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nullnullChapter 5 Frequency-response analysis Applying Frequency-response to study linear system.Reason :1.the difficulty to get the transfer function; 2.The property of the input .null（1）The frequency response has clear and definite physical meaning, it can be determined by experiment method. To the system difficult to write the differential equation, it has actual meaning of importance. (2) because the frequency response analyzes system by open -loop frequency characteristic, it is direct,simple, has little calculation. (3) the frequency response not only is applicable to the linear time-invariant system, but also the system with pure delay and part non-linear system.characteristicsnull5.1Frequency-response and description 5.1.1 Frequency-response basic conceptFrequency characteristic,also called Frequency response，is the response characteristic to sine signal of a system or component.The output amplitude and phase generally differ from input, and vary with different input signal frequency . nullt/s nullAssume system TFGiven input,its Laplace transformation A is a constant，output: (5-1)G(s) poles (5-2)To a stable system null(5-2)0,when Coefficient to be determined Owing tois a complex vector，can be expressed(5-7)(5-5)(5-6)(5-4)null (5-11)Steady-state output of a linear system is a sine signal and its frequency is the same as input sine signal frequency. the amplitude ratio of output to input isPhase deviation of output to input phase-frequency characteristicAmplitude-frequency characteristicnotenullAs a case,see R-C circuit。Fig.5-3. TF isSet input(5-15)where nullnull--------frequency characteristic of RC circuit 。andAre the function of input signal frequencyThey are called respectively amplitude-frequency characteristic and phase-frequency characteristic。Physical meaning:the output/input ratio,Or output/input amplitude ratio and output/input phase deviation . The relation between input and output when a sine signal with a frequency is exerted to a circuit.connected with construction and parameters of RC circuit, independent of the input signal。, output/input amplitude ratio when steady state, output/input phase deviation when steady statebecausenullCircuit output/input amplitude ratio(a) amplitude-frequency characteristic null(b) phase-frequency characteristic。Circuit output/input phase deviationnullphase-frequency characteristic and TF is of very similar form.。comparenull5.1.2 expressed forms (1) Bode diagram or logarithmic plot (2) Polar plot (3) Log-magnitude versus phase plotlogarithmic frequency characteristicLogarithmic amplitude-frequency characteristicLogarithmic phase-frequency characteristic ()The longitudinal coordinate is marked according to linearityHorizontal coordinate ismarked according tonullBODE diag.nullPolar plot,also amplitude-phase frequency characteristic .Can be expressed by a vector with amplitude and phase， Amplitude and phase ofthe vector G(jw) vary,the curve which its end point traces in complex plane is called polar diagram---- Nyquist curve----- Nyquist diag.When input signal frequency Nyquist analysed feedback control system stability in 1932.nullNyquist DIAG.null5.2typical component bode diag.5.2.1 proportion KPlease look at the following pagenullNumber- decibel conversionnullDiag.5-7 Number- decibel conversion straight line null5.2.2 integer and differential These amplitudes-frequency curve pass the pointInferenceDifference :signnullDiag.5-8 log－frequency characteristic of the integralnullDiag. 5-9 The log－frequency characteristic of the differentialnull-20dB/dec -40dB/dec-60dB/declog－frequency characteristic Diag. 5-10null5.2.3 first-orderfirst-orderLow frequency,Low frequency,log-amplitude frequency characteristic is a 0 decibel lineDiag.5-10 gives the asymptote and the accurate curve of first order system.high frequency,high frequency,log-amplitude frequency is a line whose slope is -20db/decPlease look at following pagephase－frequency characteristicnullAsymptote Asymptote Corner frequency Exact curveExact curveDiag.5-11 log－frequency characteristic of the 1 order systemnullDiag.5-12 log amplitude error between Asymptote and Exact curvenullDiag. 5-13 log－frequency characteristic of the 1 order factornull5.2.4 2th order factor nullLow frequency,Low frequency asymptote-20log1=0dBHigh frequency,A line whose slope is -40db/decbecause ,cross point low frequency and High frequency appears at nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullFig.5-13 amplitude frequency characteristic of 2th factornullRelation between phase frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude frequency characteristic and nullRelation between amplitude error and nullRelation between amplitude error and nullRelation between amplitude error and nullRelation between amplitude error and nullRelation between amplitude error and nullFig.5-14 log amplitude error of 2th order factornullset(5-22)(5-23)(5-25)Resonant frequencyResonant frequency and Resonant peakResonant peakwhen ，no peak ,no resonance.andIs as following page.relation betweennullFig.5-15 relation betweenand/dBnullLow frequency,Low frequency asymptote20log1=0dBHigh frequency,A line whose slope is 40db/decbecause ,cross point low frequency and High frequency appears at nullnull5.2.5 delay null5.3.1integer and differential factorSo,polar plot of is negative imaginary axis.fig5-26 integer factor polar plotpolar plot of is positive imaginary axis.5.3typical component nyquist diag.nullfig5-27 differential factor polar plotnull5.3.2first order factor Fig.5-28 first order factorPolar plotnullFig.5-29 first order factorPolar plotnull5.3.3 2-th order factor high frequency part is tangent to negative real axis.its accurate shape of the polar plot has relation with damped ratio, but for under-damped and over-damped,they are basically similar.Fig.5-30 2-th order factor polar plotnullfor under-damped,phase is 90º.the crossing point frequency atTrajectoryThe point Farthest From origin corresponds to resonant frequency,corresponds resonant peak Mr.whenand imaginary axis is nullFor over damped Increases to>>1， The trajectory of trends a semicircle。This is because to a big damped system, characteristic roots are real number,and one root is much less than another。The bigger affects system much less than the smaller. thus,the system is similar to a first order system.whennullnullnullnullnullnull low frequency part ：high frequency part ：Fig 5-31 2-th factorPolar plotnull5.3.4 pure delaywhen，whenBoth exists hypostatic differenceLow frequency part is similar to first order system,nullthanks！ end