Lecture 21: Intro to Frequency Response 1.Review of time response techniques 2.Intro to the concept...
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Transcript of Lecture 21: Intro to Frequency Response 1.Review of time response techniques 2.Intro to the concept...
Lecture 21: Intro to Frequency Response
1. Review of time response techniques2. Intro to the concept of frequency
response3. Intro to Bode plots and their
construction ME
431,
Lec
ture
21
1
Time Response Review• Previously, we have determined the time
response of linear systems to arbitrary inputs and initial conditions
• We have also studied the character of certain standard systems to certain simple inputs
• Used algebra and root locus to place dominant closed-loop poles to give desired time response − τ, Mp, tp, ts, tr, etc.
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SYSTEM
2
Time Response Review• Advantage of pole-placement approach
is that the time response can be affected directly, easiest for canonical systems
• Disadvantage of the approach is that sometimes it is difficult to determine the effect of higher-order poles and of zeros, and how the system will respond to complex inputs
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3
Frequency Response Concept• Input sine waves of different
frequencies and look at the output in steady state
• If G(s) is linear and stable, a sinusoidal input will generate in steady state a scaled and shifted sinusoidal output of the same frequency M
E 43
1, L
ectu
re 2
1
sinR t sin( )Y t
G(s) 4
Frequency Response Concept• Two primary quantities of interest that
have implications for system performance are:1. The scaling
2. The phase shift
• Important for designing controllers, filters, choosing sensors, designing mechanical systems, etc.
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R
= magnitude of G at s=jω
= angle of G at s=jω
( )G j
( )G j
5
Frequency Response Analysis• Attenuation may be
• desired: noise, disturbances• undesired: commanded reference
input• Amplification can destabilize a system
(resonance)• Phase lag means information is delayed,
can hurt performance and also destabilize a system
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Frequency Response Concept• Different ways to present this
information:• Bode diagram (two graphs)
1. magnitude vs. frequency 2. phase vs. frequency
• Nyquist plotmagnitude vs. phase (polar)
• Nichols chartmagnitude vs. phase (rectangular)
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Bode Diagram Example
• Magnitude in decibels vs. frequency in rad/sec
• Phase in degrees vs. frequency in rad/sec
ME 431, Lecture 21
-50
-40
-30
-20
-10
0
10
Mag
nitu
de (d
B)
10-1 100 101 102-180
-135
-90
-45
0
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
Other Examples• Nyquist plot
• Nichols chart
How to Plot a Bode Diagram
• Approach #1: Point by PointSubstitute s=jω into G(s) and calculate magnitude and phase for a series of different frequencies ω
where
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10( ) in dB 20log ( )M G j ( ) in deg ( )G j
2 2( ) Re( ( )) Im( ( ))G j G j G j
1 Im( ( ))( ) tanRe( ( ))
G jG jG j
10
How to Plot a Bode Diagram• Approach #2: Use asymptotic
approximationsPlot straight-line approx of components, then addEx.can add Bode plots because of mathematical props
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ture
2110( )
10sG s
s
1 2 1 2magnitude ( ) ( ) ( ) ( )G j G j G j G j
1 2 1 2log ( ) ( ) log ( ) log ( )G j G j G j G j
1 2 1 2phase ( ) ( ) ( ) ( )G j G j G j G j
1(10)( )10
ss
11
How to Plot a Bode Diagram• Need a library of components• Constant gain (K)
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ω(rad/sec)
M(dB)
ω(rad/sec)
φ(deg)
12
How to Plot a Bode Diagram2. Differentiator (s)
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ω(rad/sec)
M(dB)
ω(rad/sec)
φ(deg)
13
How to Plot a Bode Diagram3. Integrator (1/s)
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ω(rad/sec)
M(dB)
ω(rad/sec)
φ(deg)
14
How to Plot a Bode Diagram4. Simple zero (Ts+1)
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ω(rad/sec)
M(dB)
ω(rad/sec)
φ(deg)
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How to Plot a Bode Diagram5. Simple pole (1/(Ts+1))
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ω(rad/sec)
M(dB)
ω(rad/sec)
φ(deg)
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How to Plot a Bode Diagram • Will do complex poles and zeros later (2nd order)
• Approach #2:1. Put into Bode form2. Sketch straight line approximations3. Add graphs4. Try to approximate curves M
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Example• Sketch Bode diagram for
1. Put into Bode form
2. Sketch components
100( )10sG s
s
100( )110 1
10
sG ss
1( ) (10)( ) 1 110
G s ss
a b c
101 1
10
s
s
Example (continued)M(dB) φ(deg)
M(dB) φ(deg)
Sketch Requirements
Magnitude plot
• Frequency where slope changes
• Slope of each line segment
• Magnitude of at least one frequency
Phase plot• Frequency where slope
changes• Do not need to identify
slopes, but magnitudes must be relative
• Limiting phase as frequency goes to zero and infinity
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Make sure to include the following elements in your hand sketches of Bode diagrams
20