Higher Engineering Mathematics

682 FOURIER SERIES ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ f(t) = 8 π^2 ( sin ( πt 2 ) − 1 32 sin ( 3 πt 2 ) + 1 52 sin ( 5 πt 2 ) −··· ) ⎤ ⎥ ⎥ ...

L Fourier series 73 A numerical method of harmonic analysis 73.1 Introduction Many practical waveforms can be represented by sim ...

684 FOURIER SERIES Mean value= area length of base ≈ 1 2 π ( 2 π p )∑p k= 1 yk≈ 1 p ∑p k= 1 yk However,a 0 = mean value off(x) i ...

L Table 73.1 Ordinates ◦θ V cos θ V cos θ sin θ V sin θ cos 2 θ V cos 2 θ sin 2 θ V sin 2 θ cos 3 θ V cos 3 θ sin 3 θ V sin 3 θ ...

686 FOURIER SERIES Note that in equation (4), (− 46 .42 sinθ+ 69 .66 cosθ) comprises the fundamental, (4.91 sin 2θ− 6 .50 cos 2θ ...

A NUMERICAL METHOD OF HARMONIC ANALYSIS 687 L f(x) 0 π 2 πx (a) ao = 0 −π 0 π 2 πx (b) Contains no sine terms − 2 π−π 02 π πx (c ...

688 FOURIER SERIES Table 73.2 Ordinate θ i sinθ isinθ sin 3θ isin 3θ sin 5θ isin 5θ y 1 30 2 0.5 1 12 0.5 1 y 2 60 7 0.866 6.06 ...

A NUMERICAL METHOD OF HARMONIC ANALYSIS 689 L Figure 73.7 For the waveform of current shown in Fig. 73.7(b) state why only a d. ...

Fourier series 74 The complex or exponential form of a Fourier series 74.1 Introduction The form used for the Fourier series in ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 691 L Since e^0 =1, thec 0 term can be absorbed into the summation since it ...

692 FOURIER SERIES f(x) − 5 − 4 − 3 − 2 −10 1 5 2345 L = 4 x Figure 74.1 wherecnis given by: cn= 1 L ∫L 2 −L 2 f(x)e−j 2 πnx L d ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 693 L = 5 2 + 5 π (2) ( ej πx (^2) +e−j πx 2 2 ) − 5 3 π (2) ( ej 3 πx (^2) ...

694 FOURIER SERIES From equation (2), cn= ( cos 2πn−jsin 2πn −j 2 πn − cos 2πn−jsin 2πn (−j 2 πn)^2 ) + 1 (−j 2 πn)^2 However, c ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 695 L Show that the complex Fourier series for the waveform shown in Figure ...

696 FOURIER SERIES Problem 4. Obtain the Fourier series, in com- plex form, for the square wave shown in Figure 74.4. f(x) x 2 0 ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 697 L From equation (17) above,cn=−j 2 nπ ( 1 −cosnπ) Whenn=1, c 1 =−j 2 (1) ...

698 FOURIER SERIES Now try the following exercise. Exercise 249 Further problems on symme- try relationships Determine the expo ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 699 L = 20 10 ( 5 −jπn )[ e−j πn (^5) −ej πn 5 ] 20 πn [ ej πn (^5) −e−j πn ...

700 FOURIER SERIES cn − 10 − 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 3 4 2 1 02134567 8910 n Figure 74.6 6 7 8 9 10 0 1 2 3 4 − 1 1 2 ...

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 701 L either the real or the imaginary part ofVmej(ωt+α), depending on wheth ...