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