Submitted By: Alex Kremnitzer

Date: 05-11-2011

Date Performed: 05-18-2011

Lab Partners: Curtis Raatz

Sarah Woodbury

Butterworth Active Bandpass Filter

using Sallen-Key Topology Technical Report 5

Milwaukee School of Engineering

ET-3100 Electronic Circuit Design

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Abstract

The design of an Active Butterworth Filter with Sallen-Key topology was

calculated, simulated and prototyped. The simulated and tested prototyped

circuit verified the validity of the design.

Signals below the critical frequency of the High Pass Filter were

attenuated. Signals above the critical frequency of the Low Pass Filter were

attenuated. The frequency response (attenuation) in the Bandpass region was

flat. The value for Q was less than 1 which is what would be expected for a

broadband bandpass filter. All values were within expected tolerances except

the cutoff frequency of the Low Pass Filter was greater than expected.

The use of active filters has the advantage of the avoidance of inductors,

the reduction of circuit loading and the shape of the frequency response, cutoff

frequencies and Q value can be varied.

Introduction

The design constraints were to create a unity gain Active Bandpass Anti-

Aliasing Filter to be used for an audio application. A 6th order Butterworth Filter

was to be used to filter out high frequency signals above the audio range

(22kHz) and a cascaded 2nd order Butterworth Filter to reduce the low

frequency content of the signal below 40Hz. The filter input impedance was to

be greater than 1kΩ and constant from 10Hz to 40kHz.

The Butterworth filter is designed to have a flat response in the passband

region. The filter topology used was the Sallen-Key. The upper critical

frequency of the filter was chosen as 22kHz since this was just above the

maximum frequency limit of human hearing said to be at 20kHz ideal.

The frequency response of the filter was tested by applying an input

signal from a signal generator which was swept from 10Hz to 100kHz. Plots of

the simulated and measured frequency response were made and reviewed

against the design constraints.

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The circuit shown in the Appendix was constructed. Capacitor

measurements were taken using a Fluke Model PM6304 RCL Meter. Resistor

measurements were taken using an Agilent 34401A Digital Multi Meter. In the

lab, the output signal amplitude measurements were taken using a Agilent

54622D Digital Oscilloscope. The circuit power was supplied with ±10VDC

using an Agilent E3631A Power Supply. The applied input signal was a 2.0Vpp

Sinusoidal Wave from an Agilent 33220 Function Generator. The amplitude of

the output signal was measured on the oscilloscope while the frequency of the

input signal was swept from 4Hz to 100kHz. The frequency was also recorded

when the signal was at the critical upper and lower frequency as determined by

an amplitude of 0.71VDCpk (-3dB).

The following formulas were used in the design of the Butterworth

Active Band-Pass Filter Circuit:

Design Criteria:

Input Buffer Filter:

(Design Criteria: The filter input impedance was to be greater than 1kΩ and constant

from 10Hz to 40kHz.)

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LM741 Output Resistance Vs Frequency

Low Pass Filter Design Calculations:

2nd Order Unity-Gain LPF Sallen-Key Topology (For Reference)

4

6th Order Unity-Gain LPF Sallen-Key Topology

Unity Gain 6 Pole LPF Active Filter Design Using Frequency and Impedance Scaling and Look-up tables;

Butterworth Filter with Sallen-Key topology.

High Pass Filter Design Calculations:

2nd Order Unity-Gain HPF Sallen-Key Topology

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Unity Gain 2 Pole HPF Active Filter Design Using Frequency and Impedance Scaling and Look-up tables;

Butterworth Filter with Sallen-Key topology.

Bandpass Filter Calculations:

General Formula For Error Analysis:

Simulation Validation

Table 1: BPF, Calculated versus Simulated Results

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Table 2: Frequency Roll-Off, Calculated versus Simulated Results (Ideal)

Table 3: Frequency Roll-Off, Calculated versus Simulated Results (Actual)

Figure 1: Simulated Frequency Response, Ideal Values

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Analysis of Simulation Results

See Table 1 and Figure 1. The simulated results were within expected

values for the frequency response of the filter design. The upper and lower

critical frequencies were within 5% of calculated using ideal component values.

The bandpass region was flat as would be expected for a Butterworth filter.

Critical Frequency High Pass Filter. See Table 1 and Figure 1. The

calculated value for the lower critical frequency was 40Hz and the value

calculated from simulation results was 40.39Hz having an error of 0.98%. The

error increased to 3.93% when using the actual circuit component values in the

simulation.

Critical Frequency Low Pass Filter. See Table 1 and Figure 1. The

calculated value for the upper critical frequency was 22kHz and the value

calculated from simulation results was 22.05kHz having an error of 0.21%. The

error increased to 1.30% when using the actual circuit component values in the

simulation.

Bandwidth (BW): See Table 1 and Figure 1. The calculated value for the

bandwidth was 21.96kHz and the value calculated from simulation results was

22.01kHz having an error of 0.21%. The error increased to 1.30% when using

the actual circuit component values in the simulation.

Center Frequency (fcenter): See Table 1. The calculated value for the

center frequency was 938Hz and the value calculated from simulation results

was 944Hz having an error of 0.59%. The error increased to 2.61% when using

the actual circuit component values in the simulation.

Quality Factor (Q): See Table 1. The calculated value for Q from was

0.042 and the value calculated from simulation results was 0.0439 having an

error of 0.38%. The error increased to 1.29% when using the actual circuit

component values in the simulation. The Q value is less than 1 which is what

would be expected for a broadband bandpass filter.

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Frequency Roll off: See Tables 2 and 3 and Figure 1. High Pass Filter;

The roll off of the High Pass Filter was designed to be 40dB per decade and

using ideal values the simulated rolloff overall was less than 4.9% overall of

calculated. The error decreased to -3.2% using actual values in simulation. Low

Pass Filter; The roll off of the Low Pass Filter was designed to be 120dB per

decade and using ideal values the simulated rolloff overall was -5. 5% of

calculated. The error increased to -6.8% using actual values in simulation.

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Design Validation and Testing

Table 4: Component Value Error Analysis

Table 5: Calculated versus Measured Values

Table 6: Frequency Roll-Off, Calculated versus Measured Results

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Figure 2: Measured Frequency Response

Figure 3: High Pass Filter Section

Figure 4: Low Pass Filter Section

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Analysis of Testing Results

Component Values

The values of all components measurements are shown in Table 4. All

component were within their expected values except for C4 (-27.86%), C5

(15.0%), C7 (-27.32%) and C8 29.33%) All these capacitors were from the Low

Pass Filter section however their variance did not greatly affect the expected

cut-off frequency and roll off of that portion of the Band Pass Filter, as shown

in Tables 5 and 6.

Circuit Analysis

Critical Frequency High Pass Filter. See Table 5 and Figure 3. The

measured critical frequency (-3dB) of the High Pass filter section was

calculated to be 40Hz but was measured to be 90Hz, causing an error of 125%.

The design calculations of the High Pass Filter were verified to be correct and

there was small error in the actual circuit components. Table 1 verifies that there

was a minimal change in the cut-off frequency in simulation between using

ideal and actual values. After reviewing the users manuals on the Oscilloscope

and Function Generator, it was noticed that the user’s manual’s specifications

are based on a 30 minute warm-up period before measurements should be taken,

which was not observed.

Critical Frequency Low Pass Filter. See Table 5 and Figure 4. The

measured critical frequency (-3dB) of the Low Pass filter section was calculated

to be 22kHz but was measured to be 22.4kHz, having an error of 1.82% which

is acceptable.

Bandwidth (BW): See Table 5 and Figure 2. The calculated value for the

Bandwidth using measured values was 22.31kHz while it was calculated to be

21.96kHz and an error of 1.59% which is acceptable.

Center Frequency (fcenter): See Table 5. The calculated value for the

center frequency using measured values was 1420Hz while it was calculated to

be 938Hz resulting in an error of 51.36%. The variance in the center frequency

was caused by the error in the critical frequency of the High Pass Filter. Since

the center frequency is calculated as the geometric mean of the upper and lower

critical frequencies.

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Quality Factor (Q): See Table 5. The calculated value for Q from

measured values was 0.04 and was the same as calculated. It is less than 1

which is what would be expected for a broadband bandpass filter.

Frequency Roll off: See Table 6 and Figure 2. The roll off of the High

Pass Filter was designed to be 40dB per decade but was measured to be 25dB

per decade with an error of 37.5%. The roll off of the Low Pass Filter could not

be analyzed over a full decade as the output signal measurements were taken up

to 100kHz as the highest frequency. With a critical frequency of 22kHz, to

obtain a full decade, measurements should have been taken past 220kHz. With

the roll off analyzed over ¼ decade, it was measured to be -27.37 and an error of

-8.8% when compared to the calculated value of -30dB per ¼ decade. The

design criteria was reviewed to see if the requirements of the filter order were

misinterpreted (reversed) but after review, it appeared to be correct

Conclusion

The simulated and tested prototyped circuit verified the validity of the

Butterworth Sallen-Key topology Active Filter design. The calculated and

measured frequency response were of the same symmetry.

Signals below the 40Hz critical frequency of the High Pass Filter were

attenuated however the circuit measured at a critical frequency of 90Hz and an

error of 125%. Signals above the 22kHz critical frequency of the Low Pass

Filter were attenuated and the tested circuit had a maximum error of 1.82%. The

frequency response (attenuation) in the Bandpass region was flat. The frequency

rolloff was as expected. The value for Q was less than 1 which is what would be

expected for a broadband bandpass filter.

The use of active filters has the advantage of the avoidance of inductors

which tend to be large when in the audio frequency range. The use of

operational amplifiers also reduce the circuit loading which passive components

would cause. An improvement in the design would be to use tighter tolerances

on components or use variable capacitors and resistors so the shape of the

frequency response and cut-off frequencies plus the Q factor can be adjusted.