lecture 17

lecture 17 - z domain Analysis Mapping between the s plane...

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1 z domain Analysis Mapping between the s plane and the z plane ω σ jT T j T Ts e e e z j s e z = = + = = + ) (
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2 Periodic strips in the s-plane are mapped into the unit circle in the z-plane. Signals in complementary strips will be folded into primary strip after sampling. To avoid the folding, the sampling frequency must be high enough. ω σ jT T j T e e e z = = + ) (
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3 Constant-Damping Loci σ T j T e e z ω = which is a circle centered at the origin with a radius of T e Constant-Frequency Loci T j T e e z =
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4 Constant-Damping Ratio Loci ζ The s -plane loci are described by d n n n j j s ω ζω + - = - + - = 2 1 The constant loci in the z -plane are described by ) 2 1 2 exp( ) exp( 2 s d s d d n Ts j T j T e z π πζ + - - = + - = = s d s d z z 2 ; ) 1 2 exp( 2 = - - =
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5 Constant Natural-Undamped Frequency Loci n ω 2 2 1 2 2 ) 2 exp( ) 1 2 exp( ζ π πζω πζ - = = - = - - = s n s d s n s d z z
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6 Stability ) ( 1 ) ( ) ( ) ( z GH z G z R z C + = The characteristic equation: 0 ) ( 1 ) ( = + = z GH z P Note: The roots must be inside the unit circle for a stable system The system is critical stable if the roots are on the unit circle. For a stable system, its zeros can be outside of the unit circle.
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This note was uploaded on 06/02/2010 for the course HET 489 taught by Professor Zhenwei during the Three '08 term at Swinburne.

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lecture 17 - z domain Analysis Mapping between the s plane...

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