Formula Sheets: Frequency Domain Analysis (Polar & Bode Plot)

 Page 1


Control S ystems F ormula Sheet: Frequency
Domain Analysis (Polar & Bode Plots)
1. Frequency Response Basics
• Frequency Response : Response of a system to a sinusoidal input, described
b y the tr ansfer functionG(s) evaluated ats =j? :
G(j?) =|G(j?)|e
j?G(j?)
• Magnitude : |G(j?)| , expressed in absolute value or decibels (dB):
|G(j?)|
dB
= 20 log
10
|G(j?)|
• Phase :?G(j?) , measured in degrees.
2. Polar Plot
• Definition : Plot of G(j?) in the complex plane with magnitude |G(j?)| and
phase?G(j?) as? varies from 0 to8 .
• K ey Points :
– At? = 0 : Starting point isG(j0) = lim
??0
G(j?) .
– At??8 : Ending point isG(j8) = lim
??8
G(j?) .
• Type of S ystem : Determined b y the number of poles ats = 0 inG(s)H(s) .
• Low Frequency Behavior (TypeN system):
G(j?)˜
K
(j?)
N
, ?G(j?)˜-90
?
·N
• High Frequency Behavior : F orn>m (wheren is number of poles,m is num-
ber of zeros):
G(j?)˜
K
(j?)
n-m
, ?G(j?)˜-90
?
(n-m)
1
Page 2


Control S ystems F ormula Sheet: Frequency
Domain Analysis (Polar & Bode Plots)
1. Frequency Response Basics
• Frequency Response : Response of a system to a sinusoidal input, described
b y the tr ansfer functionG(s) evaluated ats =j? :
G(j?) =|G(j?)|e
j?G(j?)
• Magnitude : |G(j?)| , expressed in absolute value or decibels (dB):
|G(j?)|
dB
= 20 log
10
|G(j?)|
• Phase :?G(j?) , measured in degrees.
2. Polar Plot
• Definition : Plot of G(j?) in the complex plane with magnitude |G(j?)| and
phase?G(j?) as? varies from 0 to8 .
• K ey Points :
– At? = 0 : Starting point isG(j0) = lim
??0
G(j?) .
– At??8 : Ending point isG(j8) = lim
??8
G(j?) .
• Type of S ystem : Determined b y the number of poles ats = 0 inG(s)H(s) .
• Low Frequency Behavior (TypeN system):
G(j?)˜
K
(j?)
N
, ?G(j?)˜-90
?
·N
• High Frequency Behavior : F orn>m (wheren is number of poles,m is num-
ber of zeros):
G(j?)˜
K
(j?)
n-m
, ?G(j?)˜-90
?
(n-m)
1
3. Bode Plot
• Definition : Two separ ate plots:
– Magnitude plot: 20 log
10
|G(j?)| (in dB) vs. log
10
? .
– Phase plot:?G(j?) (in degrees) vs. log
10
? .
• Standard F orm of Tr ansfer Function :
G(s) =K
(s+z
1
)(s+z
2
)···
(s+p
1
)(s+p
2
)···
·
1
s
N
whereK is gain,z
i
are zeros,p
i
are poles,N is system type.
• Gain in dB :
20 log
10
|G(j?)| = 20 log
10
K+
?
20 log
10
|j?+z
i
|-
?
20 log
10
|j?+p
i
|-20N log
10
?
• Phase Angle :
?G(j?) =
?
tan
-1
(
?
z
i
)
-
?
tan
-1
(
?
p
i
)
-90
?
·N
• Corner Frequencies : Frequencies where poles or zeros occur (? = z
i
or? =
p
i
), causing a change in slope of the magnitude pl ot.
• Slope of Magnitude Plot :
– Pole ats =-p
i
: Slope decreases b y-20 dB/decade after? =p
i
.
– Zero ats =-z
i
: Slope increases b y +20 dB/decade after? =z
i
.
– Integr ator (1/s
N
): Slope decreases b y-20N dB/decade.
• Initial Slope (for TypeN system): -20N dB/decade.
4. Stability Margins
• Gain Crossover Frequency (?
gc
): Frequency where|G(j?)| = 1 (or 0 dB).
• Phase Crossover Frequency (?
pc
): Frequency where?G(j?) =-180
?
.
• Gain Margin (GM):
GM (dB) =-20 log
10
|G(j?
pc
)|
• Phase Margin (PM):
PM = 180
?
+?G(j?
gc
)
• Stability Condition : S ystem is stable if GM> 0 dB and PM> 0
?
. Typically , PM
between 30
?
and 60
?
is desired for good performance.
2
Page 3


Control S ystems F ormula Sheet: Frequency
Domain Analysis (Polar & Bode Plots)
1. Frequency Response Basics
• Frequency Response : Response of a system to a sinusoidal input, described
b y the tr ansfer functionG(s) evaluated ats =j? :
G(j?) =|G(j?)|e
j?G(j?)
• Magnitude : |G(j?)| , expressed in absolute value or decibels (dB):
|G(j?)|
dB
= 20 log
10
|G(j?)|
• Phase :?G(j?) , measured in degrees.
2. Polar Plot
• Definition : Plot of G(j?) in the complex plane with magnitude |G(j?)| and
phase?G(j?) as? varies from 0 to8 .
• K ey Points :
– At? = 0 : Starting point isG(j0) = lim
??0
G(j?) .
– At??8 : Ending point isG(j8) = lim
??8
G(j?) .
• Type of S ystem : Determined b y the number of poles ats = 0 inG(s)H(s) .
• Low Frequency Behavior (TypeN system):
G(j?)˜
K
(j?)
N
, ?G(j?)˜-90
?
·N
• High Frequency Behavior : F orn>m (wheren is number of poles,m is num-
ber of zeros):
G(j?)˜
K
(j?)
n-m
, ?G(j?)˜-90
?
(n-m)
1
3. Bode Plot
• Definition : Two separ ate plots:
– Magnitude plot: 20 log
10
|G(j?)| (in dB) vs. log
10
? .
– Phase plot:?G(j?) (in degrees) vs. log
10
? .
• Standard F orm of Tr ansfer Function :
G(s) =K
(s+z
1
)(s+z
2
)···
(s+p
1
)(s+p
2
)···
·
1
s
N
whereK is gain,z
i
are zeros,p
i
are poles,N is system type.
• Gain in dB :
20 log
10
|G(j?)| = 20 log
10
K+
?
20 log
10
|j?+z
i
|-
?
20 log
10
|j?+p
i
|-20N log
10
?
• Phase Angle :
?G(j?) =
?
tan
-1
(
?
z
i
)
-
?
tan
-1
(
?
p
i
)
-90
?
·N
• Corner Frequencies : Frequencies where poles or zeros occur (? = z
i
or? =
p
i
), causing a change in slope of the magnitude pl ot.
• Slope of Magnitude Plot :
– Pole ats =-p
i
: Slope decreases b y-20 dB/decade after? =p
i
.
– Zero ats =-z
i
: Slope increases b y +20 dB/decade after? =z
i
.
– Integr ator (1/s
N
): Slope decreases b y-20N dB/decade.
• Initial Slope (for TypeN system): -20N dB/decade.
4. Stability Margins
• Gain Crossover Frequency (?
gc
): Frequency where|G(j?)| = 1 (or 0 dB).
• Phase Crossover Frequency (?
pc
): Frequency where?G(j?) =-180
?
.
• Gain Margin (GM):
GM (dB) =-20 log
10
|G(j?
pc
)|
• Phase Margin (PM):
PM = 180
?
+?G(j?
gc
)
• Stability Condition : S ystem is stable if GM> 0 dB and PM> 0
?
. Typically , PM
between 30
?
and 60
?
is desired for good performance.
2
5. Bandwidth and Resonant Frequency
• Bandwidth (?
b
): Frequency where|G(j?)| drops to-3 dB from its low-frequency
value (for first-order or dominant second-order systems).
• Resonant Frequency (?
r
): Frequency where|G(j?)| peaks (for underdamped
second-order systems):
?
r
=?
n
v
1-2?
2
where?
n
is natur al frequency ,? is damping r a tio.
• Resonant Peak (M
r
):
M
r
=
1
2?
v
1-?
2
, for? < 0.707
3