# Bandpass Filter (MTB) filter circuit

Bandpass Filter (MTB) filter circuit
Introduction
In order to develop the frequency response of a second–order band pass filter, we first apply the transformation equation of   to the first order low pass transfer function:
Then replacing s with, we get the general transfer function for a second-order band pass filter which is:
………………………………….    Equation 1
In the design of band-pass filters, the gain at the mid-frequency  and the quality factor (Q) (which represents the selectivity of the band pass filter) are the main parameter of interest.
Therefore we replace  with  and  with 1/Q in Equation 1 above to obtain the following equation below:
………………………. Equation 2
Figure 1 below shows the gain responses of a second order band pass filter, which has been normalized, for different Qs. The graph also shows that the frequency response gets steeper with rising Q, thus making the second order band pass filter to be more selective.
Figure 1
A typical multiple feedback band-pass filter looks as shown in Figure 2 below.
Figure 2 (a)
The multiple feedback band-pass circuit is shown in Figure 2 (a) above has the following transfer function:
……………….. Equation 3
When the comparison of the coefficient of Equation 2 and Equation3 is done we get:
Mid-frequency:
Gain at mid-frequency:
Band pass filter quality:
Bandwidth:
The multiple feedback band-pass filter is flexible as it allows the adjustment of Q, and  independently. Its gain factor and bandwidth are also not dependent on.  This makes to be used to modify the mid frequency without affecting the bandwidth, B, or gain,   It can also be noted that this filter can work without n for low values of Q. However, this is guided by the equation:
Multiple feedback band-pass filter design equations:
Figure 2 (b)
The general transfer function of multiple band pass filter is as indicated below:
The transfer function of multiple band pass filter shown in Figure 2(b) is as indicated below:
To design the multiple band pass filter, we choose the capacitance of C3 shown in Figure 2(b) above and then calculate the other parameters as follow:

Procedure
I was allocated to use band 8, 9, and 13 as it can be seen in table 1 below.

Group

f0
Hz
Theoretical value

R1
W

R2
W

R3
W

C1, C2

Band 1

31
82k
2k7
160k
220 n

Band  2
40
82k
2k7
160k
180 n

Band  3
50
82k
2k7
160k
150 n

Band  4
63
82k
2k7
160k
120 n

Band  5
80
82k
2k7
160k
100 n

Band  6
100
82k
2k7
160k
82 n

Band  7
125
82k
2k7
160k
56 n +5n6

Band  8
160
82k
2k7
160k
47 n

Band  9
200
82k
2k7
160k
39 n

Band 10
250
82k
2k7
160k
27 n +4n7

Band 11
315
82k
2k7
160k
22 n +2n7

Band 12
400
82k
2k7
160k
18 n +1n5

Band 13
500
27k
820
56k
47 n

Band 14
630
27k
820
56k
39 n

Band 15
800
27k
820
56k
27 n +2n7

Band 16
1000
8k2
510
18k
47 n +4n7

Band 17
1k4
8k2
510
18k
39 n

Band 18
2000
8k2
510
18k
27 n

Band 19
2k8
8k2
510
18k
18 n +1n5

Band 20
4000
8k2
510
18k
12 n +1n8

Band 21
5k6
8k2
750
18k
8n2

Band 22
8000
8k2
1k2
18k
4n7

Band 23
16000
8k2
1k2
18k
2n2

Table 1
The circuits for band 8, 9, and 13 were constructed on multism by following figure 3 below.

+ ve  =  + 15 V.            C3    =  100 nF
– ve  =  – 15 V.            C4    =  100 nF

Figure 3
Band 8 circuit was achieved by constructing it in multism by following figure 3 above but the value of the following components was changed as follows: (theoretical value), , , ,  and finally
The resulting circuit looked as figure 4 shown below:
Figure 4
The second circuit of band 9 was achieved by constructing it in multism by following figure 3 above but the value of the following components was changed as follows: (theoretical value), , , ,  and finally
The resulting circuit looked as figure 5 shown below:

Figure 5
The third and last circuit of band 13 was achieved by constructing it in multism by following figure 3 above but the value of the following components was changed as follows: (theoretical value), , , ,  and finally
The resulting circuit looked as figure 6 shown below:
Figure 6
Result
After simulation in multism, the following results were observed. For Band 8, the result is as shown in figure 7 and 8 below.
Figure 7
Figure 8
For the second circuit two, Band 9, the result are as shown in figure 9, 10, 11, 12 and 13 below.
Figure 9
Figure 10
Band 9 by using Oscilloscope in the lab
Figure 11

Figure 12
Band 9 by using spectrum Device:
Figure 13
For the third and last circuit, Band 13, the result are as shown in figure 14 and 15 below.
Figure 14
Figure 15
Result Summary Table

Measured Centre Frequency (Hz)
Calculated Centre Frequency (Hz)
Measured
Bandwidth
(Hz)

Calculated
Bandwidth (Hz)

BAND 8
164.6183
165.58
(188.2073-145.4213) = 42.786
42.33

BAND 9
200.7422
199.55
(227.2422-175.5821)
= 51.6601
51.01

BAND 13
500.007
570.57
(571.6565-450.5496) = 121.1069
120.94

Table 2
Calculations
For band 8 circuit, the calculation of Q, Bandwidth and  are as shown below:
Mid-frequency:
Gain at mid-frequency:
Gain in dB:
Band pass filter quality:
Bandwidth:
For band 9 circuit, the calculation of Q, Bandwidth and  are as shown below:
Mid-frequency:
Gain at mid-frequency:
Gain in dB:
Band pass filter quality:
Bandwidth:
For band 13 circuit, the calculation of Q, Bandwidth and  are as shown below:
Mid-frequency:
Gain at mid-frequency:
Gain in dB:
Band pass filter quality:
Bandwidth:
Discussion
The calculated parameter and the measured parameters are in harmony with each other, with little variations. For example, the measured mid frequency for band 8, 9, and 13 are 164.6183 Hz, 200.7422 Hz, and 500.007 Hz in that order. On the other hand, the calculated values are 165.58 Hz, 199.55 Hz, and 570.57 Hz respectively. The slight difference between the measured and the calculated is as a result of components variation. For instance, the exact resistor may be not available and so we use the approximately available resistor. Also, resistor tolerance and temperature drift contribute to the variation of the measured and calculated values especially the Centre frequency as we have seen in our result and calculation.
Conclusion
To conclude, multiple feedback band-pass filter has an advantage of not using a bulky and expensive inductor. It also requires only one op amp device making it simple to create and use. Getting the circuit parameters by calculation and by experiment was easy and simple making the filter ideal for many occasion. The only disadvantage of this filter is that adjusting the center frequency is not independent.
References
Kugelstadt, Thomas. “Chapter 16 Active Filter Design Techniques.” Chapter 16 – Active Filter     Design Techniques – Electro. N.p., n.d. Web. 25 Mar. 2017.        .
Zumbahlen, Hank. “Multiple Feedback Band-Pass Design Example.” Multiple Feedback Band     Pass Design Example – Analog Devices. N.p., n.d. Web. 25 Mar. 2017.   .

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