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Figure 1519 Prototype low-pass lter response
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In addition to selecting a lter from a certain family, it is also possible to select the order of the lter; this is equal to the order of the differential equation that describes the input-output relationship of a given lter In general, the higher the order, the faster the transition from pass-band to stop-band (at the cost of greater phase shifts and amplitude distortion, however) Although the frequency response of Figure 1519 pertains to a low-pass lter, similar de nitions also apply to the other types of lters Butterworth lters are characterized by a maximally at pass-band frequency response characteristic; their response is de ned by a magnitude-squared function of frequency: |H (j )|2 =
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2 H0 1 + 2 2n
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(1517)
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where = 1 for maximally at response and n is the order of the lter Figure 1520 depicts the frequency response (normalized to C = 1) of rst-, second-, third-, and fourth-order Butterworth low-pass lters The Butterworth polynomials, given in Table 153 in factored form, permit the design of the lter by specifying the denominator as a polynomial in s For s = j , one obtains the frequency
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1 08 Amplitude 06 04 02 0 05 1 15 2 25 Normalized frequency 3 35 First order Second order Third order Fourth order
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Figure 1520 Butterworth low-pass lter frequency response Table 153 Butterworth polynomials in quadratic form Order n 1 2 3 4 5 Quadratic factors (s + 1) (s 2 + 2s + 1) (s + 1)(s 2 + s + 1) (s 2 + 07654s + 1)(s 2 + 18478s + 1) (s + 1)(s 2 + 06180s + 1)(s 2 + 16180s + 1)
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response of the lter Examples 154 and 155 illustrate lter design procedures that make use of these tables Figure 1521 depicts the normalized frequency response of rst- to fourthorder low-pass Chebyshev lters (n = 1 to 4), for = 106 Note that a certain amount of ripple is allowed in the pass-band; the amplitude of the ripple is de ned by the parameter , and is constant throughout the pass-band Thus, these lters are also called equiripple lters Cauer, or elliptical, lters are similar to Chebyshev lters, except for being characterized by equiripple both in the pass-band and in the stop-band Design tables exist to select the appropriate order of Butterworth, Chebyshev, or Cauer lter for a speci c application
12 1 Amplitude 08 06 04 02 0 05 1 15 2 25 3 35
First order Second order Third order Fourth order
Normalized frequency
Figure 1521 Chebyshev low-pass lter frequency response
Three common con gurations of second-order active lters, which can be used to implement second-order (or quadratic) lter sections using a single op-
Part II
Electronics
amp, are shown in Figure 1522 These lters are called constant-K, or Sallen and Key, lters (after the names of the inventors) The analysis of these active lters, although somewhat more involved than that of the active lters presented in the preceding chapter, is based on the basic properties of the ideal operational ampli er discussed earlier Consider, for example, the low-pass lter of Figure 1522 The rst unusual aspect of the lter is the presence of both negative and positive feedback; that is, feedback connections are provided to both the inverting and the noninverting terminals of the op-amp The analysis method consists of nding expressions for the input terminal voltages of the op-amp, v + and v , and using these expressions to derive the input-output relationship for the lter This analysis is left as a homework problem The frequency response of the low-pass lter is given by H (j ) = K(1/R1 R2 C1 C2 ) (j )2 +
1 R1 C1
C1 R1 vS v C2 R2 v+ + _ RA RB Low-pass filter vout
R1 C1 vS R2 C2 + _ RA RB vout
1 R2 C1
1 (K R2 C2
1) j +
1 R1 R2 C1 C2
(1518)
The above frequency response can be expressed in one of two more general forms: H (j ) = and H (j ) = K
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