Transcript 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(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