Transcript Lecture 1 - Digilent Inc.
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 • •
n
is the damping ratio (
is the natural frequency ( 0)
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:
y h ( t )
e
n t
0
2
n
0
2 2
1
n y
0
1
e
n t
2
n
2 2
1 1
n y
0
2
1
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:
y h ( t )
e
n t
y
0
0
t
• System has only a single
time constant
• Response dies out more
rapidly than any over damped system
Underdamped system natural response
•
<1:
y h ( t )
e
n t
0
n
n
1
y
0 2
sin
n t
1
2
y
0
cos
n t
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 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
Part 1: find the equation governing v
c (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 v c (0 )=1V and i L (0 )=0.01A
Simulated Response