DESIGNOFathORDERBUTTERWORTHLPF.docVIP

DESIGN OF a 5th ORDER BUTTERWORTH LOW-PASS FILTER USING SALLEN KEY CIRCUIT Background Theory: Filters are classified according to the functions that they are to perform, in terms of ranges of frequencies. We will be dealing with the low-pass filter, which has the property that low-frequency excitation signal components down to and including direct current, are transmitted, while high-frequency components, up to and including infinite ones are blocked. The range of low frequencies, which are passed, is called the pass band or the bandwidth of the filter. It extends from ω=0 to ω= ωc rad/sec (fc in Hz). The highest frequency to be transmitted is ωc, which is also called the cutoff frequency. Frequencies above cutoff are prevented from passing through the filter and they constitute the filter stopband. The ideal response of a low-pass filter is shown above. However, a physical circuit cannot realize this response. The actual response will be in general as shown below. It can be seen that a small error is allowable in the pass band, while the transition from the pass band to the stopband is not abrupt. The sharpness of the transition from stop band to pass band can be controlled to some degree during the design of a low-pass filter. The ideal low-pass filter response can be approximated by a rational function approximation scheme such as the Butterworth response. The Butterworth Response Normalizing H0=1 and Then finding the roots of D(s) Example: For n=5 All the poles are:  -1.0000 -0.8090 + 0.5878i -0.8090 - 0.5878i -0.3090 + 0.9511i -0.3090 - 0.9511i 0.3090 + 0.9511i 0.3090 - 0.9511i 1.0000 0.8090 + 0.5878i 0.8090-0.5878 POLE LOCATIONS The poles are distributed over the circle of radius 1 (). Never a pole in the imaginary axis. Finding H(s) from H(s) H(-s): H(s) is assigned all RHS poles H(-s) is assigned all LHS poles Following this procedure, the Butterworth LPF H(s) (H0=1, wc=1rad/sec) can be found for various fi