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MALVINO Electronic PRINCIPLES SIXTH EDITION

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Active Filters Chapter 21

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**Ideal filter responses**

fc A Ideal filter responses Low-pass Bandpass BW f f1 f2 A f All-pass A A High-pass Bandstop f f fc f1 f2

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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 vout/vout(mid): 3 dB = 0.5 12 dB = 0.25 20 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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**Real low-pass filter responses Inverse Chebyshev**

Butterworth f f A f Bessel Inverse Chebyshev A A Elliptic f f Note: monotonic filters have no ripple in the stopband.

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**Real bandpass filter responses Inverse Chebyshev**

Butterworth f f A f Bessel Inverse Chebyshev A A Elliptic f f

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**A second-order low-pass LC filter**

1 fr = L Q = XL R C R 600 W vout vin L C fR Q 9.55 mH mF kHz 47.7 mH nF kHz 135 mH nF kHz

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**The effect of Q on second-order response**

Q = 10 (underdamped) A Q = 2 (underdamped) 20 dB 6 dB Q = (critically damped) 0 dB a = Q 1 rolloff = 40 dB/decade f fR or f0 The Butterworth response is critically damped. The Bessel response is overdamped (Q = … not graphed). The damping factor is a.

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**Sallen-Key second-order low-pass filter**

vout vin C1 Q = 0.5 C2 C1 Av = 1 2pR C1C2 1 fp = = pole frequency

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**Second-order responses**

Butterworth: Q = 0.707; Kc = 1 Bessel: Q = 0.577; Kc = 0.786 Cutoff frequency: fc = Kcfp Peaked response: Q > 0.707 f0 = K0fp (the peaking frequency) fc = Kcfp (the edge frequency) f3dB = K3fp

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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**

vout vin C Av = R2 R1 R2 R1 Q = 1 3 - Av As Av approaches 3, this circuit becomes impractical. 2pRC 1 fp =

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**Sallen-Key second-order high-pass filter**

vout C C vin R1 Q = 0.5 R1 R2 Av = 1 2pC R1R2 1 fp =

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**Tunable MFB bandpass filter with constant bandwidth**

vout C vin R3 f0 Q BW = Av = -1 2pC 1 f0 = 2R1(R1 R3) R1+R3 Q = 0.707 R3

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**Sallen-Key second-order notch filter**

vin vout C C R R 2C R2 Av = R2 R1 R1 Q = 0.5 2 - Av 2pRC 1 f0 = As Av approaches 2, this circuit becomes impractical.

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**First-order all-pass lag filter**

vout vin R C f0 = 2pRC 1 f = -2 arctan f f0 Av = 1

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**First-order all-pass lead filter**

vout vin R C f0 f0 = 2pRC 1 Av = -1 f = 2 arctan f

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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 R2 R vin R +1 R R1 Av = Q = 3 C R C R1 R 1 f0 =**

HP output LP output f0 = 2pRC 1 R2 BP output

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