1. Recap Filters ABE425 Engineering Tony Grift, PhD Dept. of Agricultural & Biological Engineering University of Illinois

2. Agenda Recap complex numbers Relationship Laplace, frequency (Fourier) domain Relationship time, s and frequency domains decibel notation (dB) RC circuit as a Low-Pass and High-Pass filter Bode plots Combination filters

3. Complex number in complex plane Argument of s Absolute value of s (aka Modulus or Magnitude)

4. Operations on complex numbers cont. Multiplication/division using Euler’s notation

5. Operations on complex numbers cont. Complex conjugate Multiplying a complex number by its conjugate gives a real number

6. Relation Laplace and Fourier Transform s-domain (Laplace Domain) Time domain -domain (Frequency Domain) Time domain Transient response (step, impulse) Frequency response (filters)

7. Relation time, s and frequency domain i Time domain Laplace (s)-domain -domain

8. Concept of impedance (Capacitor)

9. Concept of impedance (Inductor (coil))

10. Low-Pass filter using RC network

11. Derivation transfer function with impedance

12. Decibel notation Addition is much simpler than multiplication Notation in Bel (after Alexander Graham Bell) For Power For Voltages (Power ~ Voltage 2 ) In deciBel (0.1 Bel)

13. Transfer function of RC circuit is complex number i

14. RC circuit as a Low-Pass filter Filter response has a Absolute value (Magnitude of complex number) and Phase (argument of complex number) Analyze three points: Very low frequencies ‘Corner’ frequency Very high frequencies

15. RC Filter response at very low frequencies Magnitude Magnitude in dB Phase (argument)

16. RC Filter response at corner frequency Magnitude Magnitude in dB Phase (argument)

17. RC Filter response at very high frequencies Magnitude Magnitude in dB Phase (argument)

18. Summary 1 st order low pass filter characteristics

19. RC circuit as a Low-Pass filter: Bode plot bode([0 1],[1 1])

20. High-pass filter using RC network

21. High-Pass filter characteristics

22. RC circuit as a High-Pass filter Filter response has a Absolute value (Magnitude of complex number) and Phase (argument of complex number)

23. Summary 1 st order High Pass filter characteristics

24. RC circuit as a High-Pass filter: Bode plot bode([1 0],[1 1])

25. Band-Pass filter through cascading

26. Cascade of High-Pass and Low-Pass filters to obtain a Band-Pass filter Since the sections are separated by a buffer: Add absolute values in dB;s. Add phase angles Buffer

27. The End