4/26/2011 1 1 Frequency Response Frequency Response (Bode Plot) (Bode Plot) 2 Frequency Response Frequency Response Definition The frequency response of a system is defined as the steady state response of the system to a sinusoidal input signal. t sin A) t ( u o ω = y(t) ? 2 o 2 o s A) s ( U ω + ω = ∏ = + = n 1 i i p s ) s ( m ) s ( G ) s ( G ) s ( U 3 Frequency Response Frequency Response Response to sinusoidal input ) s ( U ) s ( G ) s ( Y= 2 2 n 1 i i s Ap s ) s ( m ) s ( Yω + ω + = ∏ = o * o o n n 1 1 j s j s p s p s ) s ( Yo ω − α + ω + α + + α + + + α = L Taking the inverse Laplace transform: ( ) φ + ω α + α + + α = − − t sin A2 e e ) t ( y o o t p n t p 1 n 1 L α α = φ − ) Re( ) Im( tan o o 1 4 Frequency Response Frequency Response If the system is stable, the steady state response is ( ) φ + ω = t sin AM ) t ( y o Where ( ) { } ( ) { } 2 o 2 o j s o ) j ( G Im ) j ( G Re ) s ( G ) j ( G M o ω + ω = = ω = ω = ω ω = φ − ) j ( G Re ) j ( G Im tan o o 1 In polar form φ = ω j o Me ) j ( G Both magnitude M and phase angle φ depend on the frequency of the sinusoidal input. Response to sinusoidal input
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Display of frequency response is a problem that hasDisplay of frequency response is a problem that hasbeen studied for a long time.been studied for a long time.
The most useful technique for hand plotting wasThe most useful technique for hand plotting wasdeveloped by H.W. Bode, so called as Bode Plot.developed by H.W. Bode, so called as Bode Plot.
The idea in BodeThe idea in Bode’’s method is to plot magnitudes method is to plot magnitude
curves using a logarithmic scale and phase curvescurves using a logarithmic scale and phase curvesusing a linear scale.using a linear scale.
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Bode Plot TechniquesBode Plot Techniques
The Bode plots consists of two graphs:The Bode plots consists of two graphs:
Magnitude M (GainMagnitude M (Gain ||G(jG(jωω))||) is plotted against) is plotted againstlog(log(ωω) again is represented as logarithmic gain in) again is represented as logarithmic gain indBdB
Phase anglePhase angle ∠∠G(jG(jωω) is plotted against log() is plotted against log(ωω))
Phase is usually measured in degreesPhase is usually measured in degrees
Glog20Gdb
=
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Bode Plot TechniquesBode Plot Techniques
Advantages of Bode Plots Advantages of Bode Plots
Dynamic compensator design can be basedDynamic compensator design can be based
entirely on Bode plots.entirely on Bode plots. Bode plots can be determined experimentallyBode plots can be determined experimentally
Bode plots of system in series simply add, whichBode plots of system in series simply add, whichis quite convenient.is quite convenient.
The use of a log scale permits a much widerThe use of a log scale permits a much widerrange of frequencies to be displayed on a singlerange of frequencies to be displayed on a singleplot than is possible with linear scales.plot than is possible with linear scales.
Bode Plot TechniquesBode Plot TechniquesThe primary advantage of the logarithmic plot is theconversion of multiplicative factor into additive factorby virtue of the definition of logarithmic gain.
Consider the open-loop transfer function
( )
( ) ( ) ( )∏ ∏
∏
= =
=
ω+ωζ++
+
=M
1m
R
1k
2nk
2m
N
Q
1i
i
k nk s2spss
zs
K )s(G
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Bode Plot TechniquesBode Plot Techniques
In working with frequency response, it is moreconvenient to replace s with jω and to write thetransfer function in the Bode-form:
Since the gain is plot in dB, we can treat each pole andzero individually, and then add up the results.
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Bode Plot TechniquesBode Plot TechniquesFurthermore the phase angle is
∑∑=
−
=
− ωτ−−ωτ+=ωφM
1mm
1N
1ii
1 tan)90(Ntan0)(
∑=
−
ω−
ω
ωζ−
R
1k
2
2n
nk 1
k
k 2
tan
Similarly, the phase angle due to each pole and zerocan be computed individually, and then the results isadded together.
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Bode Plot TechniquesBode Plot Techniques
Given the response of each basic term in theGiven the response of each basic term in thetransfer function, the algebraic sum wouldtransfer function, the algebraic sum would
yield the total response.yield the total response.
If we can make an approximation of eachIf we can make an approximation of eachterm we should be able to quickly sketch theterm we should be able to quickly sketch theBode diagrams.Bode diagrams.
2020
Bode Plot SummaryBode Plot SummaryAll transfer function for kinds of systems we havetalked about, so far are composed of three classes ofterms
• Ko(jω)n
• (jωτ+1)±1
•
We can determine the logarithmic magnitude plot andphase angle for these four factors then utilize themto obtain a Bode diagram for any general form of atransfer function.
Gain margin (GM): difference (in dB) betweenGain margin (GM): difference (in dB) betweenunity gain condition (0 dB)unity gain condition (0 dB) andand magnitudemagnitude of of frequency responsefrequency response @ phase cross@ phase cross--overoverfrequencyfrequency