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

Introduction to Frequency Response

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

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.

• 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.

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.

Illustration: Gain Margin and Phase Margin

a

b

Bode Plot Constructio

n

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Constant term K

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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

S2/(0.004s2 +0.22 S +1)

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

Semilog paper

MATLAB

Phase Margin 12.6 degrees

Draw the asymptote of the Bode plot for the system having transfer function G(s)=10/s(0.1 s +1)

*****Asymptote Exercise******

Magnitude Plot

Factor 1: Magnitude and Phase

Using MATLAB

Factor 2 : Magnitude & Phase(1/s)

Factor 3 : Magnitude & Phase : 1/(1+0.1s)

Corner frequency

Using MATLAB : 1/(1+0.1s)

Resultant Plot

MATLAB: Resultant Plot

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

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

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

Factor 4:Constant= K=10

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20 log K20log 10=20 X 1= 20 dB

Factor 4 , K=10

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

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

Nyquist Stability Criterion

Week 12 Frequency Response AnalysisTopics:

1Nyquist Plot2.Nyquist Stability Criterion

Control System Engineering

PE-3032Prof. CHARLTON S. INAODefence Engineering College, Debre Zeit , Ethiopia

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

Nyquist Plot

Frequency(ω)

Magnitude Phase

0 1 0 deg

∞ 0 -90 deg

ω =1 1/√2 -45 deg

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

Using matlab; Nyquist plot of 1/(s(s+1))

STABILTY and Nyquist plot

Stable to the right of unit circle

Unstable to the left of unit circle

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

Exercises

Homework/Assignment

Populate or make a complete table for phase angle, frequency and magnitude

Make a clean and neat plot using suitable scale