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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:
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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=1F • 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 3: Is the system under-, over-, or critically damped? • In part 2, we found that = 0.2