Chapter 17 The Fourier Series EEEB123 Circuit Analysis 2 Chapter 17 The Fourier Series Materials from Fundamentals of Electric Circuits (4th Edition), Alexander & Sadiku, McGraw-Hill Companies, Inc.

The Fourier Series - Chapter 17 17.1 Introduction 17.2 Trigonometric Fourier Series 17.3 Symmetry Considerations 17.4 Circuit Applications

17.1 Introduction(1) Previously, have considered analysis of circuits with sinusoidal sources. The Fourier series provides a means of analyzing circuits with periodic non-sinusoidal excitations. Fourier is a technique for expressing any practical periodic function as a sum of sinusoids. Fourier representation + superposition theorem, allows to find response of circuits to arbitrary periodic inputs using phasor techniques.

17.2 Trigonometric Fourier Series (1) The Fourier series of a periodic function f(t) is a representation that resolves f(t) into a dc component and an ac component comprising an infinite series of harmonic sinusoids. Given a periodic function f(t)=f(t+nT) where n is an integer and T is the period of the function. where w0=2p/T is called the fundamental frequency in radians per second.

17.2 Trigonometric Fourier Series (2) and Fourier coefficients, an and bn , are: in alternative form of f(t) where

17.1 Trigometric Fourier Series (3) Conditions (Dirichlet conditions) on f(t) to yield a convergent Fourier series: f(t) is single-valued everywhere. f(t) has a finite number of finite discontinuities in any one period. f(t) has a finite number of maxima and minima in any one period. The integral

17.2 Trigometric Fourier Series (4) Example 17.1 Determine the Fourier series of the waveform shown below. Obtain the amplitude and phase spectra *Refer to textbook, pg. 760

17.2 Trigonometric Fourier Series (5) Solution: a) Amplitude and b) Phase spectrum Truncating the series at N=11

17.3 Symmetry Considerations (1) Three types of symmetry 1. Even Symmetry : a function f(t) if its plot is symmetrical about the vertical axis. In this case, Typical examples of even periodic function

17.3 Symmetry Considerations (2) 2. Odd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis. In this case, Typical examples of odd periodic function

17.3 Symmetry Considerations (3) 3. Half-wave Symmetry : a function f(t) if Typical examples of half-wave odd periodic functions

17.3 Symmetry Considerations (4) Example 17.3 Find the Fourier series expansion of f(t) given below. Ans: *Refer to textbook, pg. 771

17.3 Symmetry Considerations (5) Example 17.4 Determine the Fourier series for the half-wave cosine function as shown below. Ans: *Refer to textbook, pg. 772

17.4 Circuit Applications (1) Steps for Applying Fourier Series Express the excitation as a Fourier series. Example, for periodic voltage source: Transform the circuit from the time domain to the frequency domain. Find the response of the dc and ac components in the Fourier series. Add the individual dc and ac response using the superposition principle.

17.4 Circuit Applications (2) Example 17.6 Find the response v0(t) of the circuit below when the voltage source vs(t) is given by *Refer to textbook, pg. 775

17.4 Circuit Applications (3) Solution Phasor of the circuit For dc component, (wn=0 or n=0), Vs = ½ => Vo = 0 For nth harmonic, In time domain, Amplitude spectrum of the output voltage

Ch 17 Useful Formula Given: