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EEE445_Lectr21_chebyshev

# EEE445_Lectr21_chebyshev - 5.7 Chebyshev Multi-Section...

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EEE 591/445 Lecture 21 1 5.7 Chebyshev Multi-Section Transformers Design of multi-section transformers with equal-ripple passband responses. Chebyshev polynomials belong to orthogonal systems, including the Bessel, Legendre, Hermit, Laguerre functions and wavelets. Chebyshev minimizes maximum errors

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EEE 591/445 Lecture 21 2 Chebyshev polynomials may be obtained from the generating function equation. Chebyshev polynomials consists of two types Type I Type II Both satisfy the recurrence equations ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 2 0 2 0 n n n n n n T x xT x T x U x xU x U x + - + - - + = - + = ( 29 n T x ( 29 n U x Arfken, Mathematical methods for physicists, Academic Press, 1981
EEE 591/445 Lecture 21 3 Chebyshev Polynomials ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 0 1 2 2 3 3 4 2 4 5 3 5 1 2 2 " ' 2 1 2 1 4 3 8 8 1 16 20 5 2 1 0 n n n n n n T x T x x T x x T x x x T x x x T x x x x T x xT x T x x T x xT x n T x - - = = = - = - = - + = - + = - - - + = In this course we only use type I of the chebyshev They satisfy the ODE (5.56) (5.57)

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EEE 591/445 Lecture 21 4 Chebyshev Polynomials -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x T n (x) n=0 n=1 n=2 n=3 n=4
EEE 591/445 Lecture 21 5 Properties of Chebyshev Polynomials 1. Equal ripple property This region will be mapped to the passband of the transformer. 1. Even (odd) order polynomials possesses even (odd) symmetry. ( 29 1 1 , 1 1 For - - x T x n ( 29 ( 29 ( 29 = = - = - even , odd , n x T n x T x T n n n

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EEE 591/445 Lecture 21 6 Properties of Chebyshev Polynomials 1. This region will be mapped to the frequency range outside of the passband. ( 29 For 1, 1 and increases monotonically n x T x
EEE 591/445 Lecture 21 7 Properties of Chebyshev Polynomials 1. In filter applications, higher-order filters have faster rolloff. ( 29 For 1, increases faster with as increases n x T x x n

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EEE 591/445 Lecture 21 8 Important Identities for Chebyshev Polynomials ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 2 3 3 4 2 4 cos acos , 1 cosh acosh , 1 sec cos sec cos sec cos sec 1 cos2 1 sec cos sec cos3 3cos 3sec cos sec cos sec cos4 4cos2 3 4sec cos2 1 1 n m m m m m m m m m m n x x T x n x x T T T T θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = = = + - = + - = + + - + + ( 29 ( 29 ( 29 ( 29 1 cos cos Let s cos cos n n T n x co T x n x ϕ ϕ ϕ - = = = (5.58a) (5.58b) (5.60a) (5.60b) (5.60c) (5.60d)
EEE 591/445 Lecture 21 9 Mapping for Equal Ripple Transformer Response 1 1 1 cos sec cos cos cos cos ( ) cos cos cos cos m m m m n n m m x x x T x T n θ π θ θ θ θ θ θ θ θ θ - = - = - = = = = Map the two ends of a bandwidth (5.59)

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EEE445_Lectr21_chebyshev - 5.7 Chebyshev Multi-Section...

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