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

2. CHAPTER 16 FOURIER SERIES © 2008 Pearson Education

4. 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 16.5 An Application 16.6 Average-Power Calculations with Periodic Functions 16.7 The rms Value of a Periodic Function © 2008 Pearson Education

5. 16.8 The Exponential Form of the Fourier Series 16.9 Amplitude and Phase Spectra © 2008 Pearson Education CONTENTS

6. 16.1 Fourier Series Analysis: An Overview A periodic waveform © 2008 Pearson Education

7.   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

8. Periodic waveforms

9. 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

10. 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

11. Dirichlet’s conditio ns: Conditions on a periodic fn. f(t) that ensure expressing f(t) as a convergent Fourier series. 1.f(t) is single valued. 2.f(t) has a finite no. of discontinuities in periodic interval. 3.f(t) has a finite no. of maxima & minima in periodic interval. 4. exist. See: Dirichlet’s conds. are sufficient conds., not necessary conds.

12. 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

13. Derivation:

15. For 0 ~ T interval,

17. 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

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

19.  ( ) See :

20. 20 As before, the integration from −T/2 to 0 is identical to that from 0 to T/2. Combining Eq. 16.20 with Eq. 16.19 yields Eq. 16.15.

21. 21 All the b coefficients are zero when f (t) is an even periodic function, because the integration from −T/2 to 0 is the exact negative of the integration from 0 to T/2; that is,

23. (a) A periodic triangular wave that is neither even nor odd © 2008 Pearson Education How the choice of where t = 0 can make a periodic function even, odd, or neither.

25. 25 For example, the derivation for is

26. 26

27. 27

28. 28

29. 29 : describes a periodic fn. that has half-wave symmetry and symmetry about midpoint of the positive and negative half-cycles.

30. 30 To take advantage of quarter-wave symmetry in the calculation of Fourier coefficients, you must choose the point where t = 0 to make the function either even or odd.

31. 31

32. fn.  odd, half-wave & quarter-wave symmetry odd  all a’s = 0, that is,  

34. Derivation : See :

38.   For steady-state response, the Fourier series of response signal is determined by first finding response to each component of input signal. © 2008 Pearson Education  Individual responses are added to form Fourier series of response signal.  Response to individual terms in input series is found by either frequency domain or s-domain analysis.

39. © 2008 Pearson Education 16.5 An Application  An RC circuit excited by a periodic voltage. The RC series circuit The square-wave voltage

40. Source : odd 

41. See : vtg. S.  set of infinitely many series-connected sinusoidal S. To find output vtg., let’s use principal of superposition. Phasor of output : Phasor of fundamental freq. of S. is

42. is the following : Therefore,

43. Hence kth-harmonic component : Now output is

44. For large C, If C is so large, fundanmental C. is significant. See : 

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

46. 46

47. 47

48. 48

49. The effect of capacitor size on the steady-state response © 2008 Pearson Education

50. Finally, verify steady state resp. = Fourier series of

51.  leave you to verify.

52. 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

53. 53

54. 54 The average power is

55. 55

56. See :

57. From previous example, Therefore, See :

58. Power on resistor : Total power :  Power delivered to 1 st 5 nonzero terms : See :

60. rms value :

61. Derive : From

62. Then, see :

63. By the way, 

64. rms value of periodic fn. :  See :

65. 

66. 16.9 Amplitude and Phase Spectra The plot of C n versus n where τ = T / 5 © 2008 Pearson Education 

67. 16.9 Amplitude and Phase Spectra The plot of (sin x) / x versus x © 2008 Pearson Education

68. Phase angle of C n Phase angle : See :

69. Now when f(t) is shifted along time axis, what happened? Amplitude spectrum : Phase spectrum : See :  t 0 =,

70. EE14170 Home work Prob. 16.1 16.2 16.12 16.20 16.26 16.27 16.29 제출기한 : - 다음 요일 수업시간 까지 - 제출기일을 지키지않는 레포트는 사정에서 제외함

71. THE END © 2008 Pearson Education