1. What is a filter Passive filters Some common filters Lecture 24. Filters II 1

2. 2 Active Filters All the passive filters have a gain < 1 To achieve a gain > 1, we need to utilize an op-amp based circuit to form an active filter We can design all the common filter types using op-amp based active filters Recall for an ideal op-amp that V + = V – I + = I – = 0

3. 3 Op Amp Model + – Inverting input Non-inverting input R in V+V+ V–V– + – A(V + –V – ) VoVo R out

4. 4 Non-inverting Op-Amp Circuit + –+ – V in +–+– V out Z1Z1 Z2Z2

5. 5 Inverting Op-Amp Circuit – + V in +–+– + – V out Z2Z2 Z1Z1

6. 6 An Integrator – + V in + – V out R C +–+– Earlier in the semester, we saw this op-amp based integrator circuit. What type of filter does it create? Does this help you to understand time and frequency domain interrelations?

7. 7 A Differentiator – + V in + – V out C R +–+– What type of filtering does it produce?

8. 8 Example You are shopping for a stereo system, and the following specifications are quoted: –Frequency range: –6 dB at 65 Hz and 22 kHz –Frequency response: 75 Hz to 20 kHz ±3 dB What do these specs really mean? –Note that the human hearing range is around 20 Hz to 20 kHz (audible frequencies) Could you draw a rough Bode (magnitude) plot for the stereo system?

9. 9 Second-Order Filter Circuit C +–+– VSVS R Low Pass L H LP = (1/sC) / Z = Z = R + 1/sC + sL Example: Building Filters via Cascade of RCL circuits

10. 10 Example: Building Filters via Cascade of RCL circuits Example of last in-class exercise –4 th Order Butterworth –Cutoff- 20000 rad/sec System function: Implement each H i (s), i=1,2 using RLC (series) circuit V2V2 V3V3 V1V1 15.3Ω L=1mH C=2.5μF 37.0Ω L=1mH C=2.5μF V 2 (s)=H 1 (s)V 1 (s)V 3 (s)=H 2 (s)V 2 (s)

11. 11 Sectional 2 nd Order Filters Connected by Voltage Follower Op-Amp isolate sections of 2 nd order circuits Loading between sections can be kept low V2V2 V3V3 V1V1 R C R L C V 2 (s)=H 1 (s)V 1 (s) V 3 (s)=H 2 (s)V 2 (s) L

12. 12 Class Examples Problem 10-12 (p421).