1. Lecture 22 Second order system natural response Review Mathematical form of solutions Qualitative interpretation Second order system step response Related educational modules: –Section 2.5.4, 2.5.5

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: 1.The equation governing v c (t) 2.  n,  d, and  if L=1H, R=200 , and C=1  F 3.Whether the system is under, over, or critically damped 4.R to make  = 1 5.Initial conditions if v c (0 - )=1V and i L (0 - )=0.01A

11. Part 1: find the equation governing v c (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 v c (0 - )=1V and i L (0 - )=0.01A

16. Simulated Response