# Frequency Response Techniques

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1 4th Edition T E N Frequency Response Techniques SOLUTION TO CASE STUDY CHALLENGE Antenna Control: Stability Design and Transient Performance First find the forward transfer function, G(s). Pot: K 1 = 10 π = 3.18 Preamp: Power amp: Motor and load: G 1 (s) = K 100 s(s+100) Therefore, J = ( 1 5 )2 = 0.25 ; D = ( 1 5 )2 = 0.13; K t Ra = 1 5 ; K b = 1. G m (s) = θ m(s) E a (s) = K t R a J s(s+ 1 J (D + K tk b R a )) = 0.8 s(s+1.32). Gears: Therefore, Plotting the Bode plots for K = 1, K 2 = = 1 5 G(s) = K 1 KG 1 (s)g m (s)k 2 = 50.88K s(s+1.32)(s+100)

2 Solution to Case Study Challenge 409 a. Phase is 180 o at ω = 11.5 rad/s. At this frequency the gain is db, or K = Therefore, for stability, 0 < K < b. If K = 3, the magnitude curve will be 9.54 db higher and go through zero db at ω = 0.94 rad/s. At this frequency, the phase response is o. Thus, the phase margin is 180 o o = o. Using Eq. (10.73), ζ = Eq. (4.38) yields %OS = 14.18%. c. Program: numga=50.88; denga=poly([ ]); 'Ga(s)' Ga=tf(numga,denga); Gazpk=zpk(Ga) '(a)' bode(ga) title('bode Plot at Gain of 50.88') pause [Gm,Pm,Wcp,Wcg]=margin(Ga); 'Gain for Stability' Gm pause '(b)' numgb=50.88*3; dengb=denga; 'Gb(s)' Gb=tf(numgb,dengb); Gbzpk=zpk(Gb) bode(gb) title('bode Plot at Gain of 3*50.88') [Gm,Pm,Wcp,Wcg]=margin(Gb); 'Phase Margin' Pm for z=0:.01:1 Pme=atan(2*z/(sqrt(-2*z^2+sqrt(1+4*z^4))))*(180/pi); if Pm-Pme<=0; break end end z percent=exp(-z*pi/sqrt(1-z^2))*100 Computer response: ans = Ga(s)

3 410 Chapter 10: Frequency Response Methods Zero/pole/gain: s (s+100) (s+1.32) ans = (a) ans = Gain for Stability Gm = ans = (b) ans = Gb(s) Zero/pole/gain: s (s+100) (s+1.32) ans = Phase Margin Pm = z = percent =

4 Answers to Review Questions 411 ANSWERS TO REVIEW QUESTIONS 1. a. Transfer functions can be modeled easily from physical data; b. Steady-state error requirements can be considered easily along with the design for transient response; c. Settles ambiguities when sketching root locus; (d) Valuable tool for analysis and design of nonlinear systems. 2. A sinusoidal input is applied to a system. The sinusoidal output's magnitude and phase angle is measured in the steady-state. The ratio of the output magnitude divided by the input magnitude is the magnitude response at the applied frequency. The difference between the output phase angle and the input phase angle is

5 412 Chapter 10: Frequency Response Methods the phase response at the applied frequency. If the magnitude and phase response are plotted over a range of different frequencies, the result would be the frequency response for the system. 3. Separate magnitude and phase curves; polar plot 4. If the transfer function of the system is G(s), let s=jω. The resulting complex number's magnitude is the magnitude response, while the resulting complex number's angle is the phase response. 5. Bode plots are asymptotic approximations to the frequency response displayed as separate magnitude and phase plots, where the magnitude and frequency are plotted in db. 6. Negative 6 db/octave which is the same as 20 db/decade 7. Negative 24 db/octave or 80 db/decade 8. Negative 12 db/octave or 40 db/decade 9. Zero degrees until 0.2; a negative slope of 45 o /decade from a frequency of 0.2 until 20; a constant -90 o phase from a frequency of 20 until 10. Second-order systems require a correction near the natural frequency due to the peaking of the curve for different values of damping ratio. Without the correction the accuracy is in question. 11. Each pole yields a maximum difference of 3.01 db at the break frequency. Thus for a pole of multiplicity three, the difference would be 3x3.01 or 9.03 db at the break frequency, Z = P - N, where Z = # of closed-loop poles in the right-half plane, P = # of open-loop poles in the righthalf plane, and N = # of counter-clockwise encirclements of -1 made by the mapping. 13. Whether a system is stable or not since the Nyquist criterion tells us how many rhp the system has 14. A Nyquist diagram, typically, is a mapping, through a function, of a semicircle that encloses the right half plane. 15. Part of the Nyquist diagram is a polar frequency response plot since the mapping includes the positive jω axis. 16. The contour must bypass them with a small semicircle. 17. We need only map the positive imaginary axis and then determine that the gain is less than unity when the phase angle is 180 o. 18. We need only map the positive imaginary axis and then determine that the gain is greater than unity when the phase angle is 180 o. 19. The amount of additional open-loop gain, expressed in db and measured at 180 o of phase shift, required to make a closed-loop system unstable. 20. The phase margin is the amount of additional open-loop phase shift, Φ M, required at unity gain to make the closed-loop system unstable. 21. Transient response can be obtained from (1) the closed-loop frequency response peak, (2) phase margin 22. a. Find T(jω)=G(jω)/[1+G(jω)H(jω)] and plot in polar form or separate magnitude and phase plots. b. Superimpose G(jω)H(jω) over the M and N circles and plot. c. Superimpose G(jω)H(jω) over the Nichols chart and plot.

6 Solutions to Problems For Type zero: K p = low frequency gain; For Type 1: K v = frequency value at the intersection of the initial slope with the frequency axis; For Type 2: K a = square root of the frequency value at the intersection of the initial slope with the frequency axis. 24. No change at all 25. A straight line of negative slope, ωt, where T is the time delay 26. When the magnitude response is flat and the phase response is flat at 0 o. SOLUTIONS TO PROBLEMS 1. a. ; ; b. ; ; c. ; 2. a. ;

7 414 Chapter 10: Frequency Response Methods b. c. 3. a

8 Solutions to Problems 415 b c a

9 416 Chapter 10: Frequency Response Methods b. db c. Phase 0-40 db/dec db/dec -40 db/dec db/dec v db/dec a. System deg/dec +45 deg/dec v

10 Solutions to Problems 417 b. System 2 c. System 3 d.

11 422 Chapter 10: Frequency Response Methods 11. Stable if 0<K<1. Note: All results for this problem are based upon a non-asymptotic frequency response. System 1: Plotting Bode plots for K = 1 yields the following Bode plot, K = 1000: For K = 1, phase response is 180 o at ω = 6.63 rad/s. Magnitude response is db at this frequency. For K = 1000, magnitude curve is raised by 60 db yielding db at 6.63 rad/s. Thus, the gain margin is db.

12 Solutions to Problems 423 Phase margin: Raising the magnitude curve by 60 db yields 0 db at 9.07 rad/s, where the phase curve is o. Hence, the phase margin is 180 o o = o. K = 100: For K = 1, phase response is 180 o at ω = 6.63 rad/s. Magnitude response is db at this frequency. For K = 100, magnitude curve is raised by 40 db yielding 13.6 db at 6.63 rad/s. Thus, the gain margin is 13.6 db. Phase margin: Raising the magnitude curve by 40 db yields 0 db at 2.54 rad/s, where the phase curve is o. Hence, the phase margin is 180 o o = 72.7 o. K = 0.1: For K = 1, phase response is 180 o at ω = 6.63 rad/s. Magnitude response is db at this frequency. For K = 0.1, magnitude curve is lowered by 20 db yielding 73.6 db at 6.63 rad/s. Thus, the gain margin is 73.6 db.. System 2: Plotting Bode plots for K = 1 yields K = 1000: For K = 1, phase response is 180 o at ω = 1.56 rad/s. Magnitude response is db at this frequency. For K = 1000, magnitude curve is raised by 60 db yielding db at 1.56 rad/s. Thus, the gain

13 424 Chapter 10: Frequency Response Methods margin is db. Phase margin: Raising the magnitude curve by 54 db yields 0 db at 500 rad/s, where the phase curve is o. Hence, the phase margin is 180 o o = o. K = 100: For K = 1, phase response is 180 o at ω = 1.56 rad/s. Magnitude response is db at this frequency. For K = 100, magnitude curve is raised by 40 db yielding db at 1.56 rad/s. Thus, the gain margin is db. Phase margin: Raising the magnitude curve by 40 db yields 0 db at 99.8 rad/s, where the phase curve is o. Hence, the phase margin is 180 o o = 95.7 o. K = 0.1: For K = 1, phase response is 180 o at ω = 1.56 rad/s. Magnitude response is db at this frequency. For K = 0.1, magnitude curve is lowered by 20 db yielding db at 1.56 rad/s. Thus, the gain margin is db. Phase margin: Lowering the magnitude curve by 20 db yields 0 db at rad/s, where the phase curve is o. Hence, the phase margin is 180 o o = 80.2 o. System 3: Plotting Bode plots for K = 1 yields

14 Solutions to Problems 425 K = 1000: For K = 1, phase response is 180 o at ω = 1.41 rad/s. Magnitude response is 0 db at this frequency. For K = 1000, magnitude curve is raised by 60 db yielding 60 db at 1.41 rad/s. Thus, the gain margin is - 60 db. Phase margin: Raising the magnitude curve by 60 db yields no frequency where the magnitude curve is 0 db. Hence, the phase margin is infinite. K = 100: For K = 1, phase response is 180 o at ω = 1.41 rad/s. Magnitude response is 0 db at this frequency. For K = 100, magnitude curve is raised by 40 db yielding 40 db at 1.41 rad/s. Thus, the gain margin is - 40 db. Phase margin: Raising the magnitude curve by 40 db yields no frequency where the magnitude curve is 0 db. Hence, the phase margin is infinite. K = 0.1: For K = 1, phase response is 180 o at ω = 1.41 rad/s. Magnitude response is 0 db at this frequency. For K = 0.1, magnitude curve is lowered by 20 db yielding -20 db at 1.41 rad/s. Thus, the gain margin is 20 db. Phase margin: Lowering the magnitude curve by 20 db yields no frequency where the magnitude curve is 0 db. Hence, the phase margin is infinite.

15 446 Chapter 10: Frequency Response Methods 27. The phase margin of the given system is 20 o. Using Eq. (10.73), ζ = Eq. (4.38) yields 57% overshoot. The system is Type 1 since the initial slope is - 20 db/dec. Continuing the initial slope down to the 0 db line yields K v = 4. Thus, steady-state error for a unit step input is zero; steady state error for a unit ramp input is 1 K v = 0.25; steady-state error for a parabolic input is infinite. 28. The magnitude response is the same for all time delays and crosses zero db at 0.5 rad/s. The following is a plot of the magnitude and phase responses for the given time delays:

16 Solutions to Problems 447 a. For T = 0, Φ M = 93.3 o ; System is stable. For T = 0.1, Φ M = 55.1 o ; System is stable.

17 448 Chapter 10: Frequency Response Methods For T = 0.2, Φ M = 17 o ; System is stable. For T = 0.5, Φ M = -97 o ; System is unstable.

18 Solutions to Problems 449 For T = 1, Φ M = 72.2 o ; System is unstable because the gain margin is db. b. For T = 0, the phase response reaches 180 o at infinite frequency. Therefore the gain margin is infinite. The system is stable. For T = 0.1, the phase response is -180 o at 11.4 rad/s. The magnitude response is db at 11.4 rad/s. Therefore, the gain margin is 5.48 db. The system is stable. For T = 0.2, the phase response is -180 o at 7.55 rad/s. The magnitude response is db at 7.55 rad/s. Therefore, the gain margin is 1.09 db and the system is stable. For T =.5, the phase response is -180 o at 4.12 rad/s. The magnitude response is db at 4.12 rad/s. Therefore, the gain margin is 3.09 db and the system is unstable. For T = 1, the phase response is -180 o at 2.45 rad/s. The magnitude response is db at 2.45 rad/s. Therefore, the gain margin is db and the system is unstable. c. T = 0; T = 0.1; T = d. T = 0.5, db; T = 1, db; The Bode plots for K = 1 and 0.5 second delay is:

19 Solutions to Problems Resonance at 70 rad/s. 37. G(s) = 10 s(s+2)(s+10). Plotting the Bode plots, The gain is zero db at rad/s and the phase angle is Thus, the phase margin is 180 o o = o. Using Eq. (10.73), ζ = 0.9. Using Eq. (4.38), %OS = 0.15%. G(s) = 22.5 (s+4)(s s+9). Plotting the Bode plots, The phase response is 180 o at ω = 3.55 rad/s, where the gain is db. Thus, the gain margin is 1.17 db. Unity gain is at ω = rad/s, where the phase is o and at ω = rad/s, where

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