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
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:
yh ( t )  e
 n t


 y 0     2  1  n y0  t
n
e

2 n  2  1



 2 1
y 0     2  1  n y0
2 n   1
2
e

 n t  2 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:
yh ( t )  e  n t  y0 
 y 0  n y0 t 
• System has only a single
time constant
• Response dies out more
rapidly than any overdamped system
Underdamped system natural response
• <1:
yh ( t )  e
 n t


 y 0   n y0
2
sin

t
1




n
2
  n 1  

y0 cos nt 1   2

• 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:
1. The equation governing vc(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 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