Engineering 72 Lab 7: Filters

Aron Dobos, Tyler Strombom

03 November 2004

 

Part 1: Op-Amp Filters

 

Figure 1-First Order Low Pass Filter with Op Amp

 

The derivation for the transfer function of the op amp filter in Figure 1 is as follows

 

The following graphs are of our recorded data

Graph 1: Gain versus Frequency

Graph 2: Phase versus Frequency

 

With the given values of C, R1, & R2 =100 radians/sec. From the bode plot for a “perfect” low pass filter, (bold line on graph 3), we would expect the gain to decrease 20 dB per decade from 100 radians/sec () and a phase change of -45 degrees from 100 to 1000 radians/sec. Because a perfect low pass filter assumes exact values for resistors and capacitors and an ideal op amp, we would not expect our data to have these exact characteristics (thin line on graph 3).

Graph 3: Bode Plots for Ideal (Bold Line) and Non Ideal (Thin line) Low Pass Filters

From http://www.oz.net/~coilgun/levitation/bodeplot.htm

 

Our data coincides fairly accurately with the non ideal case in graph 3. There is a 17 dB decrease in gain from with a slight decrease in gain before. Shortly after, the decrease in gain is almost linear. The phase decreases by 38 degrees from 100 to 1000 radians/sec. Around , the decrease in phase is fairly linear.

 

Part 2: Switched Capacitor Filters

 

The LMF100 dual switched capacitor filter IC was used to implement low, band, and high pass filter circuits.  The schematic of the circuit is given below:

 

The forms of the transfer functions for the three filters are given below:

2nd Order High-Pass Filter: 

2nd Order Bandpass Filter: 

 

2nd Order Low-Pass Filter: 

 

 

By deriving the transfer function from the Mode 3 circuit schematic, we can verify that indeed the equations for f0, Q, HOLP, HOBP, and HOHP are correct.  

We can derive the transfer functions for the high pass, band pass, and low pass outputs in terms of R1, R2, R3, and R4.  The high pass output HPA can be arrived at by observing the following relation:

Rearranging the terms, we get the transfer function

Equating this transfer function with the form of the general high pass transfer function, we see that

,

.

 

These formulas are indeed what are given in the LMF100 datasheet for f0, Q, and H0HP. The band pass and low pass transfer functions can be derived in the same way.  For the band pass filter, we get:

Rearranging, we get indeed the band pass transfer function:

Equating this transfer function with the form of the general band pass transfer function, it is clear from the identical denominators that Q and f0 are the same.  By manipulating some terms, we get also that the band pass gain is

which is indeed how it is given in the datasheet.  To derive the low pass transfer function, we send the band pass output through another integrator.  The derivation is given below.

Equating this transfer function with the form of the general low pass transfer function, it is again clear that Q and f0 remain unchanged.  The low pass gain is given by

The formulas given in for f0, Q, HOLP, HOBP, and HOHP are thus shown to be correct.  Given the chosen circuit component values

we calculate that

 

The filter was originally specified in the laboratory assignment to have a center frequency of f0 = 500 Hz, but by accident the frequency knob was bumped and only after all the measurements were made was the error realized.  Instead of retaking the measurements, we assume that a filter of the above frequency was called for in the initial design.  Originally, when designing the filter, we opted for a clock frequency of 50.0 kHz to give the desired 500 Hz center frequency.  Plugging in values, the numerical transfer functions are

 

We take the magnitude of these transfer functions to calculate the theoretical gain at various frequencies .  The theoretical and measured gains for the filters at various frequencies are listed in the table below.


 

High Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain (Matlab)

36.8

231.2212

0.14

0.2

0.0087675

96.9

608.8406

0.15

0.214285714

0.06405

136

854.5132

0.17

0.242857143

0.13431

176

1105.841

0.23

0.328571429

0.24666

240

1507.964

0.45

0.642857143

0.57601

288

1809.557

0.72

1.028571429

1.0869

316

1985.487

1.07

1.528571429

1.6305

337

2117.433

1.53

2.185714286

2.2759

364

2287.079

2.35

3.357142857

3.5821

386

2425.309

2.96

4.228571429

4.7727

412

2588.672

2.95

4.214285714

4.7962

435

2733.186

2.41

3.442857143

3.948

480

3015.929

1.72

2.457142857

2.7533

541

3399.203

1.35

1.928571429

2.041

582

3656.814

1.19

1.7

1.7958

636

3996.106

1.03

1.471428571

1.5942

712

4473.628

0.96

1.371428571

1.4255

850

5340.707

0.84

1.2

1.2661

912

5730.265

0.85

1.214285714

1.2235

1058

6647.61

0.83

1.185714286

1.1573

 

Band Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain (Matlab)

74.5

468.0973

0.21

0.3

0.19558

144

904.7787

0.35

0.5

0.41937

311

1954.071

0.54

0.771428571

1.919

248

1558.23

0.77

1.1

1.0165

271

1702.743

0.88

1.257142857

1.2572

310

1947.787

1.29

1.842857143

1.8967

350

2199.115

2.06

2.942857143

3.1934

372

2337.345

2.78

3.971428571

4.3013

425

2670.354

2.54

3.628571429

4.0186

458

2877.699

1.77

2.528571429

2.7826

491

3085.044

1.23

1.757142857

2.0686

531

3336.371

1.05

1.5

1.5764

582

3656.814

0.8

1.142857143

1.2179

652

4096.637

0.68

0.971428571

0.93856

712

4473.628

0.55

0.785714286

0.79023

777

4882.035

0.52

0.742857143

0.67832

868

5453.805

0.46

0.657142857

0.56953

975

6126.106

0.37

0.528571429

0.48191

1200

7539.822

0.3

0.428571429

0.36782

1473

9255.132

0.28

0.4

0.2882

 

 

Low Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain (Matlab)

74.5

468.0973

1.2

1.714285714

1.0336

144

904.7787

1.2

1.714285714

1.1467

311

1954.071

1.3

1.857142857

2.4296

248

1558.23

1.6

2.285714286

1.6139

271

1702.743

1.2

1.714285714

1.8266

310

1947.787

1.53

2.185714286

2.4091

350

2199.115

2.29

3.271428571

3.5925

372

2337.345

2.92

4.171428571

4.5528

425

2670.354

2.36

3.371428571

3.7231

458

2877.699

1.52

2.171428571

2.3922

491

3085.044

1.05

1.5

1.6588

531

3336.371

0.77

1.1

1.1689

582

3656.814

0.56

0.8

0.82395

652

4096.637

0.39

0.557142857

0.5668

712

4473.628

0.31

0.442857143

0.43701

777

4882.035

0.26

0.371428571

0.34374

868

5453.805

0.21

0.3

0.25835

975

6126.106

0.17

0.242857143

0.19462

1200

7539.822

0.13

0.185714286

0.12069

1473

9255.132

0.11

0.157142857

0.077039

 

The bode plots for the measured gains for the high pass, band pass, and low pass filters are shown above.  Except for a few outlier data points, the graphs show clearly the correct functioning of the three filters, despite the large peak present in all of them.  Matlab generated bode plots of the filters are included below for comparison.

Theoretical Low-Pass Bode Plot (Matlab)

 

Theoretical Band-Pass Bode Plot (Matlab)

Theoretical High-Pass Bode Plot (Matlab)

 

Comparing the theoretical bode plots generated by Matlab for each of the given transfer functions with the measured results shows that the filters do indeed perform reasonably well.

 

 

The same measurements were repeated with the 50/100 pin on the LMF100 pulled to V+.  This change results in a center frequency of 789.4Hz, with .  The transfer functions for the three filters with the new center frequency are given below.

 

 

 

 

High Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain

138

867.079558

0.24

0.171428571

0.031504

220

1382.30074

0.26

0.185714286

0.084057

275

1727.87593

0.3

0.214285714

0.13769

325

2042.03519

0.36

0.257142857

0.2031

370

2324.77852

0.48

0.342857143

0.27953

440

2764.60149

0.64

0.457142857

0.44492

490

3078.76075

0.86

0.614285714

0.61441

535

3361.50408

1.1

0.785714286

0.82402

590

3707.07927

1.5

1.071428571

1.1987

660

4146.90223

2.5

1.785714286

2.0303

733

4605.57475

4.48

3.2

3.7286

755

4743.80483

5.14

3.671428571

4.368

802

5039.11453

5.88

4.2

5.0173

846

5315.57468

5.38

3.842857143

4.4043

898

5642.30031

4.32

3.085714286

3.4805

940

5906.19409

3.54

2.528571429

2.9477

1018

6396.28253

2.82

2.014285714

2.3376

1090

6848.67187

2.46

1.757142857

2.0118

1133

7118.84883

2.32

1.657142857

1.8758

1212

7615.22046

2.08

1.485714286

1.694

1288

8092.74254

1.96

1.4

1.5716

1363

8563.98143

1.86

1.328571429

1.4824

1454

9135.75128

1.76

1.257142857

1.4014

1553

9757.78662

1.76

1.257142857

1.3359

1664

10455.2202

1.68

1.2

1.2808

1734

10895.0431

1.62

1.157142857

1.2532

2037

12798.8483

1.5

1.071428571

1.1719

3250

20420.3519

1.38

0.985714286

1.0613

4115

25855.3071

1.38

0.985714286

1.0374

 

 

Band Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain

138

867.079558

0.34

0.242857143

0.18019

220

1382.30074

0.5

0.357142857

0.30158

275

1727.87593

0.6

0.428571429

0.3952

325

2042.03519

0.72

0.514285714

0.49327

370

2324.77852

0.82

0.585714286

0.59633

440

2764.60149

1.08

0.771428571

0.79815

490

3078.76075

1.35

0.964285714

0.98972

535

3361.50408

1.58

1.128571429

1.2157

590

3707.07927

1.96

1.4

1.6037

660

4146.90223

2.98

2.128571429

2.4281

733

4605.57475

4.84

3.457142857

4.0151

755

4743.80483

5.38

3.842857143

4.5665

802

5039.11453

5.76

4.114285714

4.938

846

5315.57468

5.1

3.642857143

4.1092

898

5642.30031

3.86

2.757142857

3.0593

940

5906.19409

3

2.142857143

2.4752

1018

6396.28253

2.26

1.614285714

1.8125

1090

6848.67187

1.82

1.3

1.4568

1133

7118.84883

1.64

1.171428571

1.3068

1212

7615.22046

1.42

1.014285714

1.1032

1288

8092.74254

1.28

0.914285714

0.96313

1363

8563.98143

1.12

0.8

0.85846

1454

9135.75128

1.02

0.728571429

0.76079

1553

9757.78662

1

0.714285714

0.67898

1664

10455.2202

0.9

0.642857143

0.60757

1734

10895.0431

0.84

0.6

0.57046

2037

12798.8483

0.68

0.485714286

0.45408

3250

20420.3519

0.48

0.342857143

0.25775

4115

25855.3071

0.42

0.3

0.19899

 

 

Low Pass Filter

 

 

 

Freq.

Omega

Voltage

Gain

Theoretical Gain

138

867.079558

1.32

0.942857143

1.0309

220

1382.30074

1.4

1

1.0822

275

1727.87593

1.42

1.014285714

1.1346

325

2042.03519

1.5

1.071428571

1.1982

370

2324.77852

1.6

1.142857143

1.2724

440

2764.60149

1.78

1.271428571

1.4321

490

3078.76075

1.96

1.4

1.5946

535

3361.50408

2.22

1.585714286

1.794

590

3707.07927

2.56

1.828571429

2.1459

660

4146.90223

3.5

2.5

2.9044

733

4605.57475

5.16

3.685714286

4.3245

755

4743.80483

5.6

4

4.7751

802

5039.11453

5.6

4

4.8609

846

5315.57468

4.76

3.4

3.8346

898

5642.30031

3.42

2.442857143

2.6895

940

5906.19409

2.52

1.8

2.0788

1018

6396.28253

1.74

1.242857143

1.4056

1090

6848.67187

1.36

0.971428571

1.0552

1133

7118.84883

1.1

0.785714286

0.9106

1212

7615.22046

0.96

0.685714286

0.7186

1288

8092.74254

0.84

0.6

0.59035

1363

8563.98143

0.68

0.485714286

0.49724

1454

9135.75128

0.64

0.457142857

0.41308

1553

9757.78662

0.6

0.428571429

0.34516

1664

10455.2202

0.5

0.357142857

0.28826

1734

10895.0431

0.48

0.342857143

0.25972

2037

12798.8483

0.36

0.257142857

0.17599

3250

20420.3519

0.26

0.185714286

0.062612

4115

25855.3071

0.24

0.171428571

0.038176

The bode plots for the measured gains for the high pass, band pass, and low pass filters are shown above.  The graphs show clearly the correct functioning of the three filters, despite the large peak present in all of them.  Matlab generated bode plots of the filters are included below for comparison.

 

Theoretical Low-Pass Bode Plot 2 (Matlab)

 

 

Theoretical Band-Pass Bode Plot 2 (Matlab)

 

Theoretical High-Pass Bode Plot 2 (Matlab)

 

This completes this section of the lab.

 

The discrete nature of the switching is clearly shown in the oscilloscope output below:

 

 

Applying a 101.5 Hz Square Wave Input

 

The square wave input essentially causes a step response from the system.  The filter transfer functions described in great detail above control the characteristics of the various responses from the three filters.  The high pass and band pass filters are all centered at 0 volts, since they do not pass low (DC) frequencies.  The low pass transfer function results in a DC offset of about 0.3 volts, since some of the DC is passed. 

 

All responses to square wave.

 

Response of band pass filter to square wave.

Response of high pass filter to square wave.

 

Response of low pass filter to square wave.

 

These results can swiftly be confirmed in Matlab.  Applying a step input to each of the transfer functions results in nearly identical waveform shapes, although the Matlab plots assume 0 initial conditions, when clearly the initial conditions are not zero.  Since  for all of the filters, the systems are underdamped, resulting in the resonance apparent in the oscilloscope outputs.  Looking at the high pass filter, we measure one period of the resonance to be 1.3 ms.  Calculating the frequency, we get , which is very close to the true of 789 Hz.  The discrepancy is easily accounted for in resistor tolerances, measurement error, and other errors.  Similar calculations could be carried out for the band pass and low pass responses.  The Matlab generated step responses are reproduced below for comparison’s sake.

Low Pass Step Response

 

 

Band Pass Step Response

High Pass Step Response