1、1 Chapter 5 Frequency Domain Analysis of Automatic control systems 5.1 Basic Concepts of Frequency Characteristics 5.2 Logarithmic Frequency Characteristic Plot (Bode Plot) 5.3 Polar Plot 5.4 Nyquist Stability Criterion 5.5 Stability Margins 5.6 Transient-state and Steady-state Performance Analysis
2、2 Polar plot takes the real part of the frequency characteristic of open-loop system as the abscissa of rectangular coordinate system, and the imaginary part as ordinate, reflecting the relationship between the magnitude and the phase represented by the parameter variable w. What is a polar plot ? e
3、RmI0 wA()1w()1wA()2 w()2 wGj()1wGj()2. . w 0w3 Draw polar plots manually Example1： CRV s RCsGsKVs( ) 1()()12wwj RCGj K( ) 1()wjTK1 T RC,to find out the magnitude-frequency characteristic, phase-frequency characteristic, real frequency characteristic and imaginary frequency characteristic correspondi
4、ng to G(jw). The first step: wwwTA G j K1 ( )( ) ( )2 w w wG j T( ) ( ) arctan ww jTGj K1()wwwj T j TK j T(1 )(1 )=(1 )wwTK jK T1= 22wwwTTjK K T11= 2 2 2 2 ww TP K1() 22 ww wTQ KT1() 22 1()Vs 2 ()Vs4 eRmI0. . to find the coordinates of some special points. When w =0 PKAK(0) ,(0) ,When PA( ) 0,( ) 0,
5、When w = 1/T TP KTA K2( ) ,1 2( ) ,1The second step： The third step： to connect these special points with a smooth curve. w T1 w 0w . . K2 K2Q(0) 0(0) 0 o Q( ) 0( ) 90 o TQ KT 2()1( ) 451w ww jTGjK1()5 P K Q K(2) (2)2 2 2The polar plot of the inertial element is a half circle. Its center is at , and
6、 its radius is . ( )( )QPTwwww P KTKQP1 1 ( )2 2 2PKTQKTTwwwww( )1( )12 22 2K2( ,0) K2Prove： . eRmI0 w T1w 0w . . K2 K26 The magnitude-frequency characteristic is an even function of the frequency w, while the phase-frequency characteristic is an odd function of frequency w. So the positive frequenc
7、y part and the negative frequency part of the polar plot are symmetrical to the abscissa. According to the relationship between frequency characteristic and transfer function, we know that the polar plot is actually the mapping of the variable s=jw of S-plane on G(jw)-plane when it changes along the
8、 positive imaginary axis. eRmI00wwA G j( ) ( ) w wGj( ) ( )Note： eRmI0ww 0ww 0wj S-plane G(jw)-plane 7 w w w wwwT j T jK T j( )( )= ()222 When ， When ， When ， Example 2： the frequency characteristic is www j j TGj K( 1)()magnitude-frequency characteristic ： wwww TAG K1 ( )( ) ( ) 2phase-frequency ch
9、aracteristic: o w wT( ) 90 arctan www j j TGj K( 1)() www wTP TK() 4 2 22wwwwTQK()4 2 2To obtain the coordinates of some special points： w 0 w w w wA P KT Q( ) , ( ) / 2 , ( ) , ( )ww T1/ wwwwTTK j K4 2 22 w w w wA P Q( ) 0 , ( ) , ( ) 0 , ( ) 0 w w w wA KT P KT Q KT( ) / 2 , ( ) 3 / 4 , ( ) / 2 , (
10、 ) / 2Im Re0 KT w 0w w T1real frequency characteristic imaginary frequency characteristic w w w wwwTTjTK K4 2 2 4 2 22For phase angles， In counter clockwise direction, it is positive; In clockwise direction, it is negative. 8 Let ， To find the intersection with the real axis: To find the intersectio
11、n point with the imaginary axis： w TT TT T T1/ 1 2 1 3 2 3Let ， mIeR0Example 3： The transfer function is G s KT s T s T s 3 1 2 3( )( 1)( 1)( 1)wwwwTTTA K111()1 2 32 2 2 2 2 2 w w w wT T T( ) arctan arctan arctan1 2 3）（ wwwwwTTTPK TT TT T T(1 )(1 )(1 )()1 1 2 32 2 2 2 2 21 2 1 3 2 32 wwwwwwTTTQK TT
12、T T T T(1 )(1 )(1 )()()1 2 32 2 2 2 2 21 2 3 1 2 32 w TT TT T T1 2 31 2 3wP( )=0wQ( )=0 w T T T TT T= ( )/( )1 2 3 1 2 3magnitude-frequency characteristic： phase-frequency characteristic: real frequency characteristic： imaginary frequency characteristic： w w 0w TT TT T T11 2 1 3 2 3When ， To obtain
13、the coordinates of some special points： w 0 w w w wA K P K Q( ) , ( ) 0, ( ) , ( ) 0When ， w w w w wA P Q( ) 0, ( ) 3 / 2, ( ) 0, ( ) 0Kw 09 phase-frequency characteristics: Example 4： The transfer function is s s sGs Ks( 1)(2 1)( )= (4 1) w w w w( ) arctan 4 90 arctan arctan 2eRmI01-2-3-4-5-6-5.0-1
14、-5.1-2-5.2-The polar plot has a turn. positive and negative )12)(1( )14( sss sKw 0w10 The methods of drawing polar plots manually. The first step: To calculate the magnitude-frequency characteristic, phase-frequency characteristic, real frequency characteristic and imaginary frequency characteristic
15、 of the system. The second step: To determine the starting point( ) and ending point ( )of the polar plot as well as their intersections with real axis and imaginary axis. The third step: To observe whether the phase-frequency characteristics are positive or negative to determine the direction of th
16、e polar plot. Then connect these special points in the order of frequency from low to high with a smooth curve. ww 011 The characteristics of polar plot of the minimum-phase system. wwwwjTjGj Kjjjnviim(1 )()( )=(1 )=1-=1phase-frequency characteristics: w w wvTijijm n v2( )= arctan arctan=1 =1-w 0, W
17、hen When w , m v n v2 2 2( )= ( - )The frequency characteristic of the low-frequency region is related to the type number of the system. The frequency characteristic of the high-frequency region of the polar plot is related to n-m. wwjvGK0()(0)= v 2(0)=w wGj( ) 0（ If n m） nm2= ( - )12 1. the low-fre
18、quency region ： w + + w w + w 0 0 Type w 0 Type w 0 Type Re Im 0 0mIeR2. the high-frequency region ： （ If n m） It means the ending point of the polar plot must tend toward the origin. 1=m-n2=m-n3=m-nw wjG K v()(0)0 v 2(0)=wwGj( ) 0 nm2( )= ( - )GK| (0) | , (0) 0Type 0: ，G| (0) | 2(0)Type I: ，G| (0)
19、| (0)Type II: ， nm1 ( ) / 2， nm2 ( )， nm3 ( ) 3 / 213 eRmI0The characteristics of polar plot of non-minimum-phase systems. The transfer function ： TsDsGs Nssjjniiml(1 )()() ()(1 )111w wG j j( ) 1G s Ts( ) 1/(1 )wwG j j T( ) 1/(1 )G s s( ) 1 mIeR0w 0www s1 s1+s 1Polar plot Polar plot Polar plot If th
20、e transfer function is ： frequency characteristic ： wP( ) 1magnitude-frequency characteristic ； w w A( ) 122phase-frequency characteristic ； w w( ) arctanreal frequency characteristic w wQ()imaginary frequency characteristic If the transfer function is ： frequency characteristic ： phase-frequency ch
21、aracteristic ； magnitude-frequency characteristic ； real frequency characteristic imaginary frequency characteristic ww TA 1() 1 22 w wT( ) arctan ww TP 1() 122 ww w TQ T1() 22 ww=0 w=01Ts1 1Ts1+1 Ts1 1 ss1= (1 )14 Summary The method of drawing polar plot manually The characteristics of the polar plot of minimum-phase system The characteristics of the polar plot of non-minimum-phase system
