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

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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

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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

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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

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-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

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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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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

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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)

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How to Plot a Bode Diagram2. Differentiator (s)

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ω(rad/sec)

M(dB)

ω(rad/sec)

φ(deg)

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How to Plot a Bode Diagram3. Integrator (1/s)

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ω(rad/sec)

M(dB)

ω(rad/sec)

φ(deg)

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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

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