Presentation on theme: "Frequency Response Techniques"— Presentation transcript:
1Frequency Response Techniques Chapter 10Frequency Response Techniques<<<4.1>>>###Continuous Time Signals###
2Sinusoidal frequency response: a. system; b. transfer function; c Sinusoidal frequency response: a. system; b. transfer function; c. input and output waveformsThe steady-state output sinusoid isThe system function is given by
4Frequency response plots for G(s) = 1/(s + 2): separate magnitude and phase Problem: Find the analytical expression for the magnitude and phase frequency response for a system G(s)=1/(s+2). Plot magnitude, phase and polar diagrams.Solution: substituting s=jw, we getThe magnitude frequency response isThe phase frequency response isThe magnitude diagram isAnd the phase diagram is
5Frequency response plots for G(s)= 1/(s + 2): polar plot The polar plot is a plot ofFor different values of w
6Asymptotic Approximations: Bode plots Consider the transfer functionThe magnitude frequency response isConverting the magnitude response into dB, we obtainWe could make an approximation of each term that would consists only of straight lines, then combine these approximations to yield the total response in dB
7a. magnitude plot; b. phase plot. Asymptotic Approximations: Bode plotsa. magnitude plot; b. phase plot.The straight-line approximations are called asymptotes.We have low frequency asymptoteand high frequency asymptotea is called the break frequencyBode plots for G(s) = (s + a):At low frequency when ω approaches zeroThe magnitude response in dB is20log M=20log a whereand is constant and plotted in Figure (a) from ω =0.01a to a.At high frequencies where ω ˃˃ a
8Table 10.1 Asymptotic and actual normalized and scaled frequency response data for (s + a)
9Figure 10.7 Asymptotic and actual normalized and scaled magnitude response of (s + a)
10Figure 10.8 Asymptotic and actual normalized and scaled phase response of (s + a)
11Normalized and scaled Bode plots for a. G(s) = s; b. G(s) = 1/s;
12Normalized and scaled Bode plots for c. G(s) = (s+a); d Normalized and scaled Bode plots for c. G(s) = (s+a); d. G(s) = 1/(s+a);
13Closed-loop unity feedback system Example Problem : Draw the Bode plot for the system shown in figure whereSolution: The Bode plot is the sum of the Bode plot for each first order term. To make it easeier we rewrite G(s) as
14Bode log-magnitude plot for Example: a. components; b. composite
15Table 10.2 Bode magnitude plot: slop contribution from each pole and zero in Example
16Bode phase plot for Example: a. components; b. composite
17Table 10.3 Bode phase plot: slop contribution from each pole and zero in Example 10.2
18Bode asymptotes for normalized and scaled G(s) = At low frequencies G(s) becomesThe magnitude M, in dB is therefore,At high frequenciesAndThe log-magnitude isThe log-magnitude equation is a straight line with slope 40dB/decadeThe low freq. asymptote and the high freq. asymptote are equal whenThus is called the break frequency for second order polynomial.
19Bode asymptotes for normalized and scaled G(s) = a. magnitude; b. phase
20Table Data for normalized and scaled log-magnitude and phase plots for (s2 + 2zwns + wn2). Mag = 20 log(M/wn2) (table continues)Freq. w/wnMag (dB) z = 0.1Phase (deg) z = 0.1Mag (dB) z = 0.2Phase (deg) z = 0.2Mag (dB) z = 0.3Phase (deg) z = 0.3
21Table 10.4 (continued) Data for normalized and scaled log-magnitude and phase plots for (s2 + 2zwns + wn2). Mag = 20 log(M/wn2)Freq. w/wnMag (dB) z = 0.5Phase (deg) z = 0.5Mag (dB) z = 0.7Phase (deg) z = 0.7Mag (dB) z = 1Phase (deg) z = 1
22Normalized and scaled log-magnitude response for
24Bode Plots for G(s) = 1/Bode plots for can be derived similarlyto those forWe find the magnitude curve breaks at the natural frequency and decreases at a rate of -40dB/decade.The phase plot is 0 at low frequencies. At begins a decrease of -90o/decade and continues until , where it levels off at -180o .
25Freq. w/wnMag (dB) z = 0.1Phase (deg) z = 0.1Mag (dB) z = 0.2Phase (deg) z = 0.2Mag (dB) z = 0.3Phase (deg) z = 0.3Table Data for normalized and scaled log-magnitude and phase plots for 1/(s2 + 2zwns + wn2). Mag = 20 log(M/wn2) (table continues)
26Freq. w/wnMag (dB) z = 0.5Phase (deg) z = 0.5Mag (dB) z = 0.7Phase (deg) z = 0.7Mag (dB) z = 1Phase (deg) z = 1Table (continued) Data for normalized and scaled log-magnitude and phase plots for 1/(s2 + 2zwns + wn2). Mag = 20 log(M/wn2)
27Normalized and scaled log magnitude response for
29Closed-loop unity feedback system Example Problem : Draw the Bode plot for the system shown in figure whereSolution: The Bode plot is the sum of the Bode plot for each first order term, and the 2nd order term. Convert G(s) to show the normalized components that have unity low-frequency gain. The 2nd order term is normalized by factoring out formingThus,
30Bode magnitude plot for G(s) =(s + 3)/[(s + 2)(s2 + 2s + 25)]: a Bode magnitude plot for G(s) =(s + 3)/[(s + 2)(s2 + 2s + 25)]: a. components; b. compositeThe low frequency value for G(s), found by letting s = 0, is 3/50 or dBThe Bode magnitude plot starts out at this value and continues until the first break freq at 2 rad/s.
34Introduction to the Nyquist criterion The Nyquist criterion relates the stability of a closed-loop system to the open-loop frequency response and open-loop pole location.We conclude thatThe poles of 1+G(s)H(s) are the same as the poles of G(s)H(s), the open-loop systemThe zeroes of 1+G(s)H(s) are the same as the poles of T(s), the closed-loop system
35Mapping Contours concept If we take a complex number on the s-plane and substitute it into a function, F(s), another complex number results. This process is called Mapping.Consider the collection of points, called a contour (shown in Figure as Contour A). Contour A can be mapped through F(s) Into Contour B by substituting each point in contour A into Function F(s) and plotting the resulting complex numbers.
36Examples of Contour Mapping If contour A maps clockwise then contour B maps clockwise if:F(s) has just Zeros or Poles that are not encircled by the contourThe contour B maps counterclockwise if:F(s) has just Poles that are encircled by contour A
37Examples of Contour Mapping If the Pole or Zero of F(s) is enclosed by contour A,then the mapping encircles the origin
39The Nyquist Contouris a contour which contains the imaginary axis and encloses the right half-place. The Nyquist contour is clockwise.A Clockwise CurveStarts at the origin.Travels along imaginary axis till r = ∞.At r = ∞, loops around clockwise.Returns to the origin along imaginary axis.Nyquist Contour
40Sketching a Nyquist diagram a. Turbine and generator; b. block diagram of speed control system for Example 10.4
41Vector evaluation of the Nyquist diagram for Example 10. 4 a Vector evaluation of the Nyquist diagram for Example a. vectors on contour at low frequency; b. vectors on contour around infinity; c. Nyquist diagramThe Nyquist Diagram is plotted by substituting the points of the contour shown in Figure (a) into G(s) = 500/[(s+1)(s+3)(s+10)]It’s performing complex arithmetic using vectors of G(s) drawn to points of the contour as shown in (a) and (b)The resultant vector, R, found at any point is the product of zero vectors divided by product of pole vectors as in (c)
42Vector evaluation of the Nyquist diagram for Example 10. 4 a Vector evaluation of the Nyquist diagram for Example a. vectors on contour at low frequency; b. vectors on contour around infinity; c. Nyquist diagramEvaluating G(s) at specified points we getSo from A to C the magnitude is finite goes from 50/3 to 0 and the angle goes from 0 to -270oFrom C to D the magnitude is 0 while there is an angle change of 540o so goes from -270o to 270o
43Vector evaluation of the Nyquist diagram for Example 10.4 The mapping can be explained analytically by Substituting s= jω and evaluating G(jω) at specified pointsMultiplying numerator and denominator by the complex conjugate of denominator, we obtainAt 0 freq G(jω)=500/30= 50/3. Thus Nyquist diagram starts at 50/3 at an angle 0.As ω increases, the real part remains positive while imaginary part remains negative. At ω = √30/14, the real part becomes negative. At ω=√43 the Nyquist diagram crosses the negative real axis, the value at the crossing is Continuing toward ω = ∞, the real part is negative and imaginary is positive. At ω = ∞, G(jω)≈500j/ω3 or approximately 0 at 90o
45Detouring around open-loop poles: a. poles on contour; b Detouring around open-loop poles: a. poles on contour; b. detour right; c. detour left
46a. Contour for Example 10.5; b. Nyquist diagram for Example 10.5 Nyquist Diagram for open-loop function with poles on contourProblem: Sketch the Nyquist diagram of unity feedback system where G(s)= (s+2)/ S2Solution: The system two poles at origin are on the contour and must be by-passed as in (a).Mapping DE is mirror of ABa. Contour for Example 10.5; b. Nyquist diagram for Example 10.5
47Applying the Nyquist Criterion to determine stability a Applying the Nyquist Criterion to determine stability a. contour does not enclose closed-loop poles; b. contour does enclose closed-loop polesIf a contour A, that encircles the entire right half-plane is mapped through G(s)H(s), then the number of closed loop poles, Z, in the right half-plane equals the number of open-loop poles, P, that are in the right half-plane minus the number of counterclockwise revolutions, N, around -1 of the mapping; that is Z=P-N.
48Demonstrating Nyquist stability: a. system; b. contour; c Demonstrating Nyquist stability: a. system; b. contour; c. Nyquist diagramWe could set K=1 and position the critical point at -1/K rather than -1. Thus the critical point appears to move closer to the origin as K increases.For this example P=2, and the critical point must be encircled by the Nyquist diagram to yield N=2 and a stable system.The system is marginally stable if the Nyquist diagram intersect the real axis at -1 ( frequency at this point is the frequency at which the root locus crosses the jw axis.)
49a. Contour for Example 10.6; b. Nyquist diagram Range of gain for stability via Nyquist criterionProblem: for the unity feedback system with G(s) = K/[s(s+3)(s+5)], find the range for stability, instability and the value of gain for marginal stability.Solution: first set K=1 and sketch Nyquist diagram for the system. Using the contour shown in (a), for all points on the imaginary axis,a. Contour for Example 10.6; b. Nyquist diagramNext find the point where the diagram intersects the negative real axis by setting the imaginary part equal to zero. We find and real part = FinallyFrom the contour, P=0; for stability N must equal zero. So the system is stable if the critical point lies outside the contour. K can be increased by 1/ =120.5 before the diagram encircles -1.Hence for stability K< for marginal stability K=120.5 At this gain the frequency is
50The system is stable at low values of gain, unstable at high values. Stability via Mapping Only the Positive jω-axisThe system is stable at low values of gain, unstable at high values.This system is stable for the range of loop gain, K, that ensures that the open-loop magnitude is less than unity at that frequency where the phase angle is 180o.a. Contour and root locus of system that is stable for small gain and unstable for large gain; b. Nyquist diagram
51The system is stable at high values of gain, unstable at low values. Stability via Mapping Only the Positive jω-axisThe system is stable at high values of gain, unstable at low values.This system is stable for the range of loop gain, K, that ensures that the open-loop magnitude is greater than unity at that frequency where the phase angle is 180o.a. Contour and root locus of system that is unstable for small gain and stable for large gain; b. Nyquist diagram
52Stability Design via mapping positive jw-axis Example Problem: for the unity feedback system with G(s) = K/[(s2+2s+2)(s+2)], find the range for stability, instability and the value of gain for marginal stability.Solution: since P=0, we want no encirclements of -1 for stability. Hence. A gain less than unity at +/- 180 is required. first set K=1 and sketch the portion of Nyquist diagram along the positive imaginary axis as shown in (a). We find the intersection with the negative real axis asSetting the imaginary part = zero, we find and substituting yields real part –(1/20) =This closed loop system is stable if the magnitude of the frequency response is less than unity at 180o. Hence, the system is stable foe K<20, unstable for K>20, and marginally stable for K=20 (frequency at marginal stability is )
53Nyquist diagram showing gain and phase margins Gain margin, GM, The gain margin is the change in open-loop gain, expressed in decibels (dB), required at 180o of phase shift to make the closed-loop system unstable.Phase margin, , The phase margin is the change in open-loop phase shift required at unity gain to make the closed-loop system unstable.GM = 20log a.
54Finding gain and phase margins Problem: find the gain and phase margins of system of previous example. If K = 6.Solution: to find the gain margin, first find the frequency where the Nyquist diagram crosses the negative real axis.The Nyquist diagram crosses the real axis at frequency of The real part is calculated to be Thus, the gain can be increased by (1/0.3) =3.33 before the real part becomes -1. Hence the gain margin is GM = 20log 3.33 = dB.To find the phase margin, find the frequency for which the magnitude is unity. We need to solve the equation to find frequency = rad/s. at this freq. the phase angle is o. The difference between this angle and -180o is 67.7o, which is the phase margin.
55Range of gain for stability via Bode plots Problem: use Bode plots to determine range of K within which the unity feed-back system with G(s) = K/[(s+2)(s+4)(s+5)] is stable.Solution: The closed loop system will be stable if the frequency response has a gain less than unity when the phase is +/-180oWe sketch the Bode plots as shown. The low-frequency gain of G(s)H(s) is found by setting s=0. So the Bode magnitude plot starts at K/40. For convenience let K = 40 so that the log-magnitude plot starts at 0 dB.At frequency 7rad/s, when phase is 180o, the magnitude is -20dB. So an increase in gain of +20dB is possible before system becomes unstable.Since the gain plot was scaled for a gain of 40, +20dB(a gain of 10) represents the required increase in gain above 40. Hence, the gain for instability is 40X10 = <K<400 for stability. Compare to actual results 378 at freq rad/s.
57Gain and Phase Margins from Bode plots Problem: If K =200 in the system of previous example, find the gain and phase marginsSolution: The Bode plot in Figure is scaled to a gain of 40. If K=200, the magnitude plot would be 20log5 = dB higher.At phase = 180, on the magnitude plot the gain is -20. So the gain margin = = 6.02 dBTo find phase margin, we look on the magnitude plot for freq where the gain is 0 dB ( dB normalized) occurs at 5.5 rad/s freq. At this freq the angle is 165 so the phase margin is -165-(-180) = 15 degrees
58Using the open loop transfer function Relation between Closed-Loop Transient and Closed-Loop Frequency ResponsesUsing the open loop transfer functionAnd closed-loop transfer functionWe evaluate the magnitude of the closed-loop freq response asA representative sketch of the log plot is shown in the Figure
59To relate the peak magnitude to the damping ratio we find Relation between Closed-Loop Transient and Closed-Loop Frequency ResponsesTo relate the peak magnitude to the damping ratio we findAt a frequency, ofSince damping ratio is related to percent overshoot we can plot MP vs. percent overshoot
60Response Speed and Closed-Loop Frequency Response We can relate the speed of the time response ( as measured in settling time, rising time, and peak time) and the bandwidth , of the Frequency response.The Bandwidth is defined as the frequency at which the magnitude response curve is 3 dB down from its value at zero frequency.The Bandwidth for a two-pole system can be found by finding that freq for M=1/2 (that is -3dB)
61We relate to settling time and peak time by substituting And We get, Response Speed and Closed-Loop Frequency ResponseWe relate to settling time and peak time by substitutingAnd We get,Figure Normalized bandwidth vs. damping ratio for: a. settling time; b. peak time; c. rise time
62Figure 10.42 Constant M circles Relation between Closed-Loop and Open-Loop Frequency ResponsesThese circles are called constant M circles and are the locus of the closed-loop magnitude frequency response for unity feed back systems.If the polar frequency response of an open-loop function, G(s), is plotted and superimposed on top of the constant M circles, the closed-loop magnitude frequency response is determined by each intersection of this polar plot with the M circles.Figure Constant M circles
63Figure 10.43 Constant N circles Relation between Closed-Loop and Open-Loop Frequency ResponsesThese circles are called constant N circlesIf the polar frequency response of an open-loop function, G(s), is plotted and superimposed on top of the constant N circles, the closed-loop phase response is determined by each intersection of this polar plot with the N circles.Figure Constant N circles
64Nyquist diagram for Example 10.11 and constant M and N circles Closed-Loop frequency response from Open-Loop frequency responseProblem Find the closed-loop frequency response of the unity feedback system using G(s)=50/[s(s+3)(s+6)]SolutionEvaluate the open-loop frequency functionThe Polar plot of the open-loop freq response ( Nyquist diagram) is shown superimposed over the M and N circles in the FigureNyquist diagram for Example and constant M and N circles
65Closed-loop frequency response for Example 10.11 Closed-Loop frequency response from Open-Loop frequency responseThe closed-loop magnitude frequency response can now be obtained by finding the intersection of each point of the Nyquist plot with the M circles.While the closed-loop phase response can be obtained by finding the intersection of each point of the Nyquist plot with the N circles.The result is shown in the Figure.Closed-loop frequency response for Example 10.11
66Nichols ChartsNichols Charts displays the constant M and N circles in dB, so that changes in gain are as simple to handle as in the Bode plots.The chart is a plot of the open-loop magnitude in dB vs. open-loop phase angle in degrees.
67Nichols ChartsNichols chart with frequency response for G(s) = K/[s(s + 1)(s + 2)] superimposed. Values for K = 1 and K = 3.16 are shown.