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Understanding Filters in Electrical Engineering: Analysis of RC Low-Pass and High-Pass Circuits

This educational recap covers key concepts in electrical engineering related to filters, focusing on RC circuits. It explains the relationship between complex numbers, the Laplace transform, and frequency domain analysis. The discussion includes decibel notation, Bode plots, and the characteristics of low-pass and high-pass filters. Additionally, it explores the concept of impedance in capacitors and inductors, transient and frequency responses, and the process of cascading to create band-pass filters. The material illustrates mathematical operations on complex numbers and their significance for circuit analysis.

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Understanding Filters in Electrical Engineering: Analysis of RC Low-Pass and High-Pass Circuits

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Presentation Transcript

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

  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 Time domain Time domain s-domain (Laplace Domain) -domain (Frequency Domain) Transient response (step, impulse) Frequency response (filters)

  7. Relation time, s and frequency domain Time domain i 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 ~ Voltage2) • 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 1st 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 1st 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

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