E E 2415

E E 2415

Lecture 15Introduction to Frequency Response, Poles & Zeroes, Resonant Circuit

Low-Pass Filter Example: (1/2)

Low-pass Filter:

fo

inf

f

jCV

HjV RC

Low-Pass Filter Example: (2/2)

2

2

1tan

1o

o

H

Gain in Decibels

Using the Low-pass filter example:

Drops at 20 dbper decade

Bode Plot of Low-Pass Filter

Phase Plot of Low-Pass Filter

High-Pass Filter Example: (1/2)

fo

inf

f

RVH

jV RC

1

11

o o

o

o o

j j

Hj jj

High-Pass Filter Example: (2/2)

1

2

2

tan

1

o o

o

H

2

21

gaino

o

G H

1tan Phase ShiftoP

High-Pass Gain in Decibels

2

2

20

1

odb

o

G Log

0db oG

@20 /20 Rises db decadedb o

o

G Log

Bode Plot of High-Pass Filter

Phase Plot of High-Pass Filter

Definition: Poles & Zeroes

A zero at the origin

A pole at jw1

A zero at jw1

A pole at jw2

A pole at the origin

Effect of a Pole on the Bode Plot• A pole causes the

asymptotic slope to decrease by 20 db/decade.

• A pole at the origin causes the slope to start at –20 db/decade.

• A pole not at the origin causes a corner to appear at the pole’s frequency; then the slope is 20 db/decade less for frequencies greater than the pole’s frequency.

Effect of a Zero on the Bode Plot• A zero causes the

asymptotic slope to increase by 20 db/decade.

• A zero at the origin causes the slope to start at +20 db/decade.

• A zero not at the origin causes a corner to appear at the pole’s frequency; then the slope is 20 db/decade more for frequencies greater than the zero’s frequency.

Examples: (1/3)

A zero at the origin

A pole at jw1

Examples: (2/3)

A zero at jw1

A pole at jw2

A pole at the origin

Examples: (3/3)

Resonant Bandpass Filter (1/2)

Resonant Bandpass Filter (2/2)

Resonant BandPass Poles & Zeroes

Zero at origin

Two poles

Bode Plot for Resonant Bandpass

Phase Plot for Resonant Bandpass

Bandwidth of Resonant Bandpass (1/2)

at half power

Take square and reciprocal of both sides

Need both solutions for positive values of w

Bandwidth of Resonant Bandpass (2/2)

Positive w for -1

Positive w for +1

Bandwidth for a seriesresonant bandpassfilter