Lesson+25 - 1 Lesson 25 Lesson 25 Challenge 24 Fourier...

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Lesson 25 1
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Challenge 24 Fourier Series (Properties) Ch. 16:3 Challenge 25 Lesson 25 2 Lesson 25
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Challenge 24 Lesson 25 3 -2 - 0 2 x(t)=e t/2 Compute the coefficients of a trigonometric Fourier series representation of the periodic signal shown above. 1 e - /2
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2 0 2 2 0 2 0 2 0 16 1 8 5 0 2 2 16 1 2 5 0 2 2 5 0 1 n n dt nt e B n dt nt e A dt e A t n t n t . ) sin( . ) cos( . / / / π π π π π π 2 2 p a px p px a e dx px e ax ax ) sin( ) cos( ( ) cos( 2 2 p a px p px a e dx px e ax ax ) sin( ) cos( ( ) sin( Lesson 25 4 Compute the trigonometric Fourier coefficients. Select a period (say – <t<0) How does he know these amazing facts??? CRC-Table #359 CRC-Table #358 Therefore A 1 = 1/17
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Lesson 25 5 A periodic signal can be represented as follows (shown in trigonometric form): Think superposition. How can this knowledge be used to analyze circuits?. ) cos( B ) cos( A x(t) 0 n 1 n 0 n 0 t n t n A
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Circuit Example h(t)=e -t u(t)    H(s)=1/(s+1)   x(t) 0            2   …                                   t T=2 f 0 =1/T=1/2 0 =1 r/s jkt odd k k j k k t jk k e e k e c c t x   ) / ( ) ( 2 0 0 4 2 0 4 0 X(t) L=1H R=1Ohm y(t) 6 Lesson 24 Table lookup Odd harmonics Exponential CTFS
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  k     H(s=jk 0 ) c kx         H(jk 0 ) c kx =c ky |c kx | |c ky | _0 1 2 2 2 2 _1 1.27 0.90 _3 0.42 0.13 _5 0.25 0.05     … Need to compute for all significant  k (only positive k shown). 45 2 / 1 135 2 / 4 90 / 4 5 . 71 10 / 1 7 . 78 26 / 1 90 3 / 4 90 5 / 4 5 . 161 10 3 / 4 7 . 168 26 5 / 4 Old school – The circuit only needs to be studied at the odd harmonics (exception being DC). 7 Lesson 24 jkt odd k k j e e k t x  ) 2 / ( 4 2 ) (
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Lesson 25 8 Symmetry (16.3 and Lesson 24) Even Odd Halfwave 0 0 0 ) ( ) ( t x t x ) ( ) ( t x t x ) ( ) ( 0 T t x t x
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Symmetry (16.3 also Lesson 24) Type Signal Fourier series Even x(t)=x(-t) B n =0 Odd x(t)=-x(-t) A n =0 Halfwave x(t)= -x(t-T/2) or x(t)= -x(+T/2) 9 Lesson 24 even n odd n dt t n t x T A T n ; 0 ; ) cos( ) ( / 4 2 / 0 0 even n odd n dt t n t x T B T n ; 0 ; ) sin( ) ( / 4 2 / 0 0 T n dt t n t x T A 0 0 ) cos( ) ( / 2 T n dt t n t x T B 0 0 ) sin( ) ( / 2
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Lesson 25 10 n n t n C C t x 0 0 cos ) ( A periodic signal can be represented in the following forms: ) cos( B ) cos( A x(t) 0 n 1 n 0 n 0 t n t n A
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