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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)-Part I 5.3 Polar Plot 5.4 Nyquist Stability Criterion 5.5 Stability Margins 5.6 Transient-state and Steady-state Performance Ana
2、lysis 2 1 1 2 2 Coordinate division of Bode plot Advantages of Bode plot 3 3 Bode plot of typical element Logarithmic Frequency Characteristic Plot 3 90 20 0-45 45 0-10 10 L()/dBlg1 2 1 100 10 0 lg1 2 1 100 10 0 90 20 0-45 45 0-10 10 Abscissa:LA()20lg()()Coordinate division of Bode plot L()/dBlg1 10
3、0 10 lg1 2 1 100 10 0 Linear scale Logarithmic phase-frequency characteristic ordinate:Phase shift Degree()or radians(rad)Logarithmic scale,rad/s Logarithmic magnitude-frequency characteristic ordinate:,decibel(dB),Linear scale Angular frequency 4 1 2 5 10 100 0 0.301 0.699 1 2 The relationship betw
4、een logarithmic scale and linear scale of abscissa lgCoordinate division of Bode plot lg0 1 2 100 10 5 2 1 0.301 0.699 Abscissa:LA()20lg()()Linear scale Logarithmic phase-frequency characteristic ordinate:Phase shift Degree()or radians(rad)Logarithmic scale,rad/s Logarithmic magnitude-frequency char
5、acteristic ordinate:,decibel(dB),Linear scale Angular frequency 5 The relationship between logarithmic scale and linear scale21080605040302098765310042101Note for Bode plot，10211+lglglg10121 Decade:using logarithmic division,one unit length change of the abscissa,the corresponding frequency change t
6、en times,one unit of the abscissa is called the decade.Decade Decade If Then，=41=402Decade 0 lg6 Note for Bode plot The abscissa uses a logarithmic scale,which is not even for,but even for lg.Since is scaled logarithmically,the zero frequency point is at .Decade Decade lg210605040302065310042101 une
7、ven scale lg even scale Decade:using logarithmic division,one unit length change of the abscissa,the corresponding frequency change ten times,one unit of the abscissa is called the decade.7 Note for Bode plot Generally,the magnitude-frequency characteristic and the phase-frequency characteristic are
8、 drawn on one graph,using the same abscissa(frequency axis),and the two graphs are placed one up and one down.The abscissa uses a logarithmic scale,which is not even for,but even for lg.Since is scaled logarithmically,the zero frequency point is at .Decade:using logarithmic division,one unit length
9、change of the abscissa,the corresponding frequency changes ten times,one unit of the abscissa is called the decade.8 1 1 2 2 Coordinate division of Bode plot Advantages of Bode plot 3 3 Bode plot of typical element Logarithmic frequency characteristic graph 9 Broaden the frequency range Low-frequenc
10、y Intermediate frequency High-frequency-1 0 1 2 3 0.1 1 10 100 1000 10000 4 lgBecause we use logarithmic division on the abscissa of the logarithmic frequency characteristic plot,when the frequency changes every 10 times,only one unit length is added to the abscissa.Advantages of Bode plot 0.01-2 10
11、 Easy to draw:The role of each typical link in the system can be clearly seen from the Bode plot.It is very convenient for us to analyze the performance of the system and design the controller.sT sTsT sG sKsss ejljlllnnikikkkT smmd(1)(12)()(1)(12)112211221212（jjTTj TG jKjjejlllikkkjTd()(1)(1)2()1+)(
12、1)22222Convenient operation:multiplication can be converted into addition.sjLG j()20lg()Kjjikkk20lg20lg120lg(1)222vjjTTj Tjii i20lg20lg120lg(1)222im11km12jn11ln12Advantages of Bode plot im11km12jn11ln1211 1 1 2 2 Coordinate division of Bode plot Advantages of Bode plot Bode plot of typical element 3
13、 3 Logarithmic frequency characteristic graph 12 1.Proportional element The frequency characteristics：GK(s)G jGKs j()(s)AK()The magnitude-frequency characteristics：Logarithmic magnitude-frequency characteristic：K()Logarithmic phase-frequency characteristic:K01K01K=0=1oK-1800oK00The transfer function
14、：The bode plot of each typical element =20lg|K|=constant=aufugu+Tachogenerator-+-+-+euamplifierPower amplifierVoltage-n+M13 ()180180K 0K 0L()/dBK20lgK 1K 1K20lgK 1K20lgThe bode plot of each typical element 1.Proportional element The frequency characteristics：GK(s)G jGKs j()(s)AK()The magnitude-frequ
15、ency characteristics：Logarithmic magnitude-frequency characteristic：K()Logarithmic phase-frequency characteristic:K01K01K=0=1oK-1800oK00The transfer function：=20lg|K|=constant=14 2-TsGKK(1 Ts)1(s)12.Integral element sG(s)1Example:The transfer function of the armature-controlled DC motor G jAeejj()()
16、()A()）（=The frequency characteristics：The magnitude-frequency characteristic：Phase-frequency characteristic:s1The transfer function:12-15 2.Integral element L()/dB()9020402040110100110100.LA()20lg()20lg(1/)20lgLogarithmic magnitude-frequency characteristic.ykx，L=1()0，L=10()20，L=10()40Two integral el
17、ement 16 03.Inertia element TA1()122TsG s1()1TjG j1()1 T()arctanThe frequency characteristics:The magnitude-frequency characteristic:Phase-frequency characteristic:，时当 2()，时当0(0)0，时当 TT4()11The transfer function:ssTsGKK(1 Ts)1(s)1Example:The transfer function of the armature-controlled DC motor The
18、logarithmic phase-frequency characteristic curve of the inertial element is skew-symmetric to(1/T,45)in the semi-logarithmic coordinate system.-45-90T1T101T51T21T5T10T220T17 04-03-02-01-0 T20lg 122T=20lg 122when ,kLLlglg()()2121is still a straight line.L()，through .Low-frequency band:High-frequency
19、band:when Using segments of straight lines to approximate the logarithmic magnitude-frequency characteristics.LAT()20lg()20lg 122Logarithmic magnitude-frequency characteristic：T1L()TTT20lg20lg20lg2022ybkxT,1 LTT()20lg20lg0T(,0)1L()1L()2，1021When 12 LT()20lg20lg11 LT()20lg20lg(10)21 lg10lglg20-20lg10
20、11T01 20lg 1=0 0when Slope：21Why not use l()Break frequency The biggest error occurs at w=1/T,its 3dB.3.Inertia element T1T101T51T21T5T10T2100T18 0206-04-02-04.Oscillation element When 01the magnitude-frequency characteristic T sTsG s21()122ATT()1/(1)(2)22 22The transfer function:G jTjT()1/(1)222the
21、 frequency characteristics ssnnn2222the logarithmic magnitude-frequency characteristic LTT()20lg(1)(2)2222Using segments of straight line to approximate the representation.，T01 Low-frequency band:L()TT20lg(1)(2)2222 High-frequency band:，T1L()TT20lg(1)(2)2222 1oT The two asymptotes intersect at,which
22、 is called break frequency.Note：The logarithmic magnitude-frequency characteristic curve has a peak.20lg 10TT20lg()=40lg222T1T101T51T21T5T10T2100T19 081-09-0 TT1()=-arctan222Phase-frequency characteristic:When ，T(,)1 TT()arctanarctan1122Knnnn(sj1)(sj1)22，时当 ()，时当0(0)0，时当 TT2()11j Tjj TjT K(1)(1)22ss
23、G sKnn2()22Denominator Simplification sjFactorization 4.Oscillation element T1T101T51T21T5T10T2100T20 G sT sTsG sTsG ss1/1/1/()(21)()(1)()225.Derivative element The first-order derivative element and the second-order derivative element.G ssG sTsG sT sTs22()()1()21Frequency characteristic:22()()1()12
24、G jjG jjTG jTjTThe transfer function:Integral element,the inertia element,and the oscillation element.The transfer function:Frequency characteristic:Phase-frequency characteristic:Logarithmic magnitude-frequency characteristics:Logarithmic magnitude-frequency characteristics:Phase-frequency characte
25、ristic:L()20lg+L()20lg-=()/2+()/2LTg 1+()20l22LT()20lg 122T()arctanLTT+()20lg(1)(2)2222L()20lg(1 T)(2T)22 22TT+1()arctan222G jTjTG jjTG jj()(12)1)1/1/1/()()22T()arctanTT 1()arctan22221 0202-0Logarithmic magnitude-frequency characteristics comparisons The first-order derivative element Inertia elemen
26、t Integral element Derivative element The second-order derivative element Oscillation element 0202-00202-02020-02020-00202-0T1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100T22 Phase-frequency characteristic comp
27、arisons Derivative element Integral element The first-order derivative element The second-order derivative element 081-0909-0810081-0909-081009-54-09540081-0909-0810T1T101T51T21T5T10T2100TT1T101T51T21T5T10T2100TT1T101T51T21T5T10T220TT1T101T51T21T5T10T220T23 6.Delay element The transfer function:()G
28、sesFrequency characteristic:()G jejMagnitude-frequency characteristic:A()1Phase-frequency characteristic:57.3()()(rad)09-081-072-063-054-045-001-001Logarithmic magnitude-frequency characteristic:L()0110151215102()(dB)L24 SUMMARY Coordinate division of Bode plot Advantages of Bode plot Bode plot of typical element