Post on 08-Apr-2018
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FREQUENCY RESPONSE
ANALYSIS
Submitted by,N.Aravindh
M.Tech (chemical Engg)
8/7/2019 Aravindh Seminar
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Topics focused on:
Frequency response analysis of
1. First order system
2. Second order system
BODE Diagram,
1.First order system
2.Second order system
3.Proportional Derivative controller
4.Proportional Integral controller5.Proportional Integral Derivative controller
Gain and Phase margins,
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Frequency response analysis
Introduction:
It is an alternate analysis of dynamic system.
It shows how the output response characteristics dependson frequency of input system or signal.
Its a simplified procedure to calculate frequency response
characteristics.
It is one of the powerful tool for analyzing and designing
controller.
.
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The dynamics of linear process can be entirely
characterized by Amplitude ratio (AR) and Phase
angle()over a range of values.
The frequency response of transfer function G(J) can
obtain from other models which gives information about
transfer function. No other response step, impulse, etc
provides such a complete information.
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Frequency response of First order system:
Consider the transfer function of first order is
G(s)= .(1)
Sub (S=j) in transfer function of first order
G( j)= (2)
To rationalize the equation(2) multiply numerator and
denominator by complex conjugate,
G(j)=
G(j)= (3)
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G(j)= ..(4)
Equation (4) is a complex number:
Real part of complex number
(m)= ..(5)
Imaginary part of complex number
(n)= ..(6)
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To determine the Amplitude ratio AR :=AR=
= )2 +
2
=
.(7)AR=
To determine phase angle
.....(8)
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These two relationship indicates that the response
of any linear control system can be obtained by
converting the transfer function of the system in to
a complex number and the amplitude ratio and phase
angle were obtained by applying of complex numbers.
In case of polar approach the response of the first and
second order process forced by a sinusoidal input this
response consists of,
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General sinusoidal change in input,
X(t)=A sin
Then, Y(t)= (
If the sinusoidal input continued for long time
the equation becomes,
Y(t)=
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BODE Diagram :
The Frequency response characteristics of a system is
represented by the BODE diagram.
The Bode diagram is known in the honor of H.W.Bode.
It is a convenient way for analysis of the frequency response of
linear control system.
It is useful for analysis the stability of closed loop system.
The Bode diagram consists of a pair of following graphs:
1. The variations of the logarithm of the Amplitude ratio with
Radian frequency (AR Vs t).
2. The variations of the Phase angle with Radian frequency
( Vs )
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BODE diagram for first order:
Variations of Amplitude ratio (Vs) Radian frequency:
(1) For low frequency, (or)
The Amplitude ratio AR=1 (or) log AR=1Its a horizontal line passing through the point AR=1
(2) For high frequency, (or)
The Amplitude ratio AR= (or)log AR=-log
The high frequency is a line with slope of -1 and passing
through the point AR=1 and =1
The point of intersection of low frequency asymptote(LFA) with
high frequency asymptote(HFA) is known as corner frequency
Variations of Phase angle (Vs)Radian frequency:The variations for low frequency and high frequency give he
approximate phase variation.
0
1
0
-45
-90
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BODE Diagram for first order system:
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Frequency response of second order system:
Consider the transfer function of second order system is,
G(s)= (1)
Sub(s=j ) in transfer function of second order,
G(j
G(j ...(2)
To rationalize the equation(2) multiply numerator and
denominator by complex conjugate,
G(j .(3)
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G(j (4)
Equation (4) is a complex number,
Real part of complex number,
(m)= (5)
Imaginary part of complex number,
(n)= ....(6)
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To determine Amplitude ratio, (AR) :
AR=
To determine Phase angle,( ):
.(8)
AR=
..(7)
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BODE Diagram for second order system:
(1). For low frequency : As , then (or) AR=1
(2). For high frequency :
As , then
Its a straight line with slope of -2 passing through
the point AR=1 ,
From the graph we noticed that for under damped
system the amplitude ratio can exceed significantly the
value of 1.
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BODE diagram of second order system :
AR
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Bode diagram for Proportional Derivative control :
Transfer function for PD controller is,
and rationalize the equation,
This is a equation of line passing through and
having slope +1. The corner frequency,
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The variations of phase angle obtained from the different
values, will be respectively as shown
in graph.
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Bode Diagram for Proportional Integral control :
The transfer function of PI controller is,
and rationalize the equation becomes,
Amplitude ratio :
Phase angle :
For low frequency :
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For high frequency :
This is a equation of line passing through and
having slope -1. The corner frequency
The variations of phase angle obtained from the different
values will be respectively, as
shown in graph
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Bode Diagram for Proportional Integral Derivative control :
The transfer function for PID controller is,
and rationalize the equation becomes,
Amplitude ratio :
Phase angle :
For low frequency :
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This is equation of line passing through AR/Kc=1 and
having slope -1. The corner frequency
For high frequency :
This equation of line passing through AR/Kc=1 and
having slope +1. The corner frequency
The variations of phase angle are obtained from the
different values will be -90 and +90 respectively.
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Gain and phase margins :
The bode stability criteria indicates how we can establish a
rational method for tuning the feed back controllers in
order to avoid unstable behavior by the closed loopresponse of a process.
Cross over frequency:
A control system is unstable if the open loop frequency
response exhibits an AR exceeding unity at the frequencyfor which the phase lag is 180. This frequency is called the
cross over frequency.
Consider the open-loop transfer function of a feed back
system.
Cross over frequency wco ,where = -180 ,AR=1.
Let m be the Amplitude ratio at the cross over frequency.
If M1 , the system is unstable
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Gain margin:
Gain margin = 1/M
For a stable system M
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Significance :
1. It constitutes a measure of how far the system is from
the brink of instability.2. Higher the gain margin, higher the safety factor we use
for controller tuning.
Phase margin:Besides the gain margin there is another safety
factor which is used for the design of feed back control
system.
Phase margin= 180 where, Phase lag at the frequency for which AR=1
Higher the phase margin, the larger the safety factor for
designing a controller.
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Thank you..!