Short Notes: Frequency Domain Analysis (Polar & Bode Plot) | Control Systems - Electrical Engineering (EE) PDF Download

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FrequencyDomainAnalysis: PolarandBodePlots
Frequency domain analysis evaluates a control system’s response to sinusoidal inputs, charac-
terized by gain and phase shift as functions of frequency. Polar and Bode plots are graphical
tools used to analyze system stability and performance.
1. PolarPlot
The polar plot represents the frequency response of a systems open-loop transfer function
G(j?)H(j?) in the complex plane, plotting magnitude and phase angle versus frequency?.
• De?nition: For a transfer function G(j?)H(j?), the polar plot is the locus of the com-
plex numberG(j?)H(j?) =|G(j?)H(j?)|e
j?G(j?)H(j?)
as? varies from 0 to8.
• Magnitude: |G(j?)H(j?)| =
v
Re{G(j?)H(j?)}
2
+Im{G(j?)H(j?)}
2
.
• PhaseAngle:?G(j?)H(j?) = tan
-1
(
Im{G(j?)H(j?)}
Re{G(j?)H(j?)}
)
.
• Construction:
– ComputeG(j?)H(j?) by substitutings = j?.
– Calculate magnitude and phase for various?.
– Plotpointsinthecomplexplanewithradius|G(j?)H(j?)|andangle?G(j?)H(j?).
• Stability Analysis: The Nyquist criterion uses the polar plot to assess stability. If the
plot encircles the point (-1,0) in the complex plane, the closed-loop system may be
unstable.
2. BodePlot
The Bode plot consists of two graphs: a magnitude plot (in decibels) and a phase plot (in
degrees), both versus frequency? on a logarithmic scale.
• MagnitudePlot:
– Magnitude in decibels: 20log
10
|G(j?)H(j?)|.
– Plotted against log
10
?.
– For a transfer functionG(s) =
K(j?)
n
?
m
i=1
(1+j?t
i
)
?
p
k=1
(1+j?t
k
)
, the magnitude is:
20log
10
|G(j?)| = 20log
10
K+20nlog
10
?+20
m
?
i=1
log
10
|1+j?t
i
|-20
p
?
k=1
log
10
|1+j?t
k
|.
• PhasePlot:
– Phase angle:?G(j?)H(j?) =
?
phase contributions.
– For each term:
?(1+j?t
i
) = tan
-1
(?t
i
), ?(j?)
n
= n·90
?
.
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FrequencyDomainAnalysis: PolarandBodePlots
Frequency domain analysis evaluates a control system’s response to sinusoidal inputs, charac-
terized by gain and phase shift as functions of frequency. Polar and Bode plots are graphical
tools used to analyze system stability and performance.
1. PolarPlot
The polar plot represents the frequency response of a systems open-loop transfer function
G(j?)H(j?) in the complex plane, plotting magnitude and phase angle versus frequency?.
• De?nition: For a transfer function G(j?)H(j?), the polar plot is the locus of the com-
plex numberG(j?)H(j?) =|G(j?)H(j?)|e
j?G(j?)H(j?)
as? varies from 0 to8.
• Magnitude: |G(j?)H(j?)| =
v
Re{G(j?)H(j?)}
2
+Im{G(j?)H(j?)}
2
.
• PhaseAngle:?G(j?)H(j?) = tan
-1
(
Im{G(j?)H(j?)}
Re{G(j?)H(j?)}
)
.
• Construction:
– ComputeG(j?)H(j?) by substitutings = j?.
– Calculate magnitude and phase for various?.
– Plotpointsinthecomplexplanewithradius|G(j?)H(j?)|andangle?G(j?)H(j?).
• Stability Analysis: The Nyquist criterion uses the polar plot to assess stability. If the
plot encircles the point (-1,0) in the complex plane, the closed-loop system may be
unstable.
2. BodePlot
The Bode plot consists of two graphs: a magnitude plot (in decibels) and a phase plot (in
degrees), both versus frequency? on a logarithmic scale.
• MagnitudePlot:
– Magnitude in decibels: 20log
10
|G(j?)H(j?)|.
– Plotted against log
10
?.
– For a transfer functionG(s) =
K(j?)
n
?
m
i=1
(1+j?t
i
)
?
p
k=1
(1+j?t
k
)
, the magnitude is:
20log
10
|G(j?)| = 20log
10
K+20nlog
10
?+20
m
?
i=1
log
10
|1+j?t
i
|-20
p
?
k=1
log
10
|1+j?t
k
|.
• PhasePlot:
– Phase angle:?G(j?)H(j?) =
?
phase contributions.
– For each term:
?(1+j?t
i
) = tan
-1
(?t
i
), ?(j?)
n
= n·90
?
.
1
– Total phase:?G(j?) = n·90
?
+
?
m
i=1
tan
-1
(?t
i
)-
?
p
k=1
tan
-1
(?t
k
).
• Construction:
– Approximate straight-line segments for magnitude and phase using asymptotic be-
havior (e.g., low and high frequencies).
– Identify corner frequencies where? =
1
t
i
or? =
1
t
k
.
– Magnitudeslopeschangeby±20dB/decadeforpoles/zeros;phaseshiftsby±45
?
/decade
near corner frequencies.
• KeyParameters:
– GainMargin: Theamountbywhichthesystemsgaincanincreasebeforeinstabil-
ity,foundwhere?G(j?)H(j?) =-180
?
. Gainmargin(dB)=-20log
10
|G(j?)H(j?)|.
– PhaseMargin: Theadditionalphaselagbeforeinstability,foundwhere|G(j?)H(j?)| =
1 (0 dB). Phase margin = 180
?
+?G(j?)H(j?).
Applications
• Polar plots are used for Nyquist stability analysis and to visualize frequency response in
the complex plane.
• Bode plots simplify gain and phase margin calculations, aiding in controller design and
system tuning.
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