Frequency Response Methods Bode
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Bode Graphs
Bode graphs
Bode graphs are representations of the magnitude and phase of G(j*ω) (where the frequency vector ω contains only positive
frequencies). These consist of two graphs.
The first plot is a plot of log modulus i.e.|G(j ω)| (in decibels) versus frequency. The magnitude of |G(j ω)| of any transfer function for any
value of ω is plotted on a logarithmic scale in decibels where db = 20log10| P(j ω) |
The second plot is the phase shift (in degrees) versus frequency (ω).
Both plots usually have the frequency (ω) in logarithmic scale from 0 to
∞.
Producing Bode Plots
It is very easy to produce Bode plots by hand or by using proprietary software packages such as Matlab. Links below show how Bode plots
can be produced using Excel and using Mathcad. The example plots below have been produced using Mathcad
When considering relatively complicated transfer functions which are the product of a number of terms i.e
The bode magnitude plots can be obtained by adding the bode magnitudes (20Log10|G(iω)|) plots for the individual factors i.e
20 log10|G(ω)| = 20 log10|100| + 20 log10|(1 +j ω| + 20 log10|(2 +j 3ω| + 20 log10| /(2 +j ω| + 20 log10| 1 / (3 - j ω| + 20 log10|
1 / (2 + j 2ω|
Arg G(ω) = ArgG(100) + ArgG((1 +j ω) + ArgG((2 +j 3ω) + ArgG(1 /(2 +j ω) + ArgG( 1 / (3 - j ω) + Arg(G( 1 / (2 + j 2ω)
Relative stability assessments Using Bode Plots
Two relative stability indicators "Gain Margin" and "Phase Margin" are easily obtained from Bode Plots. As 0 db corresponds to a
magnitude of 1 the gain margin is the number of db that |G(iω)| is below 0 db at the phase frequency when the phase arg(G(iω)= 180o
(Phase crossover). The phase margin in the number of degrees arg(G(iω) is above -180o frequency associated with a gain |G(iω)|= 1 (0
db) (Gain crossover)
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Frequency Response Methods Bode
If the gain margin and phase margin are both positive the system will be stable. The stability should also be checked using Nyquist or
other noted methods for complete certainty.
These are illustrated by plotting two typical Bode plots together as shown below
Typical Bode Plots
A small number of typical bode plots are provided to indicate the various shapes. Relatively complicated bode plots can easily be produced
by combining the plots below and adding the appropriate log values for constant products and divisors...
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Frequency Response Methods Bode
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Frequency Response Methods Bode
Many control systems are described by a general second order differential equation..
The resulting frequency response function and the associated normalised bode plots are show below
Sites & Links For
Control
Information
1. Michigan Tech Uni.
Paper..Tutorial On
using Mathcad to
create Bode &
Nyquist Plots
2. Michigan Tech Uni.
Paper..Tutorial On
using Excel to
create Bode &
Nyquist Plots
3. U of
Massachusetts ..
Maths Course
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Frequency Response Methods Bode
Materials Includes notes on
Frequency
Response
Methods
4. Control Hut ..
Download.. Control
Design Frequency
Domain- Course
Notes
5. Bode Plots..Very
clear pages
explaining Bode
Plots and how to
produce them
..Page is being constructed..
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Last Updated 14/02/2008
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