An Introduction to Bode Plots - Welcome to George Mason University

Dr. Guy O. Beale, ECE, George Mason University, Fairfax, VA ECE 220 Bode Plots - 1

An Introduction toBode Plots

ECE 220, Signals and Systems IMonday, November 14, 2005

Prof. Beale


Outline

What Bode plots are and how they are used.

Bode plot magnitudes: actual and asymptotic.

Bode plot phases: actual and asymptotic.

Example of constructing Bode plots.


Bode Plots and Their Use

Bode plots are a means of showing how a system responds to sinusoidal input signals.Other signals can be expressed in terms of sums or integrals of sinusoidal signals.In many applications, the frequency domain provides a more convenient way of describing the interaction between signals and systems than the time domain.



Bode plots consist of 2 graphs:• Magnitude vs. frequency;• Phase vs. frequency.

Magnitudes are often expressed in decibels (db) so that a large range of magnitudes may be easily shown on the graph.

10( ) 20log ( )db AbsVal

H j H jω ω=



Phase is generally expressed in degrees. It could also be expressed in radians.

Frequency (in Hz or rad/sec) is generally plotted on a logarithmic scale so that a large frequency range can be used while still maintaining sufficient detail.




Forms of the Transfer Function

Pole-zero format:

Bode or time constant format:

( )( )( )( )( )

1 22 2

1 2

( )2 n n

K s z s zH s

s s p s p s sζω ω+ +

=+ + + +

( ) ( )( ) ( ) ( ) ( )

( )

1 22

1 2

21 2 1 2

1 1( )

1 1 2 1

/

B

n n

B n

K s z s zH s

s s p s p s s

K Kz z p p

ω ζ ω

ω

⎡ ⎤ ⎡ ⎤+ +⎣ ⎦ ⎣ ⎦=⎡ ⎤⎡ ⎤ ⎡ ⎤+ + + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦

=


Forms of the Transfer Function

In the Bode format, each term in the transfer function except KB/s evaluates to 1 when s = 0.• Except for KB/s, each magnitude evaluates to 1

when s = 0;• Except for KB/s, each phase evaluates to 0 degrees

when s = 0.


Magnitude of a Single Real Zero

( ) ( ) ( )( ) ( )

( ) ( )

0 0 0

0

2 20

( ) 1 1

1

1

H s z s z s z

H j j z

H j z

ω ω

ω ω

= ⋅ + = +

= +

= +



At low frequencies:

At high frequencies

( )( ) ( ) ( )

0 0

10

IF , 1

1 20log 1 0 dbdb

z z

H j H j

ω ω

ω ω≈ ⇒ ≈ =

( )( ) ( ) ( ) ( )

0 0

0 10 0

IF , 1

20logdb

z z

H j z H j z

ω ω

ω ω ω ω≈ ⇒ ≈


High Frequency Slope

High frequency slope is 20 db/decade of frequency

( ) ( )1

1 10 1 0

LET

THEN 20logdb

H j z

ω ω

ω ω

=

≈

( ) ( )( ) ( ) ( ) ( )( ) ( )

2 1

2 10 2 0

2 10 1 0 10 1 0 10

2 1

NOW LET 10

THEN 20log

20log 10 20log 20log 10

20 db

db

db

db db

H j z

H j z z

H j H j

ω ω ω

ω ω

ω ω ω

ω ω

= = ⋅

≈

≈ = +

≈ +



At low frequencies, the magnitude is approximately constant at 0 db.At high frequencies, for every increase in frequency by a factor of 10, the magnitude increases by 20 db.The break between low and high frequencies occurs approximately at For the asymptotic (straight line) approximation, the break between low and high frequencies occurs exactly atThe maximum error is 3 db which occurs at the break frequency.

0.zω =

0.zω =



Phase of a Single Real Zero

( ) ( ) ( )( ) ( )( ) ( )

0 0 0

0

10

( ) 1 1

1

tan

H s z s z s z

H j j z

H j z

ω ω

ω ω−

= ⋅ + = +

= +

∠ =


Phase of a Single Real Zero

At low frequencies,

At high frequencies,

At 0 :zω =

( ) ( )1tan 0 0H jω −∠ ≈ =

( ) ( )1tan 90H jω −∠ ≈ ∞ =

( ) ( )1tan 1 45H jω −∠ = =

( )0 1:zω

( )0 1:zω


Asymptotic Approximation for the Phase

This is the most accurate three-line approximation.The maximum errors of 5.7 degrees occur at the two break points.

( ) ( )( )

( ) ( )

0

0

0 0

10 , 0

10 , 90

10 10 , slope of is 45 per decade of frequency

z H j

z H j

z z H j

ω ω

ω ω

ω ω

≤ ∠ =

≥ ⋅ ∠ =

< < ⋅ ∠



Magnitudes for Other Terms

Gain: The magnitude is constant at all frequencies at a value of Zero at s = 0: Slope of +20 db/decade at all frequencies with a magnitude of 1 (0 db) at Real Pole p0: Mirror image of a real zero; 0 db for frequencies less than p0, -20 db/decade slope for frequencies greater than p0.Pole at s = 0: Mirror image of a zero at s = 0; a slope of -20 db/decade at all frequencies with a magnitude of 1 (0 db) at

( )1020log .BK

1 / .r sω =

1 / .r sω =


Magnitudes for Other Terms

Complex conjugate poles:

• Magnitude of 1 (0 db) for frequencies• Slope of -40 db/decade for frequencies• Peak (resonant) magnitude and frequency.

( ) ( )21 , 12 1n ns s

ζω ζ ω

<+ +

;nω ω;nω ω

2

2

1 , 1 2 , 1 22 1

p r nM ω ω ω ζ ζζ ζ

= = − <−


Phases for Other Terms

Gain: Phase is constant at 0 degrees for KB > 0 and constant at 180 degrees for KB < 0.Zero at s = 0: Phase is constant at +90 degrees at all frequencies.Real Pole p0: Mirror image of a real zero; 0 degrees at low frequencies, -90 degrees at high frequencies, -45 degrees atPole at s = 0: Mirror image of a zero at s = 0; -90 degrees at all frequencies.

0 / .p r sω =


Phases for Other Terms

Complex conjugate poles:• 0 degrees at low frequencies;• -180 degrees at high frequencies;• -90 degrees atThe steepness of the change depends on the value of the damping ratio, The smaller the value, the steeper the change in phase angle.

/ .n r sω ω=

.ζ


Magnitudes and Phases

Assume KB > 0 and no poles or zeros in RHP.Let n be total number of poles, m be number of zeros, N be number of poles at s = 0.

( ) ( )

Frequency Slope of Magnitude Value of Phase0 -20 db/decade 90

20 db/decade 90N N

n m n m−

∞ − − − −


Example Transfer Function

In pole-zero format

In Bode or time constant format

( )( )( )( )( )2

50 1 2( )

0.3 8 0.2 16.01s s

H ss s s s s

+ +=

+ + + +

( )2

2.6025 1 12( )

1 1 10.3 8 4.00125 80.05

ssH s

s s s ss

⎛ ⎞+ +⎜ ⎟⎝ ⎠=

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦








Summary

Bode plots provide a convenient graphical description of the frequency response characteristics of a system.Application areas:• Amplifier design;• Filtering;• Control system analysis and design.

Documents

An Introduction to Bode Plots - Welcome to George Mason University