1 / 34

10. Frequency Response Analysis (I)

10. Frequency Response Analysis (I). Yonsei University Mechanical Engineering Prof. Hyun Seok YANG. * Frequency Response is the steady state response of a (linear) system to a sinusoidal input. (Linear) System. Dynamic behavior of the system in frequency domain - Bandwidth

afi
Download Presentation

10. Frequency Response Analysis (I)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 10. Frequency Response Analysis (I) Yonsei University Mechanical Engineering Prof. Hyun Seok YANG

  2. * Frequency Response is the steady state response of a (linear) system to a sinusoidal input. (Linear) System Dynamic behavior of the system in frequency domain - Bandwidth - System behavior to noise and disturbance - Relative stability c.f > Root Locus Analysis - Transient response (rise time, settling time, %OS, etc.)

  3. : Magnitude Ratio : Phase Angle Output Frequency = Input Frequency : Sinusoidal Transfer Function

  4. ? [Example]

  5. (1) Logarithmic Plot (Bode Plot) * Magnitude v.s. * Phase Angle v.s. (2) Polar Plot (Nyquist Plot) (3) Log-Mag v.s. Phase Plot (Nichols Chart) * Graphical Representation of Frequency Response Magnitude & Phase

  6. [dB : Decibel] The magnitudes & phase angles of the system are the sum of those of each sub-components, respectively. Therefore, in order to get the total mag. & angle, all we need is just to add them up. BODE PLOT <e.g.>

  7. For K>0 (a)

  8. [Differentiator] - 20 dB/dec + 20 dB/dec (b) [(Free) Integrator]

  9. Break Freq. - 20 dB/dec - 20 dB/dec (c)

  10. Figure 10.8 Asymptotic and actual normalized and scaled response of (s + a)

  11. Figure 10.9 Normalized and scaled Bode plots for a. G(s) = s; b. G(s) = 1/s; c. G(s) = (s + a); d. G(s) = 1/(s + a)

  12. Figure 10.16 Normalized and scaled response for

  13. Figure 10.14 Normalized and scaled response for

  14. - 20 dB/dec - 60 dB/dec [Example]

  15. 1 2 4 3 [Example]

  16. [Note] At high frequency : * For minimum phase system (i.e. no poles or zeros in RHP) - Slope of Mag. Plot @ = - 20 dB/dec x (# poles - # zeros) - Phase Angle = - 900 x (# poles - # zeros) * For non-minimum phase system (i.e. at least one poles or zeros in RHP) - Slope of Mag. Plot @ = - 20 dB/dec x (# poles - # zeros) - Phase Angle = - 900 x (# poles - # zeros)

  17. * All three have the same mag. plot. * All three have different phase angle plots. [Non-minimum phase system Bode Plot] [Example]

  18. [System Identification using Bode Plot] Unknown Linear Sys 1) Apply sinusoid input for wide range of input frequencies and mark mag. ratios & phase angles on the Bode plot. 2) Draw an asymptotic line based on the marks. 3) Find break frequencies. 4) Determine whether the sytem is min or non-min phase, base on the slope of the mag. ratio & phase angle. 5) Find the DC Gain. [Note] There are many H/W & S/W tools to do the above procedures. (e.g. Dynamics Signal Analyzer, FFT Analyzer)

  19. Extended line of - 20 dB/dec - 60 dB/dec - 40 dB/dec - 40 dB/dec

  20. [Example] Identification of VCM Actuator Dynamics Experimental Setup for VCM FRF(Bode Plot) Position FRF result at slider

  21. * 1st Order Low Pass Filter <e.g>

  22. [Note] [original signal] [noise] [signal with noise]

  23. [the original signal after noise is eliminated.] [signal with noise] • Above from which freq will be filtered out should be determined • carefully. [The choice of ] • * Phase Delay problem should also be considered.

  24. * Polar Plot of a sinusoidal transfer function - Locus of vectors in complex plane for [Example] NYQUIST PLOT

  25. + + [Nyquist Stability Criterion] Recall * Characteristic Equation

  26. which lie inside some closed contour in S-plane. [i.e. Contour does not pass through any poles or zeros of F(s).] S-plane Im F-plane Mapping Re Mapping Theorem [Note] C.W direction : + Let N be the total # of C.W. encirclements of the origin of F-plane Then, z = N + p

  27. S-plane Im F-plane Nyquist Path Mapping Re # of Unstable C.L. Poles # of Unstable O.L. Poles * Nyquist Path Then, since z = N + p

  28. F-plane -1 Nyquist Plot of O.L.T.F. [Note] Im Im GH -plane F-plane semi-circle contour with infinite radius maps into a point in GH (or F)-plane. Mapping Re Re S-plane a point Only need to examine the contour on jw-axis By the way, 0

  29. Nyquist Stability Criterion For a C.L. system to be stable, “z” should be zero. “z = N + p” means (from Nyquist plot of O.L.T.F.) z = # of Unstable C.L. Poles N = # of C.W. Encirclements of (-1+0j) Point of Nyquist Plot of O.L.T.F. p = # of Unstable O.L. Poles [Note] When the locus of GH passes through (-1+0j) point, then the C.L. poses are located on the jw-axis.

  30. [Example] S-plane 3 4 1 1 2 [Note] When G(s)H(s) has poles or zeros on the jw-axis, modify the Nyquist path not to pass through those poles or zeros.

  31. 3 2

  32. Im Symmetric on the Re-axis with Re 4 4 4 1 [Nyquist Plot] 3 2 2

  33. <K small> Im <K large> Im Re Re -1 -1 z = 2 Untable z = 0 Stable <c.f.> Root Locus [Example] Nyquist Stability Criterion p = 0, N = 2 p = 0, N = 0

More Related