This document provides an overview of Butterworth, Bessel, and Chebyshev filters in operational amplifiers. It explains their circuit diagrams, transfer functions, and characteristics. The document also includes normalized frequency response graphs and sample circuit diagrams for each type of filter.
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OPERATIONAL AMPLIFIERS By Name Course Instructor Institution Location Date
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Butterworth Filter Butterworth filter refers to a type of filter whose response frequency is flat over the region of the pass-band. The Low-pass filter which is commonly known as the LPF assist in the provision of a constant output from the DC up to the frequency of the cutoff, f(H). When this happens, all the signals above the f stated frequency are rejected. The normalized frequency response graph is thus obtained. When the operation is reversed such that the f(H) is not cut off, there will be no constant output and as such High-pass filter will be effective(Li et al 2018). Figure 1: Circuit diagram(Li et al 2018) The above shared circuit diagram illustrates a first low order-pass Butterworth filter which operates on the basis of the RC networks for the processes of filtering. The RC loading is avoided by the use of the OP-Amp whose configuration is for non-inverting mode. The two resistors R1 and Rf are used in the determination of the gain of the filter. The loading takes place at the range of the poles.
Figure 2: Pole location point (Li et al 2018) The pass band gain is thus given by the formulae: Af = 1+Rf/R1 While high cutoff frequency is given by: f (H) = 1/(2πRC) Bessel Filter Bessel filters are optimized to assist in the provision of the constant pass-band while reducing the sharpness in the response magnitude. Bessel filter pole location that has cutoff frequency of 1rad/s is outside the unit circle. It is only low pass filters selectivity which is found within the Bessel filters. This implies that high-pass selectivity will be lacking always. Considering that
they are linear phase response to the pulse unit, the flatness in the pass-band filters will definitely be lost. The normalized frequency response graph is as indicated below(Engreitz, Ollikainen and Guttman 2016). Figure 3: Normalized graph of Bessel filters(Li et al 2018) Also the sample circuit diagram of the Bessel filter is as shown below
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Figure 4: Circuit diagram of Bessel filters(Li et al 2018) The transfer function formulae for this kind of filter is dependent on the N value which ranges from 2 to 8 for example N=2;1.6221 s 2 + 2.206 s + 1.6221. The general formulae for filters applies Chebyshev Filters One of the key characteristics of the chebyshev filter is an application where it allows for steep transition between the stop-band and the pass-band. It is usually assumed that the cutoff frequency of these filters is unity and as such the poles of the gain translates to zero of the gain function denominator as can be seen in the formulae below(Malz et al 2018). Chebyshev filters have one unique characteristic that they tend to minimize existing error between actual filter and the idealized property even in the case of the high pass filters. This is the reason why Low pass filter within this component will tend to exhibit behaviour of equiripple(Dong and Peng 2015).
The sample circuit diagram for the Chebyshev filters is as shown below. Also the normalized frequency graph of the same type of filter is as shown below
Figure 5: Normalized frequency graph for Chebyshev filters(Li et al 2018)
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References Dong, L. and Peng, X., IMRA America Inc, 2015.Rare earth doped and large effective area optical fibers for fiber lasers and amplifiers. U.S. Patent 9,151,889. Engreitz, J.M., Ollikainen, N. and Guttman, M., 2016. Long non-coding RNAs: spatial amplifiers that control nuclear structure and gene expression.Nature Reviews Molecular Cell Biology,17(12), p.756. Li, J., Rehman, J., Malik, A.B. and Marsboom, G., 2018. Fibroblasts as Disease Amplifiers in Pulmonary Hypertensive Mice Carrying a BMPR2+/R899X Mutation.Circulation,138(Suppl_1), pp.A16444-A16444. Malz, D., Tóth, L.D., Bernier, N.R., Feofanov, A.K., Kippenberg, T.J. and Nunnenkamp, A., 2018. Quantum-limited directional amplifiers with optomechanics.Physical review letters,120(2), p.023601.