11. FOURIER ANALYSIS CIRCUITS by Ulaby & Maharbiz

Overview

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

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

Fourier Series Analysis Technique (details later) Example Cont.

Fourier Series Analysis Technique (cont.) Cont.

Fourier Series Analysis Technique (cont.)

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

Example Fourier series:

Example 11-1: Sawtooth Waveform

Fourier Series: Amplitude/Phase Representation

Example 11-2: Line Spectra (cont.)

Symmetry Considerations dc

Even & Odd Symmetry

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

Circuit Applications

Cont.

Example 11-5: RC Circuit cont. Cont.

Example 11-5: RC Circuit cont. Cont.

Average Power

Fourier Series: Exponential Representation

Fourier Transform Fourier Series Analysis Technique Fourier Transform Analysis Technique

Example 11-8: Pulse Train Note that:

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

Derivation Of Fourier Transform Fourier Transform Pair

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

Circuit Analysis with Fourier Transform vs(t) = cos 4t Example Cont.

Circuit Analysis with Fourier Transform Applying Inverse Fourier Transform:

The Importance of Phase Information

Summary