Aravindh Seminar

8/7/2019 Aravindh Seminar

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FREQUENCY RESPONSE

ANALYSIS

Submitted by,N.Aravindh

M.Tech (chemical Engg)


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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..!

Aravindh Seminar

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