CHAPTER 16 Fourier Series
CHAPTER CONTENTS 16.1 Fourier Series Analysis: An Overveiw 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
16.5 An Application 16.6 Average-Power Calculations with Periodic Functions 16.7 The rms Value of a Periodic Function 16.8 The Exponential Form of the Fourier Series 16.9 Amplitude and Phase Spectra
16.1 Fourier Series Analysis: An Overview A periodic waveform
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.
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.
The fundamental frequency is the frequency determined by the fundamental period . f0 = 1 / T or ω0 = 2f0 The harmonic frequency is an integer multiple of the fundamental frequency.
16.2 The Fourier Coefficients The Fourier coefficients are the constant term and the coefficient of each cosine and sine tem in the series.
16.3 The Effect of Symmetry on the Fourier Coefficients Four types of symmetry may be used to simplify the task of evaluating the Fourier coefficients: Even-function symmetry Odd-function symmetry Half-wave symmetry Quarter-wave symmetry
An even periodic function, f ( t ) = f ( -t )
An odd periodic function f ( t ) = f ( -t )
A periodic triangular wave that is neither even nor odd. The triangular wave of (a) made even by shifting the function along the t axis. The triangular wave of (a) made odd by shifting the function along the t axis.
A function that has quarter-wave symmetry. A function that does not have quarter-wave symmetry.
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 An cos ( nω0t – θn ).
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. 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. 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.
The effect of capacitor size on the steady-state response
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.
16.7 The rms Value of a Periodic Function The rms value of a periodic function can be estimate from the Fourier coefficients.
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.
16.9 Amplitude and Phase Spectra The plot of Cn versus n where τ = T/5
The plot of θ’n versus n for θ’n = - (θ’n + n / 5 )
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