ME5659_LecNotes_NyquistTech

ME5659_LecNotes_NyquistTech - Polar Plot To sketch the...

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1 Polar Plot Polar Plot • To sketch the polar plot of G ( j ω ) for the entire range of frequency , i.e., from 0 to infinity, there are four key points that usually need to be known: • the start of plot where = 0, • the end of plot where = , • where the plot crosses the real axis, i.e., Im[ G ( j )] = 0, and • where the plot crosses the imaginary axis, i.e., Re[ G ( j )] = 0.
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2 Review of System Stability and Some Concepts Related to Poles and Zeros System Stability: Consider the following closed loop system: B ( s ) _ + E ( s ) C ( s ) R ( s ) G ( s ) H ( s ) The closed-loop transfer function, T ( s ), is ( ) () ( ) () () s H s G s G s R s C s T + = = 1 ) ( where is the open-loop transfer function , i.e., the transfer function relating the error signal , E ( s ), to the feedback signal, H ( s ). s H s G
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3 Review: Some Concepts Related to Poles and Zeros For a function F ( s ) with a variable of s , Poles of F ( s ) are the values of s such that Zeros of F ( s ) are the values of s such that The closed-loop poles/zeros are the poles/zeros of the closed-loop transfer function, i.e., ( ) () () s H s G s G + 1 The open-loop poles/zeros are the poles/zeros of the open-loop transfer function, i.e., ( ) ( ) s H s G . . = ) ( s F 0 ) ( = s F
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4 Review: System Stability The closed-loop poles are the roots of the characteristic equation , i.e., ( ) ( ) 0 . 1 = + s H s G In order that the above closed-loop system remain stable, all of the closed-loop poles must be located in the left half plane (LHP) .
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5 Cauchy’s Principle of Argument The Cauchy’s Principle of Argument states that, if taking a clockwise contour in the s -plane and mapping it to the F -plane through a function of F ( s ), • The number of clockwise rotations about the origin of the contour in F -plane, N = The number of zeros of F ( s ) inside the contour in the s -plane, Z the number of poles of F ( s ) inside the contour in the s -plane, P or simply N = Z – P
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6 Cauchy’s Principle of Argument Case 1: assumes a clockwise direction for mapping the points on contour A , then contour B maps in a clockwise direction if F(s) in figure has just zeros or has just poles that are not encircled by the contour e.g. (a), (b) Case 2: the contour B maps in a counterclockwise direction if F(s) has just poles that are encircled by the contour , e.g. (d).
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This note was uploaded on 05/26/2010 for the course ME 5659 taught by Professor Jalili during the Spring '10 term at Northeastern.

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ME5659_LecNotes_NyquistTech - Polar Plot To sketch the...

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