Analog Filters Design Different Filter Responses Approximation Dr.Eng. Basem ElHalawany Analog Filters Design Frequency-Domain Filter Design Frequency-Domain Filter Design Frequency-Domain Filter Design Consider a frequency response function of a circuit in the rational form as a function of jw Zeros Poles ω We begin by defining tolerance regions on the power frequency response design parameters that select the filter attenuation at the two critical frequencies. The filter functions : The Butterworth Filter Approximation: • Butterworth filters are also known as “maximally flat” filters The Butterworth Filter Approximation: S = jω that lie on a circle with radius For stable Causal system, we only consider all poles in the left-half- plane, which allows us to take the N poles only as the poles of the filter H ( s ) ((From 1:N)) Steps for Designing a Low-Pass Butterworth Filter Approximation: Example: Design a Butterworth low-pass filter to meet the power gain specifications shown in Figure Example: Design a Butterworth low-pass filter to meet the power gain specifications shown in Figure The phase response is not linear, and the phase shift (thus, time delay) of signals passing through the filter varies nonlinearly with frequency. Designing a Butterworth Filter using Matlab To design an analog low-pass Butterworth filter using MATLAB: • The ’s’ tells MATLAB to design an analog filter. • The vectors a and b hold the coefficients of the denominator and the numerator (respectively) of the filter’s transfer function. bodemag used to plot the magnitude response from 30 rad/s out to 3,000 rad/s. At 100 rad/s the response seems to have decreased by about 3 dB From 100 rad/s to 1,000 rad/s the response seems to drop by about 80 dB. As this is a fourth order filter its rolloff should be 4 × 20 dB/decade. The Chebyshev Filter Approximation: • Filters with the Chebyshev response characteristic are useful when a rapid roll-off is required because it provides a roll-off rate greater than -20 dB/decade/pole. • This is a greater rate than that of the Butterworth, so filters can be implemented with the Chebyshev response with fewer poles and less complex circuitry for a given roll-off rate. • This type of filter response is characterized by overshoot or ripples in the pass-band or stop-band (depending on the number of poles The Chebyshev filters allow these conditions: ripple in the Pass-band. ripple in the stop-band. The required filter order may be found as follows So, The characteristic equation Take Cos of both sides, we Get: which defines an ellipse of The poles will lie on this ellipse We can substitute by s now in the original equation of TN & From characteristic equation : Compare As in the Butterworth design procedure, we select the left half-plane poles as the poles of the filter frequency response. Steps for Designing a Low-Pass Chebyshev Type 1 Filter Approximation: The pole-zero plot for the Chebyshev Type 1 filter is shown in Fig. .