1. 11. FOURIER ANALYSIS CIRCUITS by Ulaby & Maharbiz

2. Overview

3. Analysis Techniques single-sided: defined over [0,∞] double-sided: defined over [ − ∞,∞]

4. 1. Periodic Excitation: Solution Method: Fourier series + Phasor Analysis 2. Nonperiodic Excitation: Solution Method: Fourier Transform Fourier Analysis

5. Fourier Series Analysis Technique (details later) Example Cont.

6. Fourier Series Analysis Technique (cont.) Cont.

7. Fourier Series Analysis Technique (cont.)

8. Fourier Series: Cosine/Sine Representation The Fourier theorem states that a periodic function f(t) of period T can be cast in the form

9. Example Fourier series:

10. Example 11-1: Sawtooth Waveform

11. Fourier Series: Amplitude/Phase Representation

12. Example 11-2: Line Spectra (cont.)

13. Symmetry Considerations dc

14. Even & Odd Symmetry

16. This oscillatory behavior of the Fourier series in the neighborhood of discontinuous points is called the Gibbs phenomenon. Example 11-3: M-Waveform

17. Circuit Applications

18. Cont.

19. Example 11-5: RC Circuit cont. Cont.

20. Example 11-5: RC Circuit cont. Cont.

21. Average Power

22. Fourier Series: Exponential Representation

24. Fourier Transform Fourier Series Analysis Technique Fourier Transform Analysis Technique

25. Example 11-8: Pulse Train Note that:

26. Line Spectrum of Pulse Train Spacing between adjacent harmonics is : spectrum becomes continuous

27. Derivation Of Fourier Transform Fourier Transform Pair

28. Example 11-9: Rectangular Pulse The wider the pulse, the narrower is its spectrum, and vice versa

31. Circuit Analysis with Fourier Transform vs(t) = 10 + 5 cos 4t Example 11-11 Cont.

32. Circuit Analysis with Fourier Transform Applying Inverse Fourier Transform:

33. The Importance of Phase Information

34. Summary