HW#2_solution

# HW#2_solution - HW2 1(a sin cos(4 2 v t t t T π π = =...

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Unformatted text preview: HW2 1. (a) ( ) sin( ) cos(4 ) 2 v t t t T π π = + = Exponential form : 4 4 1 1 ( ) ( ) ( ) 2 2 1 [ 4 ] , [ 1] 2 2 1 [1] , [4] 2 2 j t j t j t j t v t e e e e j j V V j V V π π π π − − = − + + − = − = = = Trigonometric form: 2 2 2 2 2 2 2 2 ( )cos( ) {sin( ) cos(4 )}cos( ) cos(4 )cos( ) ( sin( )cos( ) 0) 1 1 cos(4 ) cos(4 ) 2 2 1 1 cos(4 ) ( cos(4 ) 0) 2 2 T n a v t n t dt t t n t dt T t n t dt t n t dt n t dt n t dt n t dt n t dt ω π π π π π π π π π π π π = = + = = = + + − = − + = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ If n = 4, 2 4 1 1 2 a dt = ∫ If n ≠ 4 , n a = 2 2 2 2 2 2 2 ( )sin( ) {sin( ) cos(4 )}sin( ) sin( )sin( ) cos(4 )sin( ) 1 1 { cos(1 ) cos(1 ) } ( cos(4 )sin( ) 0 ) 2 2 1 cos(1 ) 2 T n b v t n t dt t t n t dt T t n t dt t n t dt n t n t dt t n t dt n t dt ω π π π π π π π π π π π π π = = + = + = − + + − = = − ∫ ∫ ∫ ∫ ∫ ∫ ∫ If n = 1, 2 1 1 1 2 b dt = ∫ If n ≠ 1 , n b = b) ( ) |sin(2 ) | 0.5, 4 o v t t T π ω π = = Exponential form : 0.5 4 0.5 0.5 2 2 4 2 (1 2 ) 2 (1 2 ) 2 (1 2 ) 0.5 2 (1 2 ) 0.5 1 sin(2 ) 0.5 1 1 2 ( ) ( ) 2 1 1 1 [ | | ] 2 (1 2 ) 2 (1 2 ) 1 2 2 1 1 [ ] 2 (1 2 ) 2 (1 2 ) (1 2 ) (1 jk t k j t j t jk t j k t j k t j k t j k t V t e dt e e e dt e e dt j j e e j j k j k j j k j k k π π π π π π π π π π π π π π π − − − − − + − − + = = − = − = − − − + − − = + = + − + − ∫ ∫ ∫ 2 2 2 ) (1 4 ) k k π = + − Trigonometric form: 1/2 1/2 1/2 1/2 2 2 ( )cos( ) 4 |sin(2 ) | cos(4 ) 1 1 4 { sin 2(1 2 ) sin 2(1 2 ) } 2 2 1 1 cos2(1 2 ) | cos2(1 2 ) | (1 2 ) (1 2 )...
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HW#2_solution - HW2 1(a sin cos(4 2 v t t t T π π = =...

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