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Fourier Series • Fourier Transform • Laplace Transform • Applications of Laplace Transform • Z ... PDF

77 Pages·2011·1.11 MB·English
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Basics in Systems and Circuits Theory BSC Modul 4: Advanced Circuit Analysis • Fourier Series • Fourier Transform • Laplace Transform • Applications of Laplace Transform • Z-Transform Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory BSC Modul 4: Advanced Circuit Analysis • Fourier Series • Fourier Transform • Laplace Transform • Applications of Laplace Transform • Z-Transform Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory 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. ∞ f (t) = a 0 +∑(a0 cos nω0t + bn sin nω0t)  n=1 dc  ac where ω =2π/T is called the fundamental frequency in 0 radians per second. Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Trigonometric Fourier Series (2) • and a and b are as follow n n 2 T 2 T a n = T ∫0 f (t) cos(nωot)dt bn = T ∫0 f (t)sin(nωot)dt • in alternative form of f(t) ∞ f (t) = a + (c cos(nω t +φ ) 0 ∑ n 0 n  dc n=1 ac 2 2 −1 bn where cn = an + bn , φn = − tan ( ) (Inverse tangent or arctangent) an Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Fourier Series Example Determine the Fourier series of the waveform shown right. Obtain the amplitude and phase spectra. 1, 0 < t <1 f (t) =  and f (t) = f (t + 2) 0, 1< t < 2  2 / nπ ,n = odd 2 T An =  an = T ∫0 f (t) cos(nω0t)dt = 0 and  0, n = even  −90°,n = odd bn = T2 ∫0T f (t)sin(nω0t)dt = 02, / n π , n = eovdedn φn =  0, n = even ab) APhmapslei tsupdeec t raunmd ∞ 1 2 1 f (t) = + sin(nπt), n = 2k −1 ∑ 2 π k=1 n Truncating the series at N=11 Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Symmetry Considerations (1) Three types of symmetry 1.Even Symmetry : a function f(t) if its plot is symmetrical about the vertical axis. f (t) = f (−t) In this case, 2 T / 2 a0 = T ∫0 f (t)dt 4 T / 2 an = T ∫0 f (t) cos(nω0t)dt bn = 0 Typical examples of even periodic function Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Symmetry Considerations (2) 2.Odd Symmetry : a function f(t) if its plot is anti-symmetrical about the vertical axis. f (−t) = − f (t) In this case, a = 0 0 4 T / 2 b = f (t)sin(nω t)dt n T ∫0 0 Typical examples of odd periodic function Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Symmetry Considerations (3) 3.Half-wave Symmetry : a function f(t) if a0 = 0  4 T / 2 T an =  T ∫0 f (t) cos(nω0t)dt , for n odd f (t − ) = − f (t)   0 , for an even 2  4 T / 2 bn =  T ∫0 f (t)sin(nω0t)dt , for n odd   0 , for an even Typical examples of half-wave odd periodic functions Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Symmetry Considerations (4) Example 1 Find the Fourier series expansion of f(t) given below. ∞ 2 1  nπ   nπ  Ans: f (t) = ∑ 1− cos sin t  π n=1 n  2   2  Michael E.Auer 01.11.2011 BSC04 Basics in Systems and Circuits Theory Symmetry Considerations (5) Example 2 Determine the Fourier series for the half-wave cosine function as shown below. ∞ 1 4 1 Ans: f (t) = − 2 ∑ 2 cos nt, n = 2k −1 2 π k=1 n Michael E.Auer 01.11.2011 BSC04

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