l13 freq resp, nyquist
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Stability : Frequency Response Method
For a system to be stable the roots of its characteristic equation should lie
only in L.H.S. of s-plane.
The Routh-Hurwitz method of determining stability has following
limitations
1. It gives absolute stability and does not indicate about the strength of
stability.
2. The characteristic equation must be available in polynomial form.
The graphical methods based on frequency response give relative stability of
a closed-loop control system by using open-loop transfer function G(s)H(s).
System Frequency Response:
The frequency response of a control system is defined as the steady state
response of the system when a sinusoidal input is applied at the input
terminals. The sinusoidal input signal when applied to a linear system results
in an output signal which is sinusoidal in steady state and differs from the
input waveform only in amplitude and phase angle.
Frequency response method determines experimentally the properties of
complicated control systems without any difficulty as the sinusoidal test
signals for various ranges of frequencies and amplitudes are easily available.
(a) It is possible with this method to obtain experimentally the transfer
function of the system without complicated and long calculations.(b) The transfer function describing the sinusoidal steady state behaviour of
the system can be obtained by replacing s with j in the transfer function
G(s), of the system, i.e. |G(j) H(j)|
and
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The function G(j) representing the sinusoidal steady state behaviour of the
system is a function of complex variable having magnitude and phase angle
and is known as frequency function of the system.
The magnitude and phase angle of function G(j) for various frequencies are
represented by various graphical plots in different co-ordinates which give
better insight for analysis and design of control systems. The graphical plots
generally used are :
(i)Polar plot :-This is the plot of the magnitude |G(j) H(j)| versus phase
angle
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2. Therefore the frequency functions of systems are plotted in graphical
forms which indicate the system characteristics. Any curve giving
information regarding the gain or phase shift of the frequency function
is known as the frequency response curve of the system.
3. In polar plots the amplitude of G(j) is plotted as the distance from the
origin while the phase angle is plotted as angular displacement from
the right hand horizontal axis on the polar graph as shown in Fig. 7.4.
4. For negative values of() the phase angle lags while for positive
values of () the phase angle leads. Lagging phase shift is
represented by a counter-clockwise angular displacement of the
vector while leading phase shift is represented by the clockwise
angular displacement of the vector.
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5. These plots are simple to construct and easily provide the information
regarding the magnitude and phase angle of G( j) at any desired
frequency as compared to other methods. Polar plots are preferred as
compared to rectangular plots because polar plot contains the ready
information of both the parameters, amplitude and phase angle.
6. For sketching the polar plot of an open loop transfer function G(s)
the following criteria are used to determine the important position of
the complete plot.
(a)From the transfer function G(s) in general the frequency function
G(j) is obtained by substituting s = j i.e.,
From (7.8) the magnitude and phase angle at > 0 is obtained by
taking the limit of (7.8) at > 0 . Depending upon the type of
system i.e. the value of N in (7.8) the magnitude may be zero or
infinity and phase angle 90 N degrees at > 0.
(b)At higher frequencies i.e. the magnitude and phase angle
are obtained by taking the limit of magnitude and phase angle of (7.8)
at
The procedure used for sketching the polar plot of a system is asdescribed below :
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