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Automatic Control System
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Milton Kr. Das - 159CHAPTER - 9
ANAlYSIS oF FReQUeNCY ReSPoNSe & Bode Plot
9.1. INTRODUCTION :
In the time-domain analysis various standard test inputs like impulse, step and ramp functions
are used to study the performance of control systems. But from the time-domain analysis to find out
transfer function is very difficult and also laborious. On the other hand using frequency response
methods we can easily obtain the transfer function from the experimental data. Frequency response of a
system is the response of the system for sinusoidal input signal of various frequencies. In general the
sinusoidal test signal is given by,
r (t ) A sin t ………………………………………….. (9.1)
Where A is the magnitude of the input signal, is the frequency of oscillation. Normally A is
kept constant to obtain frequency response. For this standard sinusoidal test signal the system gives a
sinusoidal steady state c(t ) B sin(t ) , where B is the magnitude of output, is the frequency of
oscillation and is the phase shift between input and output. The relationship between the sinusoidal
input r(t) and steady state sinusoidal output c(t) of the system is known as frequency response.
9.2. ADVANTAGES OF FREQUENCY RESPONSE:
1. The frequency response of linear time invariant system is independent of the amplitude
and phase of the input test signal.
2. The transfer function of the system can be obtained from the frequency response.
3. The design and parameter adjustment of the open loop transfer function of a system for
specified closed loop performance can be carried out somewhat more easily in frequency
domain than in time domain.
4. The effect of noise disturbance and parameter variations in frequency domain can be easily
visualized.
5. Transient response of a first and second order systems can be obtained from frequency
response using Fourier integral (Computation is tedious for higher order systems).
6. Since there is correlation between time domain and frequency domain specifications, time
domain performance of a linear system can be easily predicted using frequency domain
specifications.
7. There are no unified methods to design a system that meet time domain specifications. On
the other hand in the frequency domain many graphical methods are available that are not
limited to lower order systems.
9.3 THE CONCEPT OF FREQUENCY RESPONSE:
Let us consider a linear time-invariant system with open loop transfer function G(s). Let r(t) & c(t)
are input and output signals of the system. Now we have to prove that for a sinusoidal input, the output
also be a sinusoidal of the same frequency but with different magnitude and phase angle.
r(t)
A
t
0
Figure 9.3.1 Typical sinusoidal input.
, Milton Kr. Das - 160
Let the input be, r (t ) A sin t ………………………………………………..… (9.2)
The steady state response of a stable LTI system to a sinusoidal input does not depend on the initial
conditions and we can write the output C(s) as
C ( s) G ( s ) R( s) ………………………………………………….. (9.3)
A
We have R( s ) 2 …………………………………………………………….... (9.4)
s 2
A
Substituting equation (9.4) in equation (9.3) we get, C ( s ) 2 G ( s ) …………………... (9.5)
s 2
Splitting equation (9.5) into partial fractions we can write,
k k Partial fraction
C ( s) ………………………………... (9.6)
s j s j terms for G(s)
Due to poles of Due to pole of
Input R(s) the transfer function
Steady state response Transient response
Note: k* is complex conjugate of k. Complex poles results complex conjugate coefficients.
The output contains transient components and steady state components. For a stable system the
transient components die out while the steady state components exist. We are interested in steady state
performance analysis so let us neglect the transient response components.
A A
*
G ( j ) e jG ( j ) G ( j ) e jG ( j )
k k 2j 2j
C ( s) ………………. (9.7)
s j s j s j s j
AG ( j ) AG ( j ) A A
Where k ( s j ) G ( j ) G ( j ) e jG ( j ) .
( s j )( s j ) s j 2 j 2j 2j
A G ( j ) A G ( j ) A A
k * ( s j ) G ( j ) G ( j ) e jG ( j )
( s j )( s j ) s j
2 j 2j 2j
Taking Laplace inverse on both sides in equation (9.7) we get,
A A
c(t ) G ( j ) e jG ( j ) e jt G ( j ) e jG ( j ) e jt ……………...…….. (9.8)
2j 2j
Since G ( j ) G ( j ) and G ( j ) G ( j )
We can write equation (9.8) as
A A
c(t ) G ( j ) e j ( j G ( j )) G ( j ) e j ( j G ( j )) ……………………. (9.9)
2j 2j
Let G ( j ) ( )
e j (t ) e j (t )
c(t ) A G ( j ) [ ] = A G ( j ) sin(t ( )) B sin(t ( )) ... (9.10)
2j
Where, B A G ( j ) .
The steady state response of the system for a sinusoidal input is a sinusoidal output with a
B
magnitude B and phase shift ( ) . The amplitude ratio is given by G ( j ) .
A
C ( j ) Amplitude ratio of the output
G ( j ) ………….. (9.11)
R ( j ) Signal to to input signal
, Milton Kr. Das - 161
C ( j ) Phase shift of the output sinusoid
( ) = G ( j ) …………. (9.12)
R( j ) With respect to the input sinusoid
C ( j )
G ( j ) …………………………………………………... (9.13)
R( j )
Sinusoidal transfer function G ( j ) can be obtained by substituting s j in G (s ) where,
G ( j ) is the sinusoidal transfer function.
In general G ( j ) and ( ) varies as frequency varies. For a constant input amplitude, when low
frequency signals have large output amplitude than high-frequency signals, then the behaviour is termed
as low-pass when the high frequency signals have large output amplitude than low-frequency signals,
then the behaviour is known as high-pass behaviour. If the amplitude is large for some frequency range,
then it is known as band-pass behaviour and if the amplitude is small for some frequency range then it is
known as band reject behaviour. Most of the control system has low pass behaviour. Communication
engineers use frequency response methods to design filters or to design circuits that shape the spectrum
of signals. But control system engineers use the frequency response methods to design stable control
systems.
9.4 PLOTTING OF FREQUENCY RESPONSE:
The frequency response of a linear time invariant system can be obtained by applying a spectrum of
input sinusoids to the system. We have already studied that when a sinusoidal signal A sin t is applied
to the system, the output also a sinusoidal signal with change in magnitude and phase given by,
y (t ) AG ( j ) sin t A G ( j ) e jG ( j ) sin t ………………………. (9.14)
Where, G ( j ) G ( j ) e jG ( j ) (in polar form), G ( j ) is magnitude and G ( j ) is phase angle.
The plot of G ( j ) vs is known as magnitude response and the plot of G ( j ) vs is known
as phase response. To plot the frequency response of a system, the frequency is varied from 0 to and
the values of G ( j ) and G ( j ) are calculated. Then these values are plotted on a linear scale for both
magnitude and phase against frequencies.
Note: Generally the value of A = 1 if it is note explicitly mentioned.
Magnitude = G ( j ) ; Phase = G ( j ) .
9.5 FREQUENCY DOMAIN SPECIFICATIONS:
The frequency domain specifications are the measures of the performance and characteristics of a
system. The specifications are: (i) Resonant Frequency ( r ). (ii) Resonant Peak ( M r ). (iii) Cut off
frequency ( c ). (iv) Bandwidth ( b ). (v) Cut-off rate. (vi) Gain margin. (vii) Phase margin.
Resonant Frequency ( r ):
The frequency at which the system has maximum magnitude is known as the resonant frequency. At
this frequency, the slope of the magnitude curve is zero.
Resonant Peak (Mr):
The maximum value of magnitude is known as the resonant peak. In general, the magnitude of Mr
gives indication on the relative stability of a stable closed-loop system. Normally, a large Mr
corresponds to a large maximum overshoot of the step response. For most control systems, it generally
accepted in practice that the desirable value of Mr should be between 1.1 and 1.5.
Cut off Frequency ( c ):
The frequency at which the magnitude G ( j ) is 1 times less than its maximum value is known
2
as cut-off frequency. In other words, the frequency at which the magnitude of the closed loop system is
3dB below its maximum value is called the cut-off frequency. Control systems (in which the maximum
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