NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Introduction to Bode Plot

D Bishakh
Dr. Bi h kh Bhattacharya
Bh tt h

Professor, Department of Mechanical Engineering

IIT Kanpur

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

ThisLectureContains

IntroductiontoBodePlot
Introduction to Bode Plot

BodePlotofaFirstOrderSystem

BodeplotofHigherOrderSystem

GainandPhaseMargin

Assignment

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Introduction to Bode Plot

So far we have designed compensators based on Root-Locus
technique and time domain response of a system.
We have used Frequency Response only to find the stability of a
dynamic system remember Nyquist Stability Criteria
Nyquist plot presents the Real vs. Imaginary plot of the open loop
transfer function, the nature of which has given us clues on stability of
y
a system. However,, we have not obtained a direct plot
p of Magnitude
g
and Phase of a Control System with respect to frequency.
Bode Plot deals with the frequency response of a system simultaneously
in terms of magnitude and phase. More precisely, the log-magnitude and
phase frequency response curves are known as Bode Plots.Plots Such plots
are useful due to the following reasons:
For designing lead compensators
For finding stability, gain and phase margin
For system identification from the frequency response
NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode Plot based on Asymptotic

A
Approximation
i ti
Consider a generalized transfer function as:

m
K ( s zi )
T (s) i 1.
nk
s k (s p j )
j 1

The magnitude (in terms of decibel) and phase of the transfer function are:
m nk
20 log T ( s ) 20 log K 20 log s zi 20 log s 20 log s p j k

i 1 j 1

T ( s ) (m n)
2

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode Plot of (s
(s+p)
p)

Consider a transfer function T(s) = s+p

The frequency response may be obtained by applying s=j.
Accordingly, we may write

T ( j ) j p p(1 j )
p
Now, for very low value of the frequency:

T ( j) p
20logT ( j) 20log(p)

Let us look at the system behavior for higher value of the frequency
(say /p >1).

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode plot for (s

(s+p)
p) contd.

For such cases

T ( j ) p ( j )
p

g(T ) 20 log(
20 log( g( p ) 20 log( g( )
g( ) 20 log(
p

In other words, the magnitude of the transfer function could be considered to

be constant (20 log(p)) till =p and then increasing at the rate of 20dB per
decade.
How about the phase. We know that the phase as tends to will be 900 . At
lower frequencies we can use the relationship stated earlier and find the phase
to be close to zero till is about 0.1a and then increase at the rate of 450
/decade till it reaches 900 .

The actual behavior is plotted in the next slide.

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode Plot for (s

(s+5)
5)

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode Plot for [1/(s+p)]

[1/(s p)]

In this case, one can follow a similar procedure to find the asymptotic behavior.
It can be
b shown
h that
th t ffor llow ffrequency th
the magnitude
it d iis close
l tto 20 llog(1/p)
(1/ ) and
d
beyond p, it decreases at the rate of 20dB per decade. The phase plot will show
that the initial phase to be close to zero and then decrease at the rate of 45
degree per decade until reaches -900 as the frequency goes beyond 10p

The actual plot for T(s) = 1/(s+5) is shown below:

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

A few more comments on Bode Plot

For a cascaded system having two poles, similar arguments could be placed
and
d the
th change
h off slope
l corresponding
di tot each
h off the
th poles
l att the
th rate
t off -20dB
20dB
per decade for the magnitude plot and -45 per decade for the phase plot could
0

be applied.

For a second order system, again asymptotic plot will show the magnitude to
be reducing at the rate of -40dB per decade and the phase plot will show -900
per decade decay from the initial value till it reaches -1800 .

A typical Bode-plot of a second order system is shown in the next slide. You
can use MATLAB for Bode plot by first defining the transfer function and then
using the command bode(transfer function).

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Bode plot for [1/(s2 + 3s + 10)]

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Gain and Phase Margin using Bode Plot

From Nyquist Criteria you know that instability occurs if there is encirclement
off -1
1 corresponding
di tot a phase
h off 1800 .

This implies that the for stability the magnitude of the transfer function must be
less than unity
y at a frequency
q y which corresponds
p to 1800 pphase. In fact the
actual Gain in dB corresponding to this frequency provides the gain margin.

In a similar manner the phase value corresponding to the frequency where gain
of the system is 0 dB provides the phase margin
margin.

You can use the margin command in MATLAB to obtain both the Bode plot,
Gain and Phase Margin.

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Assignment

Consider a unity feedback system with a plant transfer function:

( s 6)
G(s)
( s 2)( s 4)( s 7)( s 8)
Input the transfer function in MATLAB and sketch the bode plot.
Find out the Gain and Phase Margin and comment on the stability of the
system.

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NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 25

Special References for this lecture

Feedback Control of Dynamic Systems, Frankline, Powell and Emami, Pearson

Control Systems Engineering Norman S Nise, John Wiley & Sons

Design of Feedback Control Systems Stefani,

Stefani Shahian,
Shahian Savant,
Savant Hostetter

Oxford

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