Lecture 5 Active Filter (Part II) Biquadratic function filters Positive feedback active filter: VCVS Negative feedback filter: IGMF Butterworth Response Chebyshev Response Ref:080225HKN EE3110 Active Filter (Part 2)

Biquadratic function filters Realised by: Positive feedback (II) Negative feedback Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) (III) Band Pass (IV) Band Stop (V) All Pass Biquadratic functions (I) Low Pass (II) High Pass Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Low-Pass Filter Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) High-Pass Filter Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Band-Pass Filter Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Band-Stop Filter Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Voltage Controlled Votage Source (VCVS) Positive Feedback Active Filter (Sallen-Key) By KCL at Va: Therefore, we get where, Re-arrange into voltage group gives: (1) Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) But, (2) Substitute (2) into (1) gives or (3) In admittance form: (4) * This configuration is often used as a low-pass filter, so a specific example will be considered. Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) VCVS Low Pass Filter In order to obtain the above response, we let: Then the transfer function (3) becomes: (5) Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) we continue from equation (5), Equating the coefficient from equations (6) and (5), it gives: Now, K=1, equation (5) will then become, Ref:080225HKN EE3110 Active Filter (Part 2)

Simplified Design (VCVS filter) Comparing with the low-pass response: It gives the following: Ref:080225HKN EE3110 Active Filter (Part 2)

Example (VCVS low pass filter) To design a low-pass filter with and Let m = 1  n = 2 Choose Then What happen if n = 1? Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) VCVS High Pass Filter Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) VCVS Band Pass Filter Ref:080225HKN EE3110 Active Filter (Part 2)

Infinite-Gain Multiple-Feedback (IGMF) Negative Feedback Active Filter substitute (1) into (2) gives (3) Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) rearranging equation (3), it gives, Or in admittance form: Z1 Z2 Z3 Z4 Z5 LP R1 C2 R3 R4 C5 HP C1 R2 C3 C4 R5 BP Filter Value Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) IGMF Band-Pass Filter Band-pass: To obtain the band-pass response, we let *This filter prototype has a very low sensitivity to component tolerance when compared with other prototypes. Ref:080225HKN EE3110 Active Filter (Part 2)

Simplified design (IGMF filter) Comparing with the band-pass response Its gives, Ref:080225HKN EE3110 Active Filter (Part 2)

Example (IGMF band pass filter) To design a band-pass filter with and With similar analysis, we can choose the following values: Ref:080225HKN EE3110 Active Filter (Part 2)

Butterworth Response (Maximally flat) Butterworth polynomials where n is the order Normalize to o = 1rad/s Butterworth polynomials: Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Butterworth Response Ref:080225HKN EE3110 Active Filter (Part 2)

Second order Butterworth response Started from the low-pass biquadratic function For Ref:080225HKN EE3110 Active Filter (Part 2)

Bode plot (n-th order Butterworth) Butterworth response Ref:080225HKN EE3110 Active Filter (Part 2)

Second order Butterworth filter Setting R1= R2 and C1 = C2 Now K = 1 + RB/ RA Therefore, we have For Butterworth response: We define Damping Factor (DF) as:  Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Damping Factor (DF) The value of the damping factor required to produce desire response characteristic depends on the order of the filter. The DF is determined by the negative feedback network of the filter circuit. Because of its maximally flat response, the Butterworth characteristic is the most widely used. We will limit our converge to the Butterworth response to illustrate basic filter concepts. Ref:080225HKN EE3110 Active Filter (Part 2)

Values for the Butterworth response Roll-off dB/decade 1st stage 2nd stage 3rd stage Order poles DF 1 -20 optional 2 -40 1.414 3 -60 1.000 4 -80 1.848 0.765 5 -100 1.618 0.618 6 -120 1.932 0.518 Ref:080225HKN EE3110 Active Filter (Part 2)

Forth order Butterworth Filter + - R2 8.2 k C1 0.01 F Vout +15 V R1 8.2 k RB 1.5 k RA 10 k R3 8.2 k R4 8.2 k -15 V C3 0.01 F C2 0.01 F C4 RB 27 k RA 22 k 741C Ref:080225HKN EE3110 Active Filter (Part 2)

Chebyshev Response (Equal-ripple) Where  determines the ripple and is the Chebyshev cosine polynomial defined as Ref:080225HKN EE3110 Active Filter (Part 2)

Chebyshev Cosine Polynomials Ref:080225HKN EE3110 Active Filter (Part 2)

Second order Chebychev Response Example: 0.969dB ripple gives  = 0.5, Roots: Ref:080225HKN EE3110 Active Filter (Part 2)

EE3110 Active Filter (Part 2) Roots Roots of first bracketed term Roots of second bracketed term or Ref:080225HKN EE3110 Active Filter (Part 2)