Second order system response.
Im(s)
Underdamped
Overdamped or Critically damped
Underdamped
Undamped
Unstable
Re(s)
Overdamped system response
System transfer function :
Impulse response :
Step response :
Overdamped and critically damped system response.
Overdamped and critically damped system response.
Overdamped
Overdamped and critically damped system response.
Overdamped
Overdamped and critically damped system response.
Critically damped
Polar vs. Cartesian representations.
Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped
Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped
Polar vs. Cartesian representations.
System transfer function :
All 4 cases
Unless overdamped
… Cartesian overdamped
Significance of the damping ratio :
… Overdamped
… Critically damped
… Underdamped
… Undamped
Overdamped case:
Second order impulse response – Underdamped and Undamped
Impulse response :
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
-0.4
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
-0.4
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
-0.4
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
1
0.8
Amplitude
0.6
0.4
0.2
0
-0.2
-0.4
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
0.1
0.08
0.06
0.04
Amplitude
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
0.1
0.08
0.06
0.04
Amplitude
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
0.1
0.08
0.06
0.04
Amplitude
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
0.1
0.08
0.06
0.04
Amplitude
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.5
1
1.5
Time (sec)
2
2.5
3
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
6
5
Amplitude
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
10
5
0
-5
-10
-6
-4
-2
0
2
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
6
5
Amplitude
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
10
5
0
-5
-10
-6
-4
-2
0
2
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
6
5
Amplitude
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
10
5
0
-5
-10
-6
-4
-2
0
2
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
6
5
Amplitude
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
10
5
0
-5
-10
-6
-4
-2
0
2
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
5
4
3
Amplitude
2
1
0
-1
-2
-3
-4
0
2
4
6
Time (sec)
8
10
12
6
4
2
0
-2
-4
-6
-5
0
5
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
5
4
3
Amplitude
2
1
0
-1
-2
-3
-4
0
2
4
6
Time (sec)
8
10
12
6
4
2
0
-2
-4
-6
-5
0
5
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
5
4
3
Amplitude
2
1
0
-1
-2
-3
-4
0
2
4
6
Time (sec)
8
10
12
6
4
2
0
-2
-4
-6
-5
0
5
Second order impulse response – Underdamped and Undamped
Increasing
/ Fixed
Impulse Response
5
4
3
Amplitude
2
1
0
-1
-2
-3
-4
0
2
4
6
Time (sec)
8
10
12
6
4
2
0
-2
-4
-6
-5
0
5
Second order step response – Underdamped and Undamped
3
3
3
3
2
2
2
2
1
1
1
1
0
0
0
0
output
−10+j5
zeta=0.8944
−1
−2
0
time sec.
−2+j5
zeta=0.3714
−1
5
3
−2
0
time sec.
+j5
zeta=0
−1
5
−2
0
time sec.
−1
5
−2
0.2+j5
0
time sec.
5
3
40
2
2
1
1
0
0
1.5
output
30
1
−10+j0.5
zeta=0.998
−1
−2
−2+j0.5
zeta=0.97
−1
0
5
time sec.
−2
0
time sec.
0
10
+j0.5
zeta=0
0.5
5
0.2+j0.5
20
0
−10
0
10
time sec.
20
0
10
time sec.
20
Second order impulse response – Underdamped and Undamped
Higher frequency oscillations
Faster response
Slower response
Unstable
Lower frequency oscillations
Second order impulse response – Underdamped and Undamped
Less damping
Unstable
More damping
Second order step response – Time specifications.
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
Second order step response – Time specifications.
…
…
…
…
…
Steady state value.
Time to reach first peak (undamped or underdamped only).
% of
in excess of
.
Time to reach and stay within 2% of
.
Time to rise from 10% to 90% of
.
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
Second order step response – Time specifications.
… Steady state value.
More generally, if the numerator is not
, but some
:
Second order step response – Time specifications.
… Peak time.
Therefore,
is the time of the occurrence of the first peak
: