1、1Chapter 6 Frequency Response6.1 Introduction6.2 Diagram of Frequency Response 6.3 Frequency Response Function of Elementary Unit6.4 Diagram of Complex Frequency Response Function6.5 Summary26.1 IntroductionThe steady state response of a system to a sinusoidal input test signal. 3The frequency respo

2、nse of a system is defined as the steady-state response of the system to a sinusoidal input signal.The sinusoid is a unique input signal, and the resulting output signal for a linear system is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle.4Input

3、 signal isorFor a stable linear system, we have5All pi have negative real parts, so the steady-state response of the system isThe steady-state output signal depends only on the magnitude and phase of T(jw) at a specific frequency w.6DC motor7G(jw)形式：R(w)是G(jw)的實(shí)部，稱為實(shí)頻特性，X(w)是G(jw)的虛部，稱為虛頻特性思考：實(shí)際物理系統(tǒng)

4、對頻率越高的輸入響應(yīng)幅值越小，說明其傳遞函數(shù)分母階次高于分子階次，為什么？8The advantage of frequency response method:The experimental determination of the systems frequency response is easily plished, and the transfer function can be deduced from the experimental frequency response.The transfer function describing the sinusoidal stead

5、y-state behavior of a system can be obtained by replacing s with jw.9The disadvantage of frequency response method:The direct correlation between the frequency response and the corresponding transient response characteristics are somewhat tenuous.10The relationship between Fourier and Laplace transf

6、ormsLaplace: Fourier: When f(t) is defined only for t=0, as is often the case, the two equations differ only in the complex variables. Thus the Fourier transform can easily be obtained from Laplace transform just by setting s=jw.11 The Laplace transform permits us to investigate the s-plane location

7、 of the poles and zeros of a transfer G(s). However, the frequency response method allows us to consider the transfer function G(jw) and to concern ourselves with the amplitude and phase characteristics of the system. 126.2 Diagram of Frequency ResponsePolar plot (Nyquist diagram)Polar plot: The plo

8、t of frequency response G(jw) in the complex plane as w increases from 0(-) to +.13Steps：1. start point: magnitude and phase (angle). 142. end point:153. circumrotate direction:4. intersection（交點(diǎn) ）on negative real axis16Example 6.1: Polar plot of a transfer function17181. First-order Unit (a pole on

9、 the real axis)192. Integrator(poles at the origin)203. Differentiator and First-order Differential Unit 214. Oscillations Unit (complex conjugate poles)225. Delay Unit23Example 6.2: Polar plot of24 Intersection on the real axis:2526Nyquist diagram can be readily obtained by Matlab num=0 0 10;den=1

10、3 2 0;nyquist(num,den);v=-5 1 3 3;axis(v);27Example 6.3:28Notes：1.2. The polar plot is useful for investigating system stability. 3. The limitations of polar plots: the addition of poles or zeros to an existing system requires the recalculation of the frequency response, and the calculation in this

11、manner is tedious （繁瑣的）and does not indicate the effect of the individual poles or zeros.29Logarithmic plot (Bode plot)Consists of two plots: magnitude plot and phase plot.The magnitude and frequency are in logarithm scale.30Logarithmic magnitude-frequency plot the units are decibels (dB).An interva

12、l of two frequencies with a ratio equal to 10 is called a decade. 31Magnitude plot32Phase plot33Example：Bode plot for first-order system:Magnitude plot34At the asymptotic line:The frequency w=1/T is often called the break frequency or corner frequency.The slope of the asymptotic line for this first-

13、order transfer function is 20dB/decade.35Phase plot:36break frequency 37Characteristic of Bode plot 1) Expand frequency range. 2) Easy to draw (asymptote) 3) Multiply (divide) can be transformed to plus (subtract). 384)the curve move up or down.(gain)5) the curve move left or right.(time constant)39

14、6) response inverse frequency plot is the plot mirror about the horizontal axis. 40 In comparison with the Nyiquist diagrams, the Bode plots contain additional information about the system in the frequency data. Knowing the Bode plot one can construct the corresponding Nyquist diagram, but the rever

15、se is not possible.41Logarithmic Magnitude-Phase Diagram (Nichols)Nichols chartx-coordinate: phase in degree, y-coordinate: amplitude (in logarithmic scale).Nichols chart is a graphical tool for obtaining closed-loop frequency response from open-loop response. It is a popular tool before computers a

16、re widely available.426.3 Frequency Response Function of Elementary Unit1. Proportion(constant gain) Unit43442. Integrator(poles at the origin)The slope of the magnitude curve is -20dB/dec. The phase angle is 90.4546In this case the slope due to the multiple poles is 20n dB/dec. The phase angle is 9

17、0n.473. Differentiator (a zero at the origin)The slope of the magnitude curve is +20dB/dec. The phase angle is + 90.48494. First-order Unit (a pole on the real axis)50 Approximately, the magnitude response5152The actual logarithmic gain at w=1/T is 3dB. Phase response: range from 0 to 90，center symm

18、etry about the point（1/T,45）.53 5. First-order Differential Unit (a zero on the real axis) The frequency response can be obtained by the reciprocal relationship (inverse frequency response)54556. Oscillations Unit (complex conjugate poles)56The slope is -40dB/dec. 5758 =0.1 =0.3 =0.5 =0.7 =1 =0.1 =0

19、.3 =0.5 =0.7 =159Resonant frequency:When the damping ratio approaches zero, them wr approaches wn.60And the maximum value of the magnitude is 61 7. Second-order Differential Unit (complex conjugate zeros) symmetry with respect to the frequency axis.62638. Non-minimum phase system A transfer function

20、 is called a minimum phase transfer function if all its poles and zeros lie in the left-hand s-plane. It is called a non-minimum phase transfer function if it has poles and/or zeros in the right-hand s-plane, and/or has delay unit.A closed loop system, whose open-loop transfer function has poles and

21、/or zeros in the right-hand s-plane, and/or has delay unit, is called Non-minimum phase system.The range of phase shift of a minimum phase transfer function is the least possible or minimum corresponding to a given amplitude curve.64(1). A unit of delay: Transfer function of a delay of Td seconds an

22、d the frequency response is65 A time delay added into the loop does not change the magnitude but increases the phase lag of the frequency response. The main effects of time delay are to reduce phase margin and reduce the closed-loop bandwidth.66 (2). Unstable unit676.4 Diagram of Complex Frequency R

23、esponse Function68There have four different kinds of factors:Constant gain KbPoles (or zeros) at the origin (jw)Poles (or zeros) on the real axis (jwt+1)Complex conjugate poles (or zeros)691. pose transfer function to the elementary unit.702. Calculate the break frequencies.3. Calculate the slope an

24、d gain of the low frequency(only consider the proportion and the integrators)714. Obtain the asymptotic magnitude plot for the complete transfer function.*5. Furthermore, obtain the actual curves only at specific important frequencies.726. Phase plot： (1). Write the phase angle equation. (2). Calcul

25、ate the range and estimate the change direction. (3). List the phase angle of some points (the every corner frequency point and wc） (4). Draw the phase angle curve.73Example 6.1: The Bode diagram of the TF:1.2. The break frequencies3. 744. Asymptotic magnitude plot slopes: 0.4, -40dB/dec 2, -20dB/de

26、c 10, -40dB/dec 20, -60dB/dec*5. revise.756. 767. Phase plotRange from 90 to 270.7720.410203678Bode plot can be readily obtained by Matlab:num=0 0 10;den=1 3 2 0;bode(num,den)79Example 6.3 Sketching a bode plotThe factors are1. A constant gain K=52. A pole at the origin 3. A pole at =24. A zero at =

27、105. A pair of complex poles at w=wn =5080Plot the magnitude characteristic:1. The constant gain is: 20lg5=14 dB;2. The magnitude of the pole has a slope of 20 dB/decade and is 0 dB at =1.3. Beyond the break frequency at = 2, the slope is - 20 dB/decade and below the break frequency 0 dB/decade.4. B

28、eyond the break frequency at = 10, the slope is +20 dB/decade and below the break frequency 0 dB/decade.5. At = n = 50, the slope of the magnitude for the pair of complex poles is - 40 dB/decade. Because the damping ratio is = 0.3, the approximation must be corrected to the actual magnitude. 8182By adding the asymptotes due to each factor, we obtain the total asymptotic magnitude as83