1. Active Filters Conventional passive filters consist of LCR networks. Inductors are undesirable components: They are particularly non-ideal (lossy) They are bulky and expensive Active filters replace inductors using op-amp based equivalent circuits.

2. Active Filter Designs Three active filter design techniques will be covered: Synthesis by Sections Cascade of second order sections. Component Simulation Replace inductors with op-amp inductor simulations. Operational Simulation Simulate all currents and voltages in the LCR ladder using an analogue computer.

3. Analogue Filter Responses H(f)H(f) f fcfc 0 H(f)H(f) f fcfc 0 Ideal “brick wall” filterPractical filter

4. Standard Transfer Functions Butterworth Flat Pass-band. 20n dB per decade roll-off. Chebyshev Pass-band ripple. Sharper cut-off than Butterworth. Elliptic Pass-band and stop-band ripple. Even sharper cut-off. Bessel Linear phase response – i.e. no signal distortion in pass- band.

6. Analogue Transfer Functions The transfer function of any analogue filter (active or passive) can be expressed as the ratio of two polynomials : Special case when M=0, all-pole response :

7. Poles and Zeros Poles Complex values of s where the transfer function is infinite. i.e. the denominator of the transfer function is zero. Zeros Complex values of s where the transfer function is zero. An N-th order filter will have N poles and up to N zeros. Some poles may be in the same place (as may some zeros).

8. Example – Two Pole Bessel Filter Low pass, cut-off frequency = 1 rad/s, from tables :

9. Operational Amplifiers All the active filters we shall study are based on operational amplifiers (op-amps). Analysis of linear op-amp circuits is usually based on simplifying assumptions : The difference between the non-inverting and inverting inputs is zero. The input current is zero. The output voltage and current is arbitrary.

10. Op-Amp Assumptions + - V+V+ V-V- V out I+I+ I-I- I out

11. Inverting Amplifier + - Z1Z1 Z2Z2 0 V V IN V OUT

12. Non-Inverting Amplifier + - Z1Z1 Z2Z2 0 V V IN V OUT

13. Buffer Amplifier Output voltage = Input voltage Input impedance is infinite Output impedance is zero + - V IN V OUT

14. Single-Pole Passive Filter First order low pass filter Cut-off frequency = 1/CR rad/s Problem : Any load (or source) impedance will change frequency response. v in v out C R

15. Single-Pole Active Filter Same frequency response as passive filter. Buffer amplifier does not load RC network. Output impedance is now zero. v in v out C R

16. Low-Pass and High-Pass Designs High Pass Low Pass

17. Higher Order Filters You might think we could make higher order filters by simply cascading N first order filters This doesn’t work The single pole of a first order filter must be purely real (no imaginary part) The poles of a higher order filter usually need to be complex Solution: Use second order sections, each one synthesising a conjugate pair of complex poles

18. Summary Active filter designs aim to replace the inductors in passive filters. Design techniques : Synthesis by sections Component simulation Operational simulation All based on op-amps – understanding of basic op-amp circuits is essential.