Download LOW, HIGH, AND BAND PASS BUTTERWORTH FILTERS

Butterworth filters
Experiment-428
S
LOW, HIGH, AND
BAND PASS
BUTTERWORTH FILTERS
Keerthana P
Dept of Electronics, Mangalore University, Mangalagangothri-574199, INDIA
Email:[email protected]
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.
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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