����PRESENTATION ON �BODE PLOT, FILTER

• BRANCH-E & TC ENGG
• SUBJECT- CONTROL SYSTEM & COMPONENT
• CHAPTER – 8 – FREQUENCY RESPONSE ANALYSIS & BODE PLOT
• TOPIC- BODE PLOT, FILTER
• SEM-6^TH
• FACULTY – Er. MANAS RANJAN MOHANTA (Sr. Lecturer E & TC ENGG DEPARTMENT)
• AY-2021-2022, SUMMER-2022

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First-Order System Revisited

• Recall the transfer function�for a 1^st-order system:
• The frequency response of this�system is:

Bode Plot

• Asymptotic approximation to�the actual frequency response.
• Plotted on a log-log scale.
• Allows approximation of the�frequency response using�straight lines:
• Note that:

20 dB/decade

Changes by π/2

Second-Order System

• The transfer function of a �2^nd-order system:
• The frequency response of this�system can be modeled as:
• When :

40 dB/decade

Changes by π

Example: Comparison of Exact and Bode Plots

Causal Filters

• Recall that the ideal lowpass filter �is a noncausal filter and hence�unrealizable.
• The filter design problem then�becomes an optimization problem�in which we try to best meet the�user’s requirements with a filter that�can be implemented with the least number of components.
• This is equivalent to trying to minimize the number of coefficients in the (Laplace transform) transfer function. It is also comparable to minimizing the order of the filter.
• Typically the numerator order is less than or equal to the denominator order, so, building on the concept of a Bode plot, we can relate the maximum amount of attenuation desired to the order of the denominator.
• There are often no “perfect” solutions. Users must tradeoff design constraints such as passband smoothness, stopband attenuation and linearity in phase.
• Computer-aided design programs, including MATLAB, are able to design filters once the user adequately specifies the constraints. We will explore two interesting analytic solutions: Butterworth and Chebyshev filters.

Butterworth Filters

• Butterworth filters are maximally flat in the �passband. Their distinguishing characteristic is�that the poles are arranged on a semi-circle of�radius ω_c in the left-half plane.
• The filter function is an all-pole filter that exploits�the properties of Butterworth polynomials.
• Examples:
• Frequency Response:
• Design strategy: select the cutoff�frequency and the amount of �stopband attenuation, then�compute the order of the filter.

Chebyshev Filters

• Chebyshev filter, based on Chebyshev polynomials, allows ripple in the passband to achieve greater attenuation. This implies a lower filter order at the expense of smoothness of the frequency response in the passband.
• Prototype:

where:

and ε is chosen based �on the amount of passband �ripple that can be tolerated.

• Filter design theory is based�largely on the mathematical�properties of polynomials.
• An third type of filter, the �elliptical filter, attempts to�allow ripple in both bands.

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Frequency Transformations

• There are two general approaches to�filter design:
• Direct optimization of the desired �frequency response, typically�using software like MATLAB.
• Design a “normalized” lowpass�filter and then transform it to �another lowpass, highpass or�bandpass filter.
• The latter approach can be achieved �by using these simple frequency �transformations:
• This latter approach is popular�because it leverages our knowledge�of the properties of lowpass filters.

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