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Butterworth filters Experiment-428 S LOW, HIGH, AND BAND PASS BUTTERWORTH FILTERS Keerthana P Dept of Electronics, Mangalore University, Mangalagangothri-574199, INDIA Email:keerthiksd@gmail.com Abstract Using 741C Opamp low-, high-, and band pass filter circuit response to a sine wave input is studied and the pass band gain and cut-off frequencies are verified vis a vis the corresponding theoretical values. Introduction Filters are frequently used as electronic circuit elements. Active and passive filters are the two main types of filters based on the nature of the components used. Passive filters are made of passive components, such as L, C and R. An active filter along with passive components also contains one or more active components. Various combinations of L, C, and R result in LC, RC, and ̟ filters. Further, these filters are classified on the basis of the range of frequencies attenuated by them. There are four such classifications, namely Low pass filter (LPF), High pass filter (HPF), Band pass filter (BPF), and Band reject filters (BRF) [1, 2]. A passive filter has gain less than one and an active filter provides gain more than one in the output signal. This is done at the cost of the amplifier bandwidth. In most of the filter applications such trade-off is required. To obtain a large bandwidth, passive filters are used. The most important active element for designing an active filter is operational amplifier (Opamp). In this experiment active filter response is studied using an Opamp. The band reject filter requires three Opamps and hence a separate experiment is being done to design this filter [3]. Low pass filter (LPF) A low pass passive filter consists of two passive components as shown in Figure-1(a). It allows low frequency signals to pass through but attenuates high frequency signals. 1 KAMALJEETH INSTRUMENTS Butterworth filters Figure-1(b) shows a high pass filter in which low frequency signals are attenuated and high frequency signals are allowed to pass through. The frequency band for which attenuation is -3dB is known as the cut-off frequency. Hence both high pass and low pass filters have cut-off frequencies that can be determined from the transfer functions of the respective filters. Applying the voltage divider formula to the low pass filter circuit in Figure-1(a), the output voltage is given by C2 R1 V1 C1 V2 V3 R2 V4 Figure-1: (a) Low pass filter; 1(b): High pass filter V2 = V1 …1 where XC1 is the reactive impedance of capacitor C1 given by XC1 = - ω …2 Substituting for XC1 in Equation-2 and simplifying gives the transfer gain of the low pass fitter, ALP, as ALP = ω …3 The voltage gain is a function of frequency and when R C ω =1, the voltage gain becomes ALP = √ …4 This happens at a frequency ωH = …5 The frequency ωH = 2̟fH is known as the higher cut-off frequency of the low pass filter. 2 KAMALJEETH INSTRUMENTS Butterworth filters High pass filter (HPF) By interchanging the components, i.e. R and C, one gets a high pass filter as shown in Figure-1(b). Applying the same principle of voltage divider, one can write the output voltage as V = V …6 Where XC2 = - ω …7 Substituting for X C2 and simplifying, the voltage gain of a HPF is given by AHP = = …8 The voltage gain is a function of frequency and when R C ω =1, it reduces to AHP = √ …9 This happens at a frequency ωL known as the lower cut-off frequency of the high pass filter which is given by ωL = …10 This frequency ωL = 2̟fL is known as lower cut-off frequency of the high pass filter. Band pass filter (BPF) A band pass filter allows a predetermined band of frequencies to pass through. This is achieved by connecting a high pass filter with a low pass filter as shown in Figure-2. Hence the final voltage gain is a product of voltage gains of both LPF and HPF, as given by Equations- 3 and 8 respectively. R1 C2 V5 R2 V6 C1 Figure-2: Band pass filter 3 KAMALJEETH INSTRUMENTS Butterworth filters ABP = AHP x ALP = = x ω …11 As seen from the above equation, voltage gain depends on both the upper and lower cut- off frequencies as well as the cut-off frequency defined by respective parts of the filter. The Butterworth filter The Butterworth filter, an active filter used in analog signal processing, has a flat frequency response in the pass band region; hence it is also referred to as a ‘maximally flat magnitude filter’. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers", hence the name. Now it has been regarded as a useful circuit for various purposes. Chebyshev, inverse Chebyshev, Bessel, and Elliptic filters are some other active filters used in general electronic circuit applications. A single R and C network results in a first order filter (LPF and HPF in our case) and a pair of RC elements produces a second order filter. The transfer function of a second order filter is more complicated compared to that of a first order. Hence in this experiment we have selected first order low-pass and high-pass filters. Filter with 5-6 orders are used in many applications. In this experiment LPF and HPF are first order Butterworth filters whereas the BPF is a second order filter. First order low pass Butterworth filter Figure-3(a) shows a first order low pass Butterworth filter circuit. The two inputs of the Opamp are connected to passive components. The non-inverting input is fed to the RC network and the inverting input is tied with two resistors that govern the gain and stability of the circuit. Hence the total voltage gain is a product of inverting amplifier gain and non-inverting amplifier gain. The inverting gain of the amplifier is given by 4 KAMALJEETH INSTRUMENTS Butterworth filters RF +12V RI 2 7 - 6 Vo + 3 R1 Vin 4 -12V C1 Figure-3(a): Butterworth low pass filter AF = 1+ …12 The non-inverting gain is the low pass filter gain given by Equation-3, i.e. ALP = ω Hence the total gain of the first order low pass Butterworth filter is a product of both inverting- and non-inverting gains AV (LP) = ALP x AF AV (LP) = …13 ω First order high pass Butterworth filter Figure-3(b) shows a first order high pass Butterworth filter. The total gain is product inverting and non-inverting gains. AV (HP) = ω …14 5 KAMALJEETH INSTRUMENTS Butterworth filters RF +12V RI 2 7 - 6 Vo + 3 C2 4 Vin -12V R2 Figure-3(b): Butterworth high pass filter Second order band pass Butterworth filter Figure-3(c) shows a second order band pass filter. Similar to the above cases the voltage gain is the product of inverting and non-inverting gains. AV (BP) = ω x …15 ω The non-inverting gain remains the same for low-, high-, and band pass filters. Only the unit gain bandwidth (fT) of the Opamp will affect this value of non-inverting gain. Since we have used 741C Opamp, which has unity gain band frequency fT as 1MHz, the frequency is limited to 100 KHz. RF +12V RI 2 7 - 6 Vo + C2 R1 3 4 Vin -12V R2 C1 Figure-3(c): Butterworth band pass filter 6 KAMALJEETH INSTRUMENTS Butterworth filters Design considerations The non-inverting gain is constant for all the three filters. To get maximally flat response, the non-inverting gain should be 1.58 for a Butterworth filter [4]. Hence pass band gain is set to 1.58 by choosing proper values of RF and RI. Pass band gain, AF =1.58 = 1 + Taking RF = 15KΩ and R I =27KΩ gives AF =1.55 Assuming the upper cut off frequency of the low pass filter as 50 KHz, calculated values of R1 and C1 are Higher cut-off frequency fH = ̟ = 50 KHz C1R1= 3.18 µs Taking R1 = 3.3KΩ, and C1= 1nF, gives fH = 48.2 KHz Similarly for a high pass filter we select lower cut-off frequency of the high pass filter as fL =1KHz = ̟ This gives R2C2 = 1.59x10-4s To get the above value of R2C2, we have chosen R2 =15KΩ, and C2 = 0.01µF. For the band pass filter, same values of R and C are used so that the band width is BW = FH-FL = 48.1 KHz-1.06 KHz = 47.16 KHz Instruments used Opamp applications experimental set-up model LIC-201 of KamalJeeth make, consisting of: 12V split power supply; a set of capacitors and resistors; function generator; and CRO the experimental set up used is shown in Figure-4. 7 KAMALJEETH INSTRUMENTS Butterworth filters Figure-4: Experimental set-up used Experimental procedure The experiment consists of three parts, namely: Part-A: Frequency response of an LPF Part-B: Frequency response of a HPF Part-C: Frequency response of a BPF Part-A: Frequency response of an LPF 1. 2. 3. 4. 5. The low pass filter circuit is rigged as shown in Figure-3(a). A function generator is connected to input and sine wave input is selected The frequency is set to 100Hz and amplitude to 1V (PP). After the amplitude is set, it is kept constant throughout the experiment. The input is monitored on channel-1 of the CRO and output of the filter is monitored on Channel-2, as shown in Figure-5. Figure-5: Input and output waveforms of the low pass filter (Top input (input =1V), bottom output (Output=1.5V) Table-1: Frequency response of the low-pass filter 8 KAMALJEETH INSTRUMENTS Butterworth filters Frequency Output (KHz) (V) 0.1 0.2 0.4 0.5 0.8 1 2 4 5 10 20 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.5 1.45 Voltage gain Expt. Thet. 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.54 1.5 1.52 1.45 1.43 Frequency (KHz) Output (V) 25 30 35 40 45 50 55 60 70 80 100 1.4 1.3 1.25 1.2 1.13 1.1 1.02 0.98 0.9 0.8 0.6 Voltage gain Expt. Thet. 1.4 1.36 1.3 1.32 1.25 1.25 1.2 1.19 1.13 1.13 1.1 1.08 1.02 1.02 0.98 0.97 0.9 0.88 0.8 0.8 0.6 0.67 Input = 1V (PP) 6. The peak-to-peak output amplitude is noted and recorded in Table-1. The theoretical value of gain is calculated using Equation-13 as AV (LP) = ω = √×$ ."" ×%% ×%%&%%&%'( =1.55 7. The experiment is repeated by varying the frequency in suitable steps up to 100 KHz. The corresponding output is noted and presented in Table-1 and corresponding theoretical value of voltage gain is also calculated and presented in Table-1. 8. A graph is drawn taking frequency along X-axis on log scale and voltage gain on Y-axis, as shown in Figure-6. 9. From the graph the -3dB cut-off or the upper cut-off frequency of the low pass filter is noted and compared with the corresponding theoretical value. fH - as read from the graph =50.11 KHz fH - theoretical =48.2KHz 9 KAMALJEETH INSTRUMENTS Volatge gain (Av) Butterworth filters 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 1 10 100 Frequency (KHz) Figure-6: Low pass Butterworth filter response Part-B: Frequency response of a HPF 10. The high pass filter circuit is rigged as shown in Figure-3(b). 11. A function generator is connected to the input and sine wave input is selected. 12. The frequency is set to 100Hz and amplitude as 1V (PP). 13. After the amplitude is set, it is kept constant throughout the experiment. 14. The input is monitored on channel-1 of the CRO and output of the filter is monitored on Channel-2, as shown in Figure-7. 15. The peak-to-peak output amplitude is noted and recorded in Table-2 and the corresponding theoretical value of gain is also calculated using Equation-14 AV (HP) = ω = ) ."" * ×++ ×+++ ×,+.+×+' - = 0.145V 10 KAMALJEETH INSTRUMENTS Butterworth filters Figure-7: Input and Output waveforms of the high pass filter 16. The experiment is repeated by varying the frequency in suitable steps up to 100 KHz. The corresponding output is noted and tabulated in Table-2. The value of theoretical voltage gain is also calculated and presented in Table-2. 17. A graph is drawn taking frequency along X-axis on log scale and voltage gain on Y-axis, as shown in Figure-8. 18. From the graph the -3dB cut-off or the lower cut-off frequency of the high pass filter is noted and compared with the corresponding theoretical value. fL - as read from the graph =1.1 KHz fL - theoretical =1KHz Frequency (KHz) 0.1 0.2 0.3 0.4 0.5 0.8 1 1.5 2 4 Table-2: Frequency response of the high-pass filter Voltage gain Voltage gain Output Frequency Output (V) (Hz) (V) Expt. Thet. Expt. Thet. 0.15 0.28 0.41 0.52 0.62 0.9 1.04 1.28 1.45 1.55 0.15 0.28 0.41 0.52 0.62 0.9 1.04 1.28 1.45 1.5 0.145 10 0.30 15 0.44 20 0.58 30 0.69 40 0.97 50 1.09 60 1.37 80 1.39 100 1.52 150 Input= 1V (PP) 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.54 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 1.55 11 KAMALJEETH INSTRUMENTS Butterworth filters 1.8 Volatge gain (Av) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 1 10 100 1000 Frequency (KHz) Figure-8: High pass Butterworth filter response Part-C: Frequency response of a BPF 19. The band pass filter circuit is rigged as shown in Figure-3(c). 20. The function generator is connected to the input and sine wave input is selected. The frequency is set to 100Hz and amplitude as 1V (PP). 21. After the amplitude is set, it is kept constant throughout the experiment. 22. The input is monitored on channel-1 of the CRO and output of the filter is monitored on Channel-2. 23. The peak-to-peak output amplitude is noted and recorded in Table-3 and the corresponding theoretical value of gain is also calculated using Equation-15. AV (BP) = ."" ) * ×++ ×+++ ×,+.+×+'- x √×$ ×%%×%%&%%&% '( =1.45 24. The experiment is repeated by varying the frequency of the input signal in suitable steps reaching up to 100 KHz. The corresponding output is noted and presented in Table-3 and the corresponding value of theoretical voltage gain is also calculated and presented in Table-3. 25. A graph is drawn taking frequency along X-axis on log scale and voltage gain on Y-axis, as shown in Figure-9. 12 KAMALJEETH INSTRUMENTS Butterworth filters 26. From the graph the -3dB cut-off frequency of the band pass filter is noted and compared with the corresponding theoretical value. BW - as noted from the graph =50.11 KHz BW - theoretical =47.16 KHz Volatge gain (Av) Table-3: Frequency response of the band-pass filter Frequency Output Voltage gain Frequency Output Voltage gain (KHz) (V) (KHz) (V) Expt. Thet. Expt. Thet. 0.1 0.15 0.15 0.145 7 1.45 1.45 1.516 0.2 0.3 0.3 0.303 8 1.45 1.45 1.517 0.3 0.42 0.42 0.445 9 1.45 1.45 1.513 0.4 0.55 0.55 0.575 10 1.45 1.45 1.510 0.5 0.64 0.64 0.693 16 1.4 1.4 1.467 0.6 0.72 0.72 0.797 30 1.3 1.3 1.315 0.7 0.82 0.82 0.889 40 1.2 1.2 1.192 0.8 0.88 0.88 0.968 60 1 1 0.970 0.9 0.92 0.92 1.036 70 0.9 0.9 0.878 1 1 1 1.095 80 0.85 0.85 0.799 2 1.3 1.3 1.385 90 0.8 0.8 0.731 2.5 1.35 1.35 1.425 100 0.7 0.7 0.672 3 1.4 1.4 1.458 120 0.6 0.6 0.577 4 1.45 1.45 1.498 150 0.5 0.5 0.474 5 1.45 1.45 1.511 200 0.4 0.4 0.363 6 1.45 1.45 1.514 400 0.18 0.18 0.185 Input 1V (PP) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1 1 10 100 1000 Frequency (KHz) Figure-9: Band pass Butterworth filter response 13 KAMALJEETH INSTRUMENTS Butterworth filters Results The results obtained are tabulated in Table-4 Table-4: Experimental results Filters LPF HPF BPF BPF, Band width Pass band gain Expt. Thet. 1.55 1.55 1.45 - 1.55 1.55 1.55 - Cut-off frequency (KHz) Expt. 50.11 1.1 ./ = 1.1, .1 = 50.11 49.01 Thet. 48.2 1 ./ = 1, .1 = 48.2 47.2 References [1] Ramakanth A Gayakwad, Op-amps & Linear Integrated Circuits, 2nd Edn, Page272, 1988. [2] Robert F Coughlin & Frederick F Driscoll, Operational amplifiers and linear integrated circuits, 3rd Edn, Page-271, 1987. [3] http://www.ece.uah.edu/courses/ee426/Butterworth.pdf [4] http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring2011/lecture-notes/MITRES_6 14 KAMALJEETH INSTRUMENTS

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