Transfer Function and Response Analysis

This document provides an analysis of system response, focusing on transfer function, poles, overshoots, peak time, and damping ratio. It explores the relationship between damping ratio and overshoot, and peak time and natural frequency.

NAME:
NUMBER:

Q. 1. 1. 1.
Transfer Function (G) Poles Overshoots
(Mp) %
Peak Time (tp)
(Seconds)
Damping
Ratio (ζ)
933.6
---------------------
s^2 + 6.111 s + 933.6
-3.0555
+30.4018i
-3.0555 -30.4018i
72.9 0.103 0.1
933.6
---------------------
s^2 + 18.33 s + 933.6
-9.1665
+29.1476i
-9.1665 -29.1476i
37.1 0.106 0.3
933.6
---------------------
s^2 + 30.56 s + 933.6
-15.2775
+26.4614i
-15.2775 -
26.4614i
16.3 0.118 0.5
933.6
--------------------
s^2 + 43.2 s + 933.6
-21.6024
+21.6089i
-21.6024 -
21.6089i
4.33 0.145 0.707
933.6
---------------------
s^2 + 51.94 s + 933.6
-25.9717
+16.0958i
-25.9717 -
16.0958i
0.629 0.195 0.85
933.6
---------------------
s^2 + 61.11 s + 933.6
-30.5550 +
0.0000i
-30.5550 -
0.0000i
0 >3.5 1.0
933.6
---------------------
s^2 + 183.3 s + 933.6
-178.0876
-5.2424
0 >3.5 3.0
933.6
---------------------
s^2 + 366.7 s + 933.6
-364.0958
-2.5642
0 >3.5 6.0
Step Response: For constant Wn and varying Zt
Step Response
Time (seconds)
Amplitude
0 0.5 1 1.5 2 2.5 3 3.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
System: G
Peak amplitude: 1.73
Overshoot (%): 72.9
At time (seconds): 0.103
System: G
Peak amplitude: 1.37
Overshoot (%): 37.1
At time (seconds): 0.106
System: G
Peak amplitude: 1.16
Overshoot (%): 16.3
At time (seconds): 0.118
System: G
Peak amplitude: 1.04
Overshoot (%): 4.33
At time (seconds): 0.145
System: G
Peak amplitude: 1.01
Overshoot (%): 0.629
At time (seconds): 0.195
System: G
Peak amplitude: >= 1
Overshoot (%): 0
At time (seconds): > 3.5

Comment on Results:
The damping ratio of the second order system is varied to get the response curve. Therefore,
from the transfer function expression it can be seen that there is no change in the natural
frequency (Wn). In the above plotted figure we can see that the overshoot (Mp) percentage is
decreasing with increasing the damping ratio (ζ). For the ζ=0.1, 0.3, 0.5, 0.707 and 0.85 we
are getting some value of overshoot. This replies that the system is underdamped. For ζ=1.0,
the system becomes critically damped with having no overshoot. As ζ become 3.0 and 5.0
that is >1.0, the time oscillation represents no damped oscillation and such a response is
known as overdamped response.
Q. 1. 1. 2.
Plot of Overshoot (Mp) against Damping Ratio (ζ):
For ζ<1 (0.1, 0.3, 0.5, 0.707, 0.85), the response shows a damped oscillation with having
some overshoot percentage. For ζ ≥ 1.0 the response curve have zero overshoot percentage.
0 1 2 3 4 5 6
0
10
20
30
40
50
60
70
80
Plot of overshoot against damping ratio
Damping Ratio
O v e r s h o o t i n %
Plot of Peak time (tp) against Damping Ratio (ζ):
Damping ratio and peak time is directly proportional with each other. For ζ ≥ 1.0, the peak
time gets increased towards the settling time (ts) of the system.

0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
3
3.5
4
Plot of Peak time against damping ratio
Damping Ratio
P e a k t im e i n s e c
Q. 1. 1. 3.
ζ = ½ Qp , where Qp is the pole quality factor. A second order system with Qp = 0.707, will
exhibit no peak value at pole frequency. However, there will be a small overshoot in step
response. So the system will have an Underdamped response with low settling time.
Q. 1. 2. 1.
Transfer Function (G) Poles Overshoots
(Mp) %
Peak Time (tp)
(Seconds)
Natural
Frequency
(Wn)
0.25
-------------------
s^2 + 0.33 s + 0.25
-0.1650 + 0.4720i
-0.1650 -
0.4720i
33.3 0.672 0.5
1
----------------
s^2 + 0.66 s + 1
-0.3300 +
0.9440i
-0.3300 -
0.9440i
33.3 1.12 1.0
9
----------------
s^2 + 1.98 s + 9
-0.9900 +
2.8319i
-0.9900 -
2.8319i
33.3 3.35 3.0
25
----------------
s^2 + 3.3 s + 25
-1.6500 +
4.7199i
-1.6500 -
4.7199i
33.3 6.72 5.0

Step Response: For constant Zt and varying Wn
Step Response
Time (seconds)
Amplitude
0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
1.2
1.4
System: G
Peak amplitude: 1.33
Overshoot (%): 33.3
At time (seconds): 0.67
System: G
Peak amplitude: 1.33
Overshoot (%): 33.3
At time (seconds): 6.7
System: G
Peak amplitude: 1.33
Overshoot (%): 33.3
At time (seconds): 3.35
Comment on Results: Here damping ratio is constant at 0.330. This signifies the response
having a damping oscillation. In the above figure we can see that for all the values of Wn the
responses are Underdamped. As the ζ<1 i.e. constant the overshoot in all the responses are
fixed at 33.3%. The natural frequency controls the peak time rise time and the settling time of
the system. With the increments of Wn, the time response parameters (tp, ts, tr) and steady
state value increases. From the above figure it can be concluded that the first response curve
has a good steady state response compared to others.
Q. 1. 2. 2
Plot of Peak time against natural frequency: Peak response time increases with the
increasing value of natural frequency, keeping the damping ratio constant. This curve shows
almost a linear relationship of the peak time with the natural frequency.
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
1
2
3
4
5
6
7
Plot of Peak time against natural frequency
natural frequency in rad per sec
P e a k t i m e i n s e c