Polar Plot

Report
By: Nafees Ahmed
Asstt. Prof., EE Deptt,
DIT, Dehradun
By: Nafees Ahmed, EED, DIT, DDun
Introduction
 The polar plot of sinusoidal transfer function G(jω) is a plot
of the magnitude of G(jω) verses the phase angle of G(jω)
on polar coordinates as ω is varied from zero to infinity.
 Therefore it is the locus of G ( j  )  G ( j  ) as ω is varied from
zero to infinity.
 As G ( j  )  G ( j  )  Me j (  )
 So it is the plot of vector Me j  (  ) as ω is varied from zero
to infinity
By: Nafees Ahmed, EED, DIT, DDun
Introduction conti…
 In the polar plot the magnitude of G(jω) 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=jω in the G(s)
& lim G ( j  )
 Step 4: At ω=0 & ω=∞ find  G ( j  ) by lim  G ( j ) & lim  G ( j )
 Step 5: Rationalize the function G(jω) and separate the real
and imaginary parts
 Step 6: Put Re [G(jω) ]=0, determine the frequency at which
plot intersects the Im axis and calculate intersection value by
putting the above calculated frequency in G(jω)
 Step 3: At ω=0 & ω=∞ find G ( j  ) by
By: Nafees Ahmed, EED, DIT, DDun
lim

G ( j )
0

0

Steps to draw Polar Plot conti…
 Step 7: Put Im [G(jω) ]=0, determine the frequency at
which plot intersects the real axis and calculate
intersection value by putting the above calculated
frequency in G(jω)
 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
G (s) 
K
(1  sT 1 )( 1  sT 2 )
 Step 1: Put s=jω
G ( j ) 

K
(1  j  T1 )( 1  j  T 2 )
K
1   T1 
2
1   T 2 
  tan
2
1
 T1  tan
1
 T2
 Step 2: Taking the limit for magnitude of G(jω)
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti…
lim

G ( j ) 
lim

G ( j ) 
0

K
1   T1 
1  j  T 2 
2
 K
2
K
1   T1 
2
1  j  T 2 
0
2
 Step 3: Taking the limit of the Phase Angle of G(jω)
lim

 G ( j  )    tan
1
 T1  tan
1
 T2  0
lim

 G ( j  )    tan
1
 T1  tan
1
 T 2   180
0

By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti…
 Step 4: Separate the real and Im part of G(jω)
K (1   T1T 2 )
K  (T1  T 2 )
2
G ( j ) 
1   T1   T   T1T 2
2
2
2
2
2
4
 j
1   T1   T 2   T1T 2
2
2
2
2
 Step 5: Put Re [G(jω)]=0
K (1   T1T 2 )
2
1   T1   T
2
2
2
2
2
  T1 T 2
4
 0  
1
&   
T1 T 2
So When
 
1
 G ( j ) 
T1 T 2
&   
K
T1 T 2
  90
T1  T 2
 G ( j  )  0   180
By: Nafees Ahmed, EED, DIT, DDun
0
0
4
Type 0 system conti…
 Step 6: Put Im [G(jω)]=0
K  (T1  T 2 )
1   T1   T   T1T 2
2
2
2
2
2
4
0 0 & 
So When
  0  G ( j )  K  0
0
    G ( j  )  0  180
0
By: Nafees Ahmed, EED, DIT, DDun
Type 0 system conti…
By: Nafees Ahmed, EED, DIT, DDun
Polar Plot for Type 1 System
 Let G ( s )  s (1  sT K)(1  sT
1
2
)
 Step 1: Put s=jω
K
G ( j ) 
j  (1  j  T 1 )( 1  j  T 2 )

K
 1   T 1 
2
1  j  T 2 
  90
2
0
 tan
1
 T 1  tan
By: Nafees Ahmed, EED, DIT, DDun
1
T2
Type 1 system conti…
 Step 2: Taking the limit for magnitude of G(jω)
lim

G ( j ) 
lim

G ( j ) 
0

K
 1   T1 
2
1  j  T 2 
 
2
K
 1   T1 
2
1  j  T 2 
 0
2
 Step 3: Taking the limit of the Phase Angle of G(jω)
lim

 G ( j  )    90
0
 tan
1
 T1  tan
1
 T 2   90
lim

 G ( j  )    90
0
 tan
1
 T1  tan
1
 T 2   270
0
0

By: Nafees Ahmed, EED, DIT, DDun
0
Type 1 system conti…
 Step 4: Separate the real and Im part of G(jω)
G ( j ) 
  K (T 1  T 2 )
2
   (T 1  T 2   T 1 T 2 )
3
2
2
j ( K  T1T 2  K )
2
2
2
 j
   (T 1  T 2   T 1 T 2 )
3
2
 Step 5: Put Re [G(jω)]=0
  K (T1  T 2 )
   (T1  T
3
2
2
2
  T1 T )
2
2
2
2
 0   
So at
 
 G ( j  )  0   270
By: Nafees Ahmed, EED, DIT, DDun
0
2
2
2
2
Type 1 system conti…
 Step 6: Put Im [G(jω)]=0
j ( K  T1T 2  K )
2
   (T1  T
3
2
2
2
  T1 T )
2
2
2
2
1
 0  
T1T 2
So When
 
1
T1T 2
  

G ( j )  
K
T1T 2
T1  T 2
G ( j  )   0
0
0
0
By: Nafees Ahmed, EED, DIT, DDun
&   
Type 1 system conti…
By: Nafees Ahmed, EED, DIT, DDun
Polar Plot for Type 2 System
 Let
G (s) 
K
s (1  sT 1 )( 1  sT 2 )
2
 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(jω) 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=jω
G ( j ) 

20
j  ( j   1)( j   2 )
20
  1   4
2
  90
0
 tan
1
  tan
2
By: Nafees Ahmed, EED, DIT, DDun
1
 /2
 Step 2: Taking the limit for magnitude of G(jω)
lim

G ( j ) 
lim

G ( j ) 
20
0

 
  1   4
2
2
20
 0
  1   4
2
2
 Step 3: Taking the limit of the Phase Angle of G(jω)
lim

 G ( j  )    90
0
 tan
1
  tan
1
 / 2   90
lim

 G ( j  )    90
0
 tan
1
  tan
1
 / 2   270
0
0

By: Nafees Ahmed, EED, DIT, DDun
0
 Step 4: Separate the real and Im part of G(jω)
G ( j ) 
 60 
(
4
(
4
2
 j
(
4
  )( 4   )
2
2
  )( 4   )
2
3
  )( 4   )
2
 60 
j 20 (  2 )
2
2
0  
So at
 
 G ( j  )  0   270
By: Nafees Ahmed, EED, DIT, DDun
0
2
 Step 6: Put Im [G(jω)]=0
j 20 (
(
4
3
 2 )
  )( 4   )
2
2
 0     2 &   
So for positive value of 
 
2
G ( j )  
10
0
0
G ( j )  0  0
0
3
  
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 (ωp) : The frequency where a
polar plot intersects the –ve real axis is called phase
crossover frequency
 Gain Crossover Frequency (ωg) : The frequency
where a polar plot intersects the unit circle is called
gain crossover frequency
So at ωg
G ( j  )  Unity
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)
Φm=1800+Φ
Where
Φ=∠G(jωg)
if
Φm>0 => +PM
(Stable System)
if
Φm<0 => -PM
(Unstable System)
By: Nafees Ahmed, EED, DIT, DDun
 Gain Margin (GM):
 The gain margin is the reciprocal of magnitude G ( j  ) at
the frequency at which the phase angle is -1800.
GM 
1

| G ( jwc ) |
1
x
In terms of dB
GM in dB  20 log
1
10
  20 log
| G ( jwc ) |
By: Nafees Ahmed, EED, DIT, DDun
10
| G ( jwc ) |  20 log
10
( x)
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(jω) is a graph of 1/G(jω) as
a function of ω.
 Ex: if G(jω) =1/jω then 1/G(jω)=jω
lim

G ( j )
1
0
lim

G ( j )
1

0

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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