L13 Freq Resp, Nyquist

8/3/2019 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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L13 Freq Resp, Nyquist