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

Zeros

Poles

Consider a frequency response function of a circuit in the rational form as a

function of jw

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 :

ω

The Butterworth Filter Approximation:

• Butterworth filters are also known as “maximally flat” filters

The Butterworth Filter Approximation:

that lie on a circle with radius

S = jω

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.

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

stop-band.

ripple in the

Pass-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.