Butterworth Active Bandpass Filter using Sallen-Key Topology Technical Report 5 Milwaukee School of Engineering ET-3100 Electronic Circuit Design Submitted By: Alex Kremnitzer Date: 05-11-2011 Date Performed: 05-18-2011 Lab Partners: Curtis Raatz Sarah Woodbury
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 6 th order Butterworth Filter was to be used to filter out high frequency signals above the audio range (22kHz) and a cascaded 2 nd 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. 1
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.) 2
LM741 Output Resistance Vs Frequency Low Pass Filter Design Calculations: 2 nd Order Unity-Gain LPF Sallen-Key Topology (For Reference) 3
6 th 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: 2 nd Order Unity-Gain HPF Sallen-Key Topology 4
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 5
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 6
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 (f center ): 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. 7
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. 8
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 9
Figure 2: Measured Frequency Response Figure 3: High Pass Filter Section Figure 4: Low Pass Filter Section 10
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 (f center ): 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. 11
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. 12