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Unformatted text preview: EEE 591/445 Lecture 21 1 5.7 Chebyshev MultiSection Transformers Design of multisection transformers with equalripple passband responses. Chebyshev polynomials belong to orthogonal systems, including the Bessel, Legendre, Hermit, Laguerre functions and wavelets. Chebyshev minimizes maximum errors 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 2 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 1 2 2 3 3 4 2 4 5 3 5 1 2 2 " ' 2 1 2 1 4 3 8 8 1 1 6 2 0 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 x T x T x x T x x T x n T x  = = =  =  =  + =  + =   + = In this course we only use type I of the chebyshev They satisfy the ODE (5.56) (5.57) EEE 591/445 Lecture 21 4 Chebyshev Polynomials21.510.5 0.5 1 1.5 221.510.5 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 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 m onotonically n x T x EEE 591/445 Lecture 21 7 Properties of Chebyshev Polynomials 1. In filter applications, higherorder filters have faster rolloff. ( 29 For 1, increases faster with as increases n x T x x n 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 = = = +  = +  = + +  + + ( 29 ( 29 ( 29 ( 29 1 cos cos Let s cos cos n n T n x co T x n x  = = = (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...
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This note was uploaded on 12/14/2010 for the course EEE 445 taught by Professor Georgepan during the Fall '10 term at ASU.
 Fall '10
 GEORGEPAN

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