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# Active filter1 - Active Filters Active-RC Filters Real...

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Active Filters : Active-RC Filters : Real frequency characteristics of (a) a low-pass and (b) a band-pass filter.

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Filter Specification : We use the low-pass response in figure to illustrate the information that must be available to the designer of a filter. At a minimum, the designer requires the following specifications: 1.The cutoff frequency f H (or f C ), that is, the range of passband frequencies. 2.The stopband attenuation H 0 – H 2 . 3.The stopband frequency range. That is, f S is specified. 4.The allowable passband ripple γ = H 0 – H 1 . If no ripple is permitted, γ = 0 and H 1 = H 0 . Butterworth and Chebyshev Filter Functions : The general form of the transfer function H(s) can be expressed as ) ( ) ( ) ( s B s A s H = Where A(s) and B(s) are polynomials in the frequency variables.
Type of characteristic Form of transfer function H(s) = A(s)/B(s) Low-pass 2 2 ) / ( o o s Q s K ϖ ϖ + + 2 2 ) / ( ) ( o o s Q s z s K ϖ ϖ + + + High-pass 2 2 2 ) / ( o o s Q s Ks ϖ ϖ + + 2 2 ) / ( ) ( o o s Q s z s Ks ϖ ϖ + + + Band-pass 2 2 ) / ( o o s Q s Ks ϖ ϖ + + Band-elimination 2 2 2 2 ) / ( ) ( o o r s Q s s K ϖ ϖ ϖ + + + Biquadratic Transfer Functions

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The Biquadratic Function : Consider H(s) to be of the form 0 1 2 0 1 2 2 ) ( b s b s a s a s a s H + + + + = The expression for H(s) in the above equation is called a biquadratic function or simply a biquad because both numerator and denominator are quadratics in s. The low-pass function in the left-hand column in Table 5-1 can be written as 1 ) / )( / 1 ( ) / ( ) ( 0 2 0 2 0 + + = ϖ ϖ s Q s H s H 2 0 / ) ( ϖ K s H = where
Butterworth Polynomials : The use of Butterworth polynomials is a common all-pole approximation of the low-pass characteristic. Hence H(s) = H 0 /B(s), where B(s) is a Butterworth polynomial whose magnitude is given by n B 2 0 2 1 ) ( + = ϖ ϖ ϖ Filters that use Butterworth polynomials are called Butterworth filters.

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Normalized Butterworth Polynomials n Factor of polynomial B n (s) 1 (s +1) 2 (s 2 + 1.414s + 1) 3 (s + 1)(s 2 + s + 1) 4 (s 2 + 0.765s + 1)(s 2 + 1.848s + 1) 5 (s + 1)(s 2 + 0.618s + 1)( s 2 + 1.618s + 1) 6 (s 2 + 0.518s + 1)(s 2 + 1.414s + 1)(s 2 + 1.932s + 1) 7 (s +1)( s 2 + 0.445s + 1)( s 2 + 1.802s + 1) 8 (s 2 + 0.390s + 1)( s 2 + 1.111s + 1)( s 2 + 1.663s + 1)( s 2 + 1.962s +1)
Example Determine the order of a low-pass Butterworth filter that is to provide 40-dB attenuation at ϖ / ϖ 0 = 2. Solution : The normalized magnitude of the filter transfer function is n H j H 2 0 2 0 ) / ( 1 1 ) ( ϖ ϖ ϖ + = An attention of 40 dB corresponds to H(j ϖ )/H 0 = 0.01 , and hence ( 29 ( 29 n 2 2 2 1 / 1 010 . 0 + = 1 10 2 4 2 - = n or Solving for n by taking the logarithm of both sides gives 2 log ) 1 10 log( 2 4 - = n and n = 6.64 Since the order of the filter must be an integer, n = 7.

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Chebyshev Filters : When specifications permit a small amount of passband ripple, a frequently used all-pole approximation is the Chebyshev filter. The transfer function is of the form ) / ( 1 ) ( 2 2 2 0 2 C n C H j H ϖ ϖ ε ϖ + = where C n ( ϖ / ϖ C ) are Chebyshev polynomials defined by = - C C n n C ϖ ϖ ϖ ϖ 1 cos cos 1 0 C ϖ ϖ = - C n ϖ ϖ 1
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