1 / 17

Lecture 22

Lecture 22. Second order system natural response Review Mathematical form of solutions Qualitative interpretation Second order system step response Related educational materials: Chapter 8.4. Second order input-output equations. Governing equation for a second order unforced system:

Download Presentation

Lecture 22

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. Lecture 22 Second order system natural response Review Mathematical form of solutions Qualitative interpretation Second order system step response Related educational materials: Chapter 8.4

  2. Second order input-output equations • Governing equation for a second order unforced system: • Where •  is the damping ratio (  0) • n is the natural frequency (n  0)

  3. Homogeneous solution – continued • Solution is of the form: • With two initial conditions: ,

  4. Damping ratio and natural frequency • System is often classified by its damping ratio, : •  > 1  System is overdamped (the response has two time constants, may decay slowly if  is large) •  = 1  System is critically damped (the response has a single time constant; decays “faster” than any overdamped response) •  < 1  System is underdamped (the response oscillates) • Underdamped system responses oscillate

  5. Overdamped system natural response • >1: • We are more interested in qualitative behavior than mathematical expression

  6. Overdamped system – qualitative response • The response contains two decaying exponentials with different time constants • For high , the response decays very slowly • As  increases, the response dies out more rapidly

  7. Critically damped system natural response • =1: • System has only a single time constant • Response dies out more rapidly than any over-damped system

  8. Underdamped system natural response • <1: • Note: solution contains sinusoids with frequency d

  9. Underdamped system – qualitative response • The response contains exponentially decaying sinusoids • Decreasing  increases the amount of overshoot in the solution

  10. Example • For the circuit shown, find: • The equation governing vc(t) • n, d, and  if L=1H, R=200, and C=1F • Whether the system is under, over, or critically damped • R to make  = 1 • Initial conditions if vc(0-)=1V and iL(0-)=0.01A

  11. Part 1: find the equation governing vc(t)

  12. Part 2: find n, d, and  if L=1H, R=200 and C=1F

  13. Part 3: Is the system under-, over-, or critically damped? • In part 2, we found that  = 0.2

  14. Part 4: Find R to make the system critically damped

  15. Part 5: Initial conditions if vc(0-)=1V and iL(0-)=0.01A

  16. Simulated Response

More Related