09-10Chapter 7-3 Direct polar plots

09-10Chapter 7-3 - Principle of Automatic Control Direct polar plots CHAPTER 7 Frequency Response(P.244 Ch.8 Ch By Hui Wang Outline of Chapter 7

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Direct polar plots Principle of Automatic Control CHAPTER 7 requency Response (P244 Ch 8) By Hui Hui Wang Wang Frequency Response (P.244, Ch.8) 浙江大学控制科学与工程学系
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Direct polar plots Outline of Chapter 7 9 Introduction 9 Bode Plots (Logarithmic plots) 9 Direct Polar Plots 9 Nyquist’s Stability Criterion 9 Phase Margin and Gain Margin and Their elation to Stability Relation to Stability 9 ……… Plum blossom
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Direct polar plots Frequency Response: 3. Direct polar plots qy p pp The Lm and the angle of G(j ω )vs . ω , plotted on semilog graph paper can be constructed easily and rapidly. The polar plot of G(j ω ) (called the direct polar plot ) can be obtained by the following methods: 1) the magnitude and angle of G(j ω ), for sufficient frequency points, re adily btainable om e mG(j ω ) nd (j ω ) s g ω rves are readily obtainable from the LmG(j ) and G(j ) vs log curves. 2) calculating |G(j ω )| and Φ ( ω ) analytically for each frequency point desired, unless a CAD program is available. 3) the data for drawing the polar plot of the frequency response can also be obtained from the pole-zero diagram . The polar plot of [G(j ω )] -1 is called the inverse polar plot .(Chap.9) 3
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Direct polar plots Frequency Response qy p 3. Direct polar plots: Typical factors K j G = ) ( ω Constant K (K>0) The frequency response K j G = ) ( 0 0 ) ( = j G The magnitude and angle of G(j ω ) are as following Im Constant -K (K>0) . K -K K j G = ) ( The magnitude and angle of G(j ω ) re s llowing The frequency response 0 Re K j G = ) ( 0 180 ) ( = j G are as following 4
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Direct polar plots Frequency Response qy p 3. Direct polar plots: Typical factors ω j j G 1 ) ( = Factor: 1/s The frequency response The magnitude and angle of G(j ω ) are as following 1 1 ) ( = = j G 0 90 ) ( = j G j . Im Factor: s ω Re ω→∞ 0 j j G = ) ( The magnitude and angle of G(j ω )areas llowing The frequency response ω =0 ω 0 = = j j G ) ( 0 90 ) ( = j G following 5
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Direct polar plots Frequency Response t 1/(T +1) qy p 3. Direct polar plots: Typical factors Factor: 1/(Ts+1) The frequency response ) ( ) ( 1 1 1 1 ) ( 2 2 ω jV U T T j T j j G + = + = + = The magnitude and angle of G(j ω ) are following 2 2 1 ) ( j G = arctgT j G = ) ( 1 T + If ω =0 ,then 1 ) 0 ( = j G ° = 0 ) 0 ( j G . Im T j 1 1 If ω =1/T 2 2 ) 1 ( = T j G ° = 45 ) 1 ( T j G If ω = 0 ) ( = j G 0 90 ) ( = j G Re 1 ω→∞ ω = 0 I can be proved the polar plot Factor: 1/(-Ts+1) 1 ) ( j G = rctgT T j + 1 1 tca bep ovedt epoa pot is a half circle 2 2 1 T + a ctg j G ) (
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Direct polar plots Frequency Response t T + 1 qy p 3. Direct polar plots: Typical factors Factor: Ts+1 The frequency response T j j G ω + = 1 ) ( The magnitude and angle of G(j ω ) are following 2 2 1 ) ( T j G + = arctgT j G = ) ( Im ω . T j + 1 If ω =0, then 1 ) 0 ( = j G ° = 0 ) 0 ( j G 1 1 ω =0 If ω =1/T, then 2 ) ( = T j G ° = 45 ) ( T j G If ω = ,then = ) ( j G 0 90 ) ( = j G Re 1 T j 1 Factor: -Ts+1 2 2 1 ) ( T j G + = arctgT j G = ) ( 7
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Direct polar plots Frequency Response qy p 3. Direct polar plots: Typical factors Factor: [1+ 2 ζ s/ ω n +(s/ ω n ) 2 ] -1 The frequency response 2 2 2 2 2 1 1 1 ) ( n n j j G ω ζ = = = ( ) 2 2 2 2 2 2 2 2 4 1 2 1 1 2 1 n
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This note was uploaded on 12/01/2011 for the course EE 400 taught by Professor Hui during the Spring '11 term at Zhejiang University.

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09-10Chapter 7-3 - Principle of Automatic Control Direct polar plots CHAPTER 7 Frequency Response(P.244 Ch.8 Ch By Hui Wang Outline of Chapter 7

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