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: Where  is the damping ratio (  0) n is the natural frequency (n  0)

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

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

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

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

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

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

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

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

Part 1: find the equation governing vc(t)

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

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

Part 4: Find R to make the system critically damped

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

Simulated Response