ELECTRIC CIRCUITS EIGHTH EDITION JAMES W. NILSSON & SUSAN A. RIEDEL

CHAPTER 16 FOURIER SERIES © 2008 Pearson Education

CONTENTS 16.1 Fourier Series Analysis: An Overview 16.2 The Fourier Coefficients 16.3 The Effect of Symmetry on the Fourier Coefficients 16.4 An Alternative Trigonometric Form of the Fourier Series © 2008 Pearson Education

16.5 An Application 16.6 Average-Power Calculations with Periodic Functions 16.7 The rms Value of a Periodic Function © 2008 Pearson Education CONTENTS

16.8 The Exponential Form of the Fourier Series 16.9 Amplitude and Phase Spectra © 2008 Pearson Education CONTENTS

16.1 Fourier Series Analysis: An Overview A periodic waveform © 2008 Pearson Education

  A periodic function is a function that repeats itself every T seconds.   A period is the smallest time interval ( T ) that a periodic function can be shifted to produce a function identical to itself. © 2008 Pearson Education 16.1 Fourier Series Analysis: An Overview

  The Fourier series is an infinite series used to represent a periodic function.   The series consists of a constant term and infinitely many harmonically related cosine and sine terms. © 2008 Pearson Education

16.1 Fourier Series Analysis: An Overview   The fundamental frequency is the frequency determined by the fundamental period. or   The harmonic frequency is an integer multiple of the fundamental frequency. © 2008 Pearson Education

16.2 The Fourier Coefficients   The Fourier coefficients are the constant term and the coefficient of each cosine and sine term in the series. © 2008 Pearson Education

16.3 The Effect of Symmetry on the Fourier Coefficients   Five types of symmetry are used to simplify the computation of the Fourier coefficients: Even-function symmetry Odd-function symmetry Half-wave symmetry Quarter-wave, half-wave, even symmetry Quarter-wave, half-wave, odd symmetry © 2008 Pearson Education

An even periodic function, f (t) = f(-t) 16.3 The Effect of Symmetry on the Fourier Coefficients

An odd periodic function, f (t) = f(-t) © 2008 Pearson Education 16.3 The Effect of Symmetry on the Fourier Coefficients

(a) A periodic triangular wave that is neither even nor odd © 2008 Pearson Education 16.3 The Effect of Symmetry on the Fourier Coefficients How the choice of where t = 0 can make a periodic function even, odd, or neither.

b) The triangular wave of (a) made even by shifting the function along the t axis © 2008 Pearson Education 16.3 The Effect of Symmetry on the Fourier Coefficients

c) The triangular wave of (a) made odd by shifting the function along the t axis © 2008 Pearson Education 16.3 The Effect of Symmetry on the Fourier Coefficients

(a) A function that has quarter-wave symmetry (b) A function that does not have quarter-wave symmetry © 2008 Pearson Education 16.3 The Effect of Symmetry on the Fourier Coefficients

16.4 An Alternative Trigonometric Form of the Fourier Series   In the alternative form of the Fourier Series, each harmonic represented by the sum of a cosine and sine term is combined into a single term of the form: © 2008 Pearson Education

16.5 An Application   For steady-state response, the Fourier series of the response signal is determined by first finding the response to each component of the input signal. © 2008 Pearson Education

  The individual responses are added (super-imposed) to form the Fourier series of the response signal.   The response to the individual terms in the input series is found by either frequency domain or s-domain analysis. © 2008 Pearson Education 16.5 An Application

© 2008 Pearson Education 16.5 An Application  An RC circuit excited by a periodic voltage. The RC series circuit The square-wave voltage

  The waveform of the response signal is difficult to obtain without the aid of a computer.   Sometimes the frequency response (or filtering) characteristics of the circuit can be used to ascertain how closely the output waveform matches the input waveform. © 2008 Pearson Education 16.5 An Application

The effect of capacitor size on the steady-state response © 2008 Pearson Education 16.5 An Application

16.6 Average-Power Calculations with Periodic Functions   Only harmonics of the same frequency interact to produce average power.   The total average power is the sum of the average powers associated with each frequency. © 2008 Pearson Education

16.7 The rms Value of a Periodic Function   The rms value of a periodic function can be estimated from the Fourier coefficients. © 2008 Pearson Education

16.8 The Exponential Form of the Fourier Series   The Fourier series may also be written in exponential form by using Euler’s identity to replace the cosine and sine terms with their exponential equivalents. © 2008 Pearson Education

16.9 Amplitude and Phase Spectra The plot of C n versus n where τ = T / 5 © 2008 Pearson Education

16.9 Amplitude and Phase Spectra The plot of (sin x) / x versus x © 2008 Pearson Education

The plot of θ’ n versus n for θ’ n = - (θ n + nπ / 5 ) © 2008 Pearson Education 16.9 Amplitude and Phase Spectra

THE END © 2008 Pearson Education