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MALVINO Electronic PRINCIPLES SIXTH EDITION

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Chapter 21 Active Filters

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A f fcfc A f f1f1 A f A f fcfc A f f2f2 BW f1f1 f2f2 Bandstop Bandpass High-pass Low-pass All-pass Ideal filter responses

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Real filter response Ideal (brickwall) filters do not exist. Real filters have an approximate response. The attenuation of an ideal filter is in the stopband. Real filter attenuation is v out /v out(mid) : 3 dB = dB = dB = 0.1

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The order of a filter In an LC type, the order is equal to the number of inductors and capacitors in the filter. In an RC type, the order is equal to the number of capacitors in the filter. In an active type, the order is approximately equal to the number of capacitors in the filter.

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Filter approximations Butterworth (maximally flat response): rolloff = 20n dB/decade where n is the order of the filter Chebyshev (equal ripple response): the number of ripples = n/2 Inverse Chebyshev (rippled stopband). Elliptic (optimum transition) Bessel (linear phase shift)

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A f A f A f A f A f Chebyshev Inverse Chebyshev Butterworth Bessel Real low-pass filter responses Elliptic Note: monotonic filters have no ripple in the stopband.

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A f A f A f A f A f Chebyshev Inverse Chebyshev Butterworth Bessel Real bandpass filter responses Elliptic

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22 LC 1 f r = L C R v out v in Q = XLXL R A second-order low-pass LC filter 600 L C f R Q 9.55 mH 2.65 F 1 kHz mH 531 nF 1 kHz mH 187 nF 1 kHz 0.707

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A f f R or f 0 6 dB 0 dB 20 dB Q = 10 (underdamped) Q = 2 (underdamped) Q = (critically damped) The Butterworth response is critically damped. The Bessel response is overdamped (Q = … not graphed). The damping factor is . The effect of Q on second-order response = Q 1 rolloff = 40 dB/decade

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Sallen-Key second-order low-pass filter v out v in R C1C1 C2C2 R Q = 0.5 C2C2 C1C1 2R2R C1C2C1C2 1 f p = = pole frequency A v = 1

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Second-order responses Butterworth: Q = 0.707; K c = 1 Bessel: Q = 0.577; K c = Cutoff frequency: f c = K c f p Peaked response: Q > *f 0 = K 0 f p (the peaking frequency) *f c = K c f p (the edge frequency) *f 3dB = K 3 f p

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Higher-order filters Cascade second-order stages to obtain even-order response. Cascade second-order stages plus one first-order stage to obtain odd-order response. The dB attenuation is cumulative. Filter design can be tedious and complex. Tables and filter-design software are used.

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Sallen-Key equal-component filter v out v in R C C R 2 RC 1 f p = R2R2 R1R1 A v = + 1 R2R2 R1R1 Q = A v As A v approaches 3, this circuit becomes impractical.

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Sallen-Key second-order high-pass filter v out v in R2R2 CC R1R1 Q = 0.5 R1R1 R2R2 2C2C R1R2R1R2 1 f p = A v = 1

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Tunable MFB bandpass filter with constant bandwidth v out v in 2R 1 C C R3R3 R 1 +R 3 Q = R3R3 A v = -1 R1R1 2C2C 1 f 0 = 2R 1 (R 1 R 3 ) f0f0 Q BW =

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Sallen-Key second-order notch filter v out v in R 2C C R 2 RC 1 f 0 = R2R2 R1R1 A v = + 1 R2R2 R1R1 Q = A v As A v approaches 2, this circuit becomes impractical. C R/2

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R’R’ First-order all-pass lag filter v out v in R R’R’ C = -2 arctan f f0f0 A v = 1 f 0 = RC 1

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R’R’ First-order all-pass lead filter v out v in R R’R’ C = 2 arctan f f0f0 A v = -1 f 0 = RC 1

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Linear phase shift Required to prevent distortion of digital signals Constant delay for all frequencies in the passband Bessel design meets requirements but rolloff might not be adequate Designers sometimes use a non-Bessel design followed by an all-pass filter to correct the phase shift

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State variable filter v in R1R1 C R R R R R R2R2 C HP output BP output LP output R2R2 R1R A v = Q = f 0 = RC 1

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