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Frequency Response Analysis
Consider
let the input be in the form
Assume that the system is stable and the steady state response of the system to a sinusoidal inputdoes not depend on the initial conditions
We have PFEyields
Complex conjugate of
Inverse Laplace transform yields
At steady state (when ) we have
where
● Since is a complex quantity, we have
where
and similarly
Hence
That is a sinusoidal input generates a sinusoidal output
Note that
– Negative phase is called phase lag
– Positive phase is called phase lead
There are many ways representing these, We will concentrate on 3 common graphical representations
Amplitude ratios of the output sinusoid to the input
Phase shift of the output with respect to the input
Graphical Representations
1) Bode Plot
– A plot of magnitude in decibels (dB) vs in semi-logaritmic coordinates. The Phase angle is also plotted against in semilogarithmic coordinates
2) Polar Plot
– A plot of magnitude vs phase in polar coordinates as from zero to infinity
3) Magnitude vs Phase plots
– A plot of magnitude in dB vs the phase on the rectangular coordinates with as a varying parameter on the curve
Bode Plot Construction
Consider the transfer function of the system given in the form
where are positive real values.
In order to put the given tranfer function into “Bode“ form let and normalize
Magnitude PlotMagnitude Plot Phase PlotPhase Plot
Curve Breaks {up/down} by 20 dB/dec at {a,b / c,d,e}
Curve breaks {up/down} by 45 degrees one decade before {a,b / c,d,e} and breaks {down/up} by 45 degrees one decade after {a,b / c,d,e}
Initial Value,
● For n=0 Calculate and convert to dBusing
● For n>0 Calculate -intercept =
Calculate the initial slope -20n dB /dec Final magnitude slope -20 RD dB /dec
Initial and final values
● Plot a zero / pole plot for the original transfer function
● Calculate the angle from the figure at = 0
An illustrative example
The
Initial angle
Final angle
-10 -8 -6 -4 -2 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Axis
Imag
Axi
s
Bode Plot Example
Construct the Bode Plot for
Start with putting the transfer function into Bode form
Magnitude PlotMagnitude Plot
– Initial Magnitude (note that n=0)
– Use “up“ arrow to denote a slope of +20 dB/dec and “down“ arrow to denote a slope of -20 db/dec
110 110 210010
-20
-40Decade Decade
Initial Magnitude
RD = 0
Final slope is zero
Semi log
plot
– Start with the initial magnitude and use arrows to draw the asymptotic plot
110 110 210010
-20
-40
+ 20 dB/dec
dB
Phase plot :
Start with the pole / zero Plot
Initial angle :
Final angle :
-10 -8 -6 -4 -2 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real AxisIm
ag A
xis
Use the negative of the poles and the zeros on the jw axis (zero at -1 and pole at -10)
Use an “up“ arrow to denote a slope of +45 deg/dec and a “down“ arrow to denote a slope of -45 deg/dec
Frequency (rad/sec)
110 110 210010
45
90Decade
Initial Phase Final Phase
- 45 deg/dec 45 deg/dec
deg
The actual Bode plot is obtain using the following code
% (s+1)/(s+10)
figure;
num = [1 1];
denum = [1 10];
SYS=tf(num,den);
bode(SYS);
Bode plot construction
Another example : This time sketch the Bode plot for
In Bode form
Magnitude PlotMagnitude Plot● Initial Magnitude (note that n=1) Calculate the intercept
Initial slope = -20.n dB/dec = -20 dB/dec
Plot the negative poles and zeros on the jw axis and use “up“ arrows to denote a slope of -20 dB/dec and “down“ arrows to denote a slope of +20 dB/dec
Finally use the initial value and the arrows to draw the asymptotic plot
Mag
nitu
de (
dB)
Frequency (rad/sec)
110 210010
-40
-20
-20 dB/dec
-20 dB/dec
-40 dB/dec
-40 dB/dec-60
0
20
Phase plot :
Start with the pole / zero Plot
Initial angle :
Final angle :
Use the negative of the poles and the zeros on the jw axis (zero at -10 and poles at 0,-1,-100)
Use an “up“ arrow to denote a slope of +45 deg/dec and a “down“ arrow to denote a slope of -45 deg/dec
Pha
se (
deg)
Frequency (rad/sec)
110 110 210010
-90
-45
-135
-180
Initial Phase
Final Phase
- 45 deg/dec
- 45 deg/dec
310
-225
The actual Bode plot
figure;
num = [10 10];
denum = [1 101 100 0];
SYS=tf(num,den);
bode(SYS);
Bode Plot for repeated roots
Sketch the Bode plot of
In Bode form
Magnitude PlotMagnitude Plot● Initial Magnitude (note that n=0)
dB conversion
Plot the negative poles on jw axis (2 poles at )
use “up“ arrows to denote a slope of -20 dB/dec and “down“ arrows to denote a slope of +20 dB/dec
Finally use the initial value and the arrows to draw the asymptotic plot
Bode Diagramsn n1010/n
-20
-40
Decade20
0
n100
Initial Magnitude ( )
2
1log20
n
- 40 dB/dec
Phase plot :
Start with the pole / zero Plot
Initial angle :
Final angle :
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Real Axis
Imag
Axi
s
Use the negative of the poles the jw axis
Use an “up“ arrow to denote a slope of +45 deg/dec and a “down“ arrow to denote a slope of -45 deg/dec
n
-90
-180
90
0
-270
Initial PhaseFinal Phase
n 10 n 10010
n
The actual Bode plot when
figure;
num = [1];
denum = [1 2 1];
SYS=tf(num,den);
bode(SYS);
Example
Use the Bode plot to determine the stability of a closed loop system
-120 -100 -80 -60 -40 -20 0 20-100
-80
-60
-40
-20
0
20
40
60
80
100
Real Axis
Imag
Axi
s
dj
djc 100 10
Transfer Function
Root locus form of the denominator
For the closed loop system is marginally stable. At the closed loop poles are at (from Root Locus)
Angle Condition
Magnitude Condition
Bode Solution for finding and
Note that
Might not be easy to solve for and
● A Bode plot is the graph of magnitude and phase of the transfer function with respect to
– Hence we can directly obtain (read) the value of the desired frequency from the phase plot
– Using the desired frequency we can read the corresponding value of the magnitude of the transfer function and obtain the value of as
or in “normal“ units
in dB
● The Bode plot gives us a graphical way of finding the values of the and
● Specific to our problem, the closed loop system is defined by
Construct the Bode Plot
-120
-100
-80
-60
-40
-20
0
10-1
100
101
102
103
-250
-200
-150
-100
-50
0
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
- 40 dB
-180 deg
20d
Then using
That is for the stability of the system we need
Bode Compensator Design
● Similar to the root locus design we can use the characteristic equation to examine the stability and/or the performance of the closed loop system using Bode plot
10-1
100
101
102
103
Frequency (rad/sec)
Ph
ase
(deg
)M
agn
itu
de
(dB
)
0
-180
|KH ( jωφ )|
∠KH ( jωp )
ωφω p
: Gain cross over freq.● The value of frequency
when
: phase cross over freq.● The value of frequency
when
Gain margin and Phase margin
● These points on the Bode plot leads us to 2 new definitions
Phase Margin
Gain Margin
The system is unstableif this value is less than zero
Bode plot – Root locus● The Bode plot parameters are related to the root locus
parameters according to
● Phase margin is related to the damping ratio as follows
Note that damping ratio is a measure of relative stability
Root locus design parameters
Damping ratio
Natural frequency
for
Hence phase cross over frequency is a measure of relative stability as well
Bode Design Example
Given
with Find K so that the phase margin is 45 degrees for the closed loop system
Solution :
Design Steps
● Find the characteristic equation in the form
● Plot and asuming
● From calculate the desired
i.e.
● From the desired , and the plot, find the desired
● Raise or lower the magnitude plot so that it crosses the 0 dB level at the desired value of
● Find the shift between the desired plot and the original amgnitude plot at the desired value of
● Calculate the value of using
Desired + original = shift in dB
● Step 1 :
● Step 2 :
● Step 3 :
Desired
● Given the desired value find from the plot
● Step 5 :
● Step 6 :
The difference between the desired magnitude plot and the actual magnitude plot is -9 dB , thefore
● Step 7 :
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