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# 4945427.ppt

Sc Pattar
Associate Professor em BLDE Association's College
29 de Mar de 2023
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### 4945427.ppt

1. Fourier Series
2. Fourier Series 2 0 0 0 2 0 0 0 1 ( )cos(2 ) ( ) 1 ( )sin(2 ) ( ) n n a x t nf t d t b x t nf t d t             0 0 0 0 0 0 0 0 1 2 1 2 ( ) 2 2 f T t f t d t f dt dt dt T T            is the “fundamental frequency” 0 0 1 1 ( ) cos(2 ) sin(2 ) 2 N n i n n x t a a nf t b nf t       
3. Fourier Series 2 0 0 0 2 0 0 0 1 ( )cos(2 ) ( ) 1 ( )sin(2 ) ( ) n n a x t nf t d t b x t nf t d t             0 0 0 0 0 0 0 0 1 2 1 2 ( ) 2 2 f T t f t d t f dt dt dt T T            is the “fundamental frequency” 0 0 1 1 ( ) cos(2 ) sin(2 ) 2 N n i n n x t a a nf t b nf t       
4. Fourier Series Integration limits: when 0 2 t    , then 0 0 0 2 2 1 2 / t T T        so we get: 0 0 1 1 ( ) cos(2 ) sin(2 ) 2 N n i n n x t a a nf t b nf t        0 0 0 0 0 0 0 0 2 ( )cos(2 ) 2 ( )sin(2 ) T n T n a x t nf t dt T b x t nf t dt T      
5. Fourier Series Euler: 0 0 1 1 ( ) cos(2 ) sin(2 ) 2 N n i n n x t a a nf t b nf t        2 cos(2 ) sin(2 ) i j f t i i e f t j f t      0 2 ( ) jn f t n n x t c e     
6. Fourier Series 0 2 ( ) jn f t n n x t c e      0 0 0 2 0 2 1 ( ) T jn t n T c x t e dt T      We can show 2 2 n n n c a b   1 tan ( / ) n n b a    ; recall that 2 2 1 cos( ) sin( ) cos( tan ( )) b a b a b a        
7. Phasors: Phasors 2 2 a b 
8. Symmetry Odd f(-t) =-f(t) Fourier: sine terms only Even f(t) = f(-t) Fourier: cosine terms only Neither
9. Half-wave symmetry: 0 ( ) ( ) 2 T f t f t    has no even harmonics | | t t+T/2
10. Example of non-symmetric waveform: 0 ( ) ( ) 2 T f t f t   
11. Fundamental Signals Unit Step: 1, 0 ( ) 0, 0 t t t          
12. Fundamental Signals Unit Step: 1, 0 ( ) 0, 0 t t t           Unit Impulse 0, 0 ( ) , 0 t t undefined t           ( ) ( ) t t dt t     
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