circuit analyses and the given operational amplifier component, the students were able to calculate the necessary resistor values and required capacitor/capacitors for active filter circuits. The results were close in comparison and confirmed the data.

b. The hand calculations done to create the filter circuits proved to correlate with the required data and graphs produced in Multisim, MATLAB, and Excel. The measured values for each test run for the filters were plotted on one graph in Excel with the Multisim and MATLAB data. All of the data matched for each filter on the graphs that can be seen above in the Implementation.

c. The objective of creating three filter circuits was achieved. All three filters were additionally tested out with a speaker while playing music and correctly filtered the sound. By completing this project, a better knowledge of how to design filters and circuits was gained.

ReferencesResistors Color Code available on Blackboard or Dorf - Appendix D, page 889

Second-Order RLC Filters on Dorf – Table 16.4-1, page 809

Sallen-Key Filters on Dorf – Table 16.4-2, page 810, 811

First-Order Filter Stages on Dorf – Table 16.5-2, page 819

Band Pass Filter Circuit Design and network function on Dorf – Table 13.3-2, page 632

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Capacitor Code Guide and speaker information – EGR 232 Project S16 Document available on Blackboard

Appendix AMATLAB Code

SallenkeyBWHPF_2 = tf([1.58579 0 0],[1 31095.4 21991.1^2]);figure(1)bode(SallenkeyBWHPF_2)BWLPF_1 = tf([0 0 -1028.31],[0 1 816.814]);figure(2)bode(BWLPF_1)BWBPF_2= tf([0 -81681.4 0],[1 22808 1.796e+7]);figure(3)bode(BWBPF_2)BW_HPF = SallenkeyBWHPF_2 * 28183.8;figure(4)bode(BW_HPF)BW_LPF = BWLPF_1 * 35481.3;figure(5)bode(BW_LPF)BW_BPF = BWBPF_2 * 11220.2;figure(6)bode(BW_BPF)

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