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.