Polar Plot - Department Of Electrical Engineering -...
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Polar Plot By: Nafees Ahmed Asstt. Prof., EE Deptt, DIT, Dehradun By: Nafees Ahmed, EED, DIT, DDun
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Transcript of Polar Plot - Department Of Electrical Engineering -...
Polar Plot
Polar Plot
By: Nafees Ahmed
Asstt. Prof., EE Deptt,
DIT, Dehradun
By: Nafees Ahmed, EED, DIT, DDun
Introduction
The polar plot of sinusoidal transfer function G(j) is a plot of the magnitude of G(j) verses the phase angle of G(j) on polar coordinates as is varied from zero to infinity.
Therefore it is the locus of as is varied from zero to infinity.
As
So it is the plot of vector as is varied from zero to infinity
By: Nafees Ahmed, EED, DIT, DDun
Introduction conti
In the polar plot the magnitude of G(j) is plotted as the distance from the origin while phase angle is measured from positive real axis.
+ angle is taken for anticlockwise direction.
Polar plot is also known as Nyquist Plot.
By: Nafees Ahmed, EED, DIT, DDun
Steps to draw Polar Plot
Step 1: Determine the T.F G(s)
Step 2: Put s=j in the G(s)
Step 3: At =0 & = find by &
Step 4: At =0 & = find by &
Step 5: Rationalize the function G(j) and separate the real and imaginary parts
Step 6: Put Re [G(j) ]=0, determine the frequency at which plot intersects the Im axis and calculate intersection value by putting the above calculated frequency in G(j)
By: Nafees Ahmed, EED, DIT, DDun
Steps to draw Polar Plot conti
Step 7: Put Im [G(j) ]=0, determine the frequency at which plot intersects the real axis and calculate intersection value by putting the above calculated frequency in G(j)
Step 8: Sketch the Polar Plot with the help of above information
By: Nafees Ahmed, EED, DIT, DDun
Polar Plot for Type 0 System
Let
Step 1: Put s=j
Step 2: Taking the limit for magnitude of G(j)
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti
Step 3: Taking the limit of the Phase Angle of G(j)
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti
Step 4: Separate the real and Im part of G(j)
Step 5: Put Re [G(j)]=0
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti
Step 6: Put Im [G(j)]=0
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti
By: Nafees Ahmed, EED, DIT, DDun
Polar Plot for Type 1 System
Let
Step 1: Put s=j
By: Nafees Ahmed, EED, DIT, DDun
Type 1 system conti
Step 2: Taking the limit for magnitude of G(j)
Step 3: Taking the limit of the Phase Angle of G(j)
By: Nafees Ahmed, EED, DIT, DDun
Type 1 system conti
Step 4: Separate the real and Im part of G(j)
Step 5: Put Re [G(j)]=0
By: Nafees Ahmed, EED, DIT, DDun
Type 1 system conti
Step 6: Put Im [G(j)]=0
By: Nafees Ahmed, EED, DIT, DDun
Type 1 system conti
By: Nafees Ahmed, EED, DIT, DDun
Polar Plot for Type 2 System
Let
Similar to above
By: Nafees Ahmed, EED, DIT, DDun
Type 2 system conti
By: Nafees Ahmed, EED, DIT, DDun
Note: Introduction of additional pole in denominator contributes a constant -1800 to the angle of G(j) for all frequencies. See the figure 1, 2 & 3
Figure 1+(-1800 Rotation)=figure 2
Figure 2+(-1800 Rotation)=figure 3
By: Nafees Ahmed, EED, DIT, DDun
Ex: Sketch the polar plot for G(s)=20/s(s+1)(s+2)
Solution:
Step 1: Put s=j
By: Nafees Ahmed, EED, DIT, DDun
Step 2: Taking the limit for magnitude of G(j)
Step 3: Taking the limit of the Phase Angle of G(j)
By: Nafees Ahmed, EED, DIT, DDun
Step 4: Separate the real and Im part of G(j)
By: Nafees Ahmed, EED, DIT, DDun
Step 6: Put Im [G(j)]=0
By: Nafees Ahmed, EED, DIT, DDun
By: Nafees Ahmed, EED, DIT, DDun
Gain Margin, Phase Margin & Stability
By: Nafees Ahmed, EED, DIT, DDun
Phase Crossover Frequency (p) : The frequency where a polar plot intersects the ve real axis is called phase crossover frequency
Gain Crossover Frequency (g) : The frequency where a polar plot intersects the unit circle is called gain crossover frequency
So at g
By: Nafees Ahmed, EED, DIT, DDun
Phase Margin (PM):
Phase margin is that amount of additional phase lag at the gain crossover frequency required to bring the system to the verge of instability (marginally stabile)
m=1800+
Where
=G(jg)
if m>0 => +PM(Stable System)
if m -PM(Unstable System)
By: Nafees Ahmed, EED, DIT, DDun
Gain Margin (GM):
The gain margin is the reciprocal of magnitude at the frequency at which the phase angle is -1800.
In terms of dB
By: Nafees Ahmed, EED, DIT, DDun
Stability
Stable: If critical point (-1+j0) is within the plot as shown, Both GM & PM are +ve
GM=20log10(1 /x) dB
By: Nafees Ahmed, EED, DIT, DDun
Unstable: If critical point (-1+j0) is outside the plot as shown, Both GM & PM are -ve
GM=20log10(1 /x) dB
By: Nafees Ahmed, EED, DIT, DDun
Marginally Stable System: If critical point (-1+j0) is on the plot as shown, Both GM & PM are ZERO
GM=20log10(1 /1)=0 dB
By: Nafees Ahmed, EED, DIT, DDun
MATLAB Margin
By: Nafees Ahmed, EED, DIT, DDun
Inverse Polar Plot
The inverse polar plot of G(j) is a graph of 1/G(j) as a function of .
Ex: if G(j) =1/j then 1/G(j)=j
By: Nafees Ahmed, EED, DIT, DDun
Books
Automatic Control system By S. Hasan Saeed
Katson publication
By: Nafees Ahmed, EED, DIT, DDun
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![1 Convolutional Polar Codes - arXiv · 1 Convolutional Polar Codes Andrew James Ferris, Christoph Hirche and David Poulin Abstract Arikan’s Polar codes [1] attracted much attention](https://static.fdocument.org/doc/165x107/5f07505c7e708231d41c5eb5/1-convolutional-polar-codes-arxiv-1-convolutional-polar-codes-andrew-james-ferris.jpg)
