wk 12 fr bode plot nyquist may 9 2016
TRANSCRIPT
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Chapter 5Week 12 -Frequency Response Analysis
Topics:
1. Frequency Response Introduction2. Bode Plot
3. Bode Plot Construction4. Nyquist Stability Criterion
5. Control System Design Using Frequency Response
Control System Engineering
PE-3032Prof. CHARLTON S. INAODefence Engineering College, Debre Zeit , Ethiopia
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Introduction to Frequency Response
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Terminologies- Frequency , Amplitude , Phase
• Frequency, or its inverse, the period- is the number of occurrences of a repeating event per unit time. The number of cycles per unit of time is called the frequency.
The hertz (symbol Hz) is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications, such as the frequency of musical tones. The word "hertz" is named for Heinrich Rudolf Hertz, who was the first to conclusively prove the existence of electromagnetic wave.
• Period is the inverse of frequency
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Bode Plot :Introduction
• The plot of magnitude as well as phase angle versus frequency may represent a sinusoidal transfer function.
• Hendrik Wade Bode used the logarithmic scale extensively for the study of the magnitude of the transfer function and the frequency variable. The logarithmic plot is called Bode Plot.
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• A Bode plot, named after Hendrik Wade Bode, is usually a combination of a Bode magnitude plot and Bode phase plot:
• A Bode magnitude plot is a graph of log magnitude against log frequency often used in signal processing to show the transfer function or frequency response of a linear, time-invariant system.
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Phase and Gain Cross Over Frequency
The gain cross over frequency is the frequency at which the magnitude of the open loop transfer function is UNITY.
The phase cross over frequency is the frequency at which the phase of the open loop transfer function is 180o.
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Illustration: Gain Margin and Phase Margin
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a
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b
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Bode Plot Constructio
n
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Constant term K
20 log K=20log 10=20 dB05/02/2023 18
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Zeros and poles at the origin
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Simple Zero
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Simple Pole
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Quadratics/2nd order
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Bode Plot Example
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S2/(0.004s2 +0.22 S +1)
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Frequency VS Magnitudeω (rad/sec) A(magnitu
de)0.5 -12 dB5 28 dB
50 48 dB100 48 dB
First corner frequency, ωc1
second corner frequency, ωc2
Chosen Lower limit frequency, ωL
Chosen pper limit limit frequency, ωh
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Semilog paper
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MATLAB
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Phase Margin 12.6 degrees
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Draw the asymptote of the Bode plot for the system having transfer function G(s)=10/s(0.1 s +1)
*****Asymptote Exercise******
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Magnitude Plot
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Factor 1: Magnitude and Phase
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Using MATLAB
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Factor 2 : Magnitude & Phase(1/s)
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Factor 3 : Magnitude & Phase : 1/(1+0.1s)
Corner frequency
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Using MATLAB : 1/(1+0.1s)
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Resultant Plot
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MATLAB: Resultant Plot
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Example: Bode diagram of the open loop systemsG(s)H(s) could be regarded as:
Then we have:101.0
1s11)(s10
)101.0()1(10)()( 22
sssssHsG
① ② ③ ④
0dB, 0o
1001010.1)(log
)( ),( L
③
④
②①
20dB, 45o
-20dB, -45o
-40dB, -90o
40dB, 90o
-80dB,-180o
-60dB.-135o
-40dB/dec
- 20dB/dec
20dB/dec
- 40dB/dec
- 20dB/dec
- 40dB/dec
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101.01
s
11)(s10
)101.0(
)1(10)()( 22
sss
ssHsG
23 101.0
10100)()(
ss
ssHsG
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Solved Problems
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Factor No. 1= 1/s (pole at the origin)=s-1
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Factor No. 2= 1/(1+0.5s)= (simple pole)=(1+0.5s)-1
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Factor No. 2= 1/(1+0.5s)= 1/(1+1/2s) = (simple pole)=(1+0.5s)-1
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Factor No. 3= 1/(1+0.1s)= (simple pole)=(1+0.1s)-1
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Factor No. 3= 1/(1+0.1s)= (simple pole)=(1+0.1s)-1
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Factor 4:Constant= K=10
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20 log K20log 10=20 X 1= 20 dB
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Factor 4 , K=10
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Phase Angles
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Computing the phase angle using MS Excel
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Using Matlab
s0.6s0.5s
10G(s)H(s) 23
Num=10Den=[0.5 0.6 1 0 ]r=tf(num,den)Bode(r)
commands
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s0.6s0.5s
10G(s)H(s) 23
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x
x
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(rad/sec)
dB MagPhase (deg)
1 1 1 1 1 1
wlg
This is a sheet of 5 cycle, semi-log paper.This is the type of paper usually used forpreparing Bode plots.
Semi log paper used in Bode Plots
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Nyquist Stability Criterion
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Week 12 Frequency Response AnalysisTopics:
1Nyquist Plot2.Nyquist Stability Criterion
Control System Engineering
PE-3032Prof. CHARLTON S. INAODefence Engineering College, Debre Zeit , Ethiopia
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Instructional Objectives
At the end of this lecture, the students shall be able to:1. Conduct review of Frequency Response
fundamentals2. Discuss the parameters in Nyquist Plot construction3. Understand Nyquist Stability Criterion
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Nyquist Plot
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Frequency(ω)
Magnitude Phase
0 1 0 deg
∞ 0 -90 deg
ω =1 1/√2 -45 deg
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ωT Frequency(ω) Magnitude Phase
0 0 1 0 deg
∞ ∞ 0 -90 deg
0.577 0.866 -30 deg
1ω =1 0.707 -45 deg
2 0.447214 -63.4396215
5 0.196116 -78.6958639
10 0.099504 -84.2956157
20 0.049938 -87.1440135
100 0.010000 -89.4336486
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Using matlab; Nyquist plot of 1/(s(s+1))
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STABILTY and Nyquist plot
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Stable to the right of unit circle
Unstable to the left of unit circle
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Summary
Gain Margin= reciprocal of the magnitude of the locus of frequency response as it first touch the real axis, i. e.,at the phase cross over frequency. Kg=1/G(jw)pc
Phase Margin= the angle through which the gain cross over line must be rotated to reach the real axis and pass through the unit circle(gain cross over frequency) (-1, j0). γ= 180 +φgc
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Exercises
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Homework/Assignment
Populate or make a complete table for phase angle, frequency and magnitude
Make a clean and neat plot using suitable scale