frequency respose of control system T.F.

find the frequency respose of a closed loop transfer function

1. SPC318: System Modeling and Linear Systems
Lecture 10: Frequency Response Methods
Dr. Haitham El-Hussieny
Adjunct Lecturer
Space and Communication Engineering
Zewail City of Science and Technology
Fall 2016
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

2. Table of Contents
1
Introduction.
2
Bode Plot of Frequency Response.
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

3. Introduction:
In previous lectures, we see the physical system response to one of thetest signalssuch
as animpulse, astepand arampsignal.
Frequency Response Analysis:
Frequency response is the steady-state response of a system to asinusoidal input.
In frequency-response methods, the frequency of the input signal is varied over a certain
range and the resulting response is studied.
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

4. Introduction:
The Concept of Frequency Response:
In the steady state, sinusoidal inputs to a
linear system generate sinusoidal
responses of the same frequency.
Even though these responses are of the
same frequency as the input, they dier in
amplitude and phase anglefrom the
input.
The dierences in the amplitude and the
frequency are functions of the input
frequency.
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

5. Introduction:
The Concept of Frequency Response:
Suppose a sinusoidal signal with amplitudeXand
phase shift.
x(t) =Xsin(!t+)
A sinusoidal signal can be represented as complex
numbers calledphasor.
The magnitude of the complex number is the
amplitudeXof the sinusoid, and the angle of the
complex number is the phase angleof the
sinusoid.
Thus, a sinusoid signal can be represented as:
X\
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

6. Introduction:
The Concept of Frequency Response:
Input phasor:Mi\i
Output phasor:Mo\o
So, the system TF can be represented as:
G(s) =M\=
Mo\o
Mi\i
such thatMi\iM\=Mo\o
Puts=j!inG(s):
G(j!) =jG(j!)j\G(j!)
M=
Mo
Mi
=jG(j!)jMagnitude
=\G(j!)Phase Shift
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

7. Introduction:
The Concept of Frequency Response:
Example:G(s) =
1
s+ 1
s G(j!) M=jG(j!)j =\G(j!)
j0
1
j0 + 1
j
1
1
j= 1 \
1
j0 + 1
= 0
o
j1
1
j1 + 1
j0:5j0:5j=
p
2=2 \0:5j0:5 = 45
o
jBjB!1
1
jB+ 1
jB!1 j
1
jB+ 1
jB!1= 0\
1
jB+ 1
jB!1=90
o
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

8. Table of Contents
1
Introduction.
2
Bode Plot of Frequency Response.
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

9. Bode Plot of Frequency Response:
We need to plot the change of theMagnitudeM(!)
and thePhase(!) separately when we change the
input frequency!:
Magnitude vs. Frequency: log-log (dB vs. decade)
Phase vs. Frequency: linear-log scale (deg vs. decade)
Example:G(s) =s
s MjdB= 20 logM
j1 20 log 1 = 0dB 90
o
j10 20 log 10 = 20dB 90
o
j100 20 log 100 = 40dB 90
o
j0.120 log 0:1 =20dB90
o
j0.0120 log 0:01 =40dB90
o
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

10. Bode Plot:
Example:G(s) =
1
s
s MjdB= 20 logM
j1 20 log
1
1
= 0dB 90
o
j10 20 log
1
10
=20dB90
o
j10020 log
1
100
=40dB90
o
j0.1 20 log
1
0:1
= 20dB 90
o
j0.0120 log
1
0:01
= 40dB90
o
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

11. Approximation of Bode Plot:
Example:G(s) =Ts+ 1
It will be dicult to try some points.
[1] whens=j! <<1=T(low frequencies):
G(j!)
1
T
M= 20logj
1
T
j= 0
o
[2] whens=j! >>1=T(high frequencies):
G(j!)s
M= 20logjj!j= 90
o
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

12. Approximation of Bode Plot:
Example:G(s) =
1
Ts+ 1
[1] whens=j! <<1=T(low frequencies):
G(j!)T
M= 20logjTj= 0
o
[2] whens=j! >>1=T(high frequencies):
G(j!)
1
s
M= 20logj
1
j!
j=90
o
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

13. Approximation of Bode Plot:
Example:G(s) =
!
2
n
s
2
+ 2!ns+!
2
n
Puts=j!
G(j!) =
1
1 + 2(j
!
!n
) + (j
!
!n
)
2
G(j!) =
1
(1(
!
!n
)
2
) +j(
2!
!n
)
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

14. Bode Plot with MATLAB:
Example
MATLAB Code:
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

15. PR01IP
Diaa Zekry
Mohamed Ashraf
PR02IP
Mohammed Nady
Mai Mahmoud
Mennah Mahmoud
PR03BB
Esraa Magdy
Mohamed Elsayed
Omar El-Geddawy
PR04IP
Ahmed Elsherif
Ahmed Abbdein
Muhammed Helmy
Osama Eldeeb
PR05BAB
Ahmed Bayoumy
Ahmed Magdy
Abdelrahman Bedeir
Ziad Saad
PR06IP
Bassma Hossny
PR07BP
Maha Ezzat
ShroukShalaby
PR08BT
Yomna Sherif
Eman Abulmagd
Yara Alaa
Mai Tarek
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems

16. End of Lecture
Best Wishes
[email protected]
Dr. Haitham El-Hussieny SPC318: System Modeling and Linear Systems