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Filters and Tuned Amplifiers

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Passive LC Filters Inductorless Filters Active-RC Filters Switched Capacitors

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Filters and Tuned Amplifiers

  1. 1. Chapter 11 Filters and Tuned Amplifiers Passive LC Filters Inductorless Filters Active-RC Filters Switched Capacitors
  2. 2. Filter Transmission, Types and Specification Linear Filters Transfer Function T s( ) Vo s( ) Vi s( ) The Filter Transmisson found by evaluating T(s) for physical frequencies s j ω⋅ T jω( ) T jω( ) e jφ ω( ) ⋅ Gain Function G ω( ) 20 log T jω( )( )⋅ dB Attenuation Function A ω( ) 20− log T jω( )( )⋅ dB
  3. 3. Specification of the transmission characteristics of a low- pass filter. The magnitude response of a filter that just meets specifications is also shown. Filter Specification Frequency-Selection function Passing Stopping Pass-Band Low-Pass High-Pass Band-Pass Band-Stop Band-Reject Summary – Low-pass specs -the passband edge -the maximum allowed variation in passband, Amax -the stopband edge -the minimum required stopband attenuation, Amin Passband ripple Ripple bandwidth
  4. 4. Transmission specifications for a bandpass filter. The magnitude response of a filter that just meets specifications is also shown. Note that this particular filter has a monotonically decreasing transmission in the passband on both sides of the peak frequency. Filter Specification
  5. 5. Exercises 11.1 and 11.2
  6. 6. Pole-zero pattern for the low-pass filter whose transmission is shown. This filter is of the fifth order (N = 5.) The Filter Transfer Function transfer function zeros or transmission zeros T s( ) aM s z1−( )⋅ s z2−( )⋅ s z3−( )⋅ ⋅ ⋅ ⋅ s zM−( )⋅ s p1−( ) s p2−( )⋅ s p3−( )⋅ ⋅ ⋅ ⋅ s pN−( )⋅ transfer function poles or the natural poles
  7. 7. Pole-zero pattern for the bandpass filter whose transmission is shown. This filter is of the sixth order (N = 6.) The Filter Transfer Function
  8. 8. The magnitude response of a Butterworth filter. Butterworth Filters
  9. 9. Magnitude response for Butterworth filters of various order with ε = 1. Note that as the order increases, the response approaches the ideal brickwall type transmission. Butterworth Filters
  10. 10. Graphical construction for determining the poles of a Butterworth filter of order N. All the poles lie in the left half of the s-plane on a circle of radius ω0 = ωp (1/ε)1/N , where ε is the passband deviation parameter : (a) the general case, (b) N = 2, (c) N = 3, (d) N = 4. ∈= −10 110Amax / e j Butterworth Filters
  11. 11. Sketches of the transmission characteristics of a representative even- and odd-order Chebyshev filters. Chebyshev Filters
  12. 12. First-Order Filter Functions
  13. 13. First-Order Filter Functions
  14. 14. Fig. 11.14 First-order all-pass filter. First-Order Filter Functions
  15. 15. Second-Order Filter Functions
  16. 16. Second-Order Filter Functions
  17. 17. Second-Order Filter Functions
  18. 18. Realization of various second-order filter functions using the LCR resonator of Fig. 11.17(b): (a) general structure, (b) LP, (c) HP, (d) BP, (e) notch at ω0 , (f) general notch, (g) LPN (ωn ≥ ω0 ), (h) LPN as s → ∞, (i) HPN (ωn < ω0 ). The Second-order LCR Resonator
  19. 19. The Antoniou inductance-simulation circuit. (b) Analysis of the circuit assuming ideal op amps. The order of the analysis steps is indicated by the circled numbers. The Second-Order Active Filter – Inductor Replacement
  20. 20. Realizations for the various second-order filter functions using the op amp-RC resonator of Fig. 11.21 (b). (a) LP; (b) HP; (c) BP, (d) notch at ω0 ; The Second-Order Active Filter – Inductor Replacement
  21. 21. (e) LPN, ωn ≥ ω0 ; (f) HPN, ωn ≥ ω0 ; (g) all-pass. The circuits are based on the LCR circuits in Fig. 11.18. Design equations are given in Table 11.1. The Second-Order Active Filter – Inductor Replacement
  22. 22. The Second-Order Active Filter – Two-Integrator-Loop Vhp Vi K s 2 ⋅ s 2 s ωo Q       ⋅+ ωo 2 + Two integrations of signal with time constant ωo s Vhp⋅ Vhp 1 Q ωo s Vhp⋅       ⋅+ ωo 2 s 2 Vhp⋅         + K Vi⋅ 1 ωo Vhp K Vi⋅ 1 Q ωo s ⋅ Vhp⋅− ωo 2 s 2 Vhp⋅− Summing Point
  23. 23. The Second-Order Active Filter – Two-Integrator-Loop Circuit Implementation
  24. 24. The Second-Order Active Filter – Two-Integrator-Loop Circuit Design and Performance T a 1 2, 40..:= b 1 2, 20..:= j 1−:= w0 2 π⋅ 10 3 ⋅:= K 3:= w a 100 700 a⋅+:= Q b 0.1 0.2 b⋅+:= T a b, K j w a ⋅( )2 ⋅ j w a ⋅( )2 j w a ⋅( ) w0 Q b ⋅      + w0 2 + :=
  25. 25. The Second-Order Active Filter – Two-Integrator-Loop Exercise 11.21
  26. 26. Derivation of an alternative two-integrator-loop biquad in which all op amps are used in a single-ended fashion. The resulting circuit in (b) is known as the Tow-Thomas biquad. The Second-Order Active Filter – Two-Integrator-Loop
  27. 27. Fig. 11.26 The Tow-Thomas biquad with feedforward. The transfer function of Eq. (11.68) is realized by feeding the input signal through appropriate components to the inputs of the three op amps. This circuit can realize all special second-order functions. The design equations are given in Table 11.2.
  28. 28. Fig. 11.37 A two-integrator-loop active-RC biquad and its switched-capacitor counterpart.
  29. 29. Fig. 11.47 Obtaining a second-order narrow-band bandpass filter by transforming a first-order low-pass filter. (a) Pole of the first- order filter in the p-plane. (b) Applying the transformation s = p + jω0 and adding a complex conjugate pole results in the poles of the second-order bandpass filter. (c) Magnitude response of the firs-order low-pass filter. (d) Magnitude response of the second-order bandpass filter.
  30. 30. Fig. 11.48 Obtaining the poles and the frequency response of a fourth-order stagger-tuned narrow-band bandpass amplifier by transforming a second-order low-pass maximally flat response.

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