lecture 19lecture 19 bode plot,,g , high pass filter ... 19.pdf · lecture 19lecture 19 bode...

Lecture 19Lecture 19Bode Plot, High Pass Filter, , g ,

Series Resonance

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.

First-Order Low Pass Filter

( ) ∠fH 011 o

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∠+

=+

=

BB

B

ffff

ffjfH

arctan11 2

⎠⎝

( )( )21

1

fffH = ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=∠

fffH arctan


( )21 Bff+ ⎠⎝ Bf

Bode Plot for a First-Order Low-Pass Filter

A B d l t h th it d f t kA Bode plot shows the magnitude of a network function in decibels versus frequency using a l ith i l f flogarithmic scale for frequency.


Bode Plot for a First-Order Low-Pass Filter

( ) 1fH ( )( )

( ) ( )

2

1

1 BfffH

⎟⎞

⎜⎛

+=

( ) ( )( )

( )2

2

1l20)1l (20

1

1log20log20B

dB

ff

fffHfH

⎟⎞⎜⎛

⎟⎟⎟

⎠⎜⎜⎜

⎝ +==

( )

( )2

2

1log20

1log20)1log(20

B

B

ff

ff

⎟⎠⎞⎜

⎝⎛ +−=

⎟⎠⎞⎜

⎝⎛ +−=

( )( ) 2/121log20 Bff+−=

⎠⎝


( )( )21log10 Bff+−=

Asymptotic Behavior of Magnitudefor Low and High Frequencies

( ) ⎥⎤

⎢⎡

⎟⎟⎞

⎜⎜⎛

+−=ffH 1log10

2

( )

( ) =−≈<<

⎥⎥

⎦⎢⎢

⎣⎟⎟⎠

⎜⎜⎝

+

B

B

fHffFor

ffH

0)1log(10

1log10dB

( )

( ) ⎟⎟⎞

⎜⎜⎛

−=⎟⎟⎞

⎜⎜⎛

−≈>>

≈<<

B

B

fffHffFor

fHffFor

log20log10

0)1log(102

dB

( ) ⎟⎟⎠

⎜⎜⎝

⎟⎟⎠

⎜⎜⎝

≈>>BB

B fffHffFor log20log10dB


Magnitude Bode Plot for First-Order Low Pass Filter


Asymptotic Behavior of Phasefor Low and High Frequencies

( ) ( ) ( )B ffffffjfH

∠

∠=

+=

)t (1

011

12

o

( ) ( )( ) B

BBB

fffHffffffj

−=∠

∠++

)arctan()arctan(11 2

B

B

ffff

>>−=<<=

900

B

B

ff =−= 45


1. A horizontal line at zero for f < fB /10.

2. A sloping line from zero phase at fB /10 to –90° at 10fB.

3 A h i t l li t 90° f f > 10f


3. A horizontal line at –90° for f > 10fB.

Exercise 6.11

Sketch the Bode plots for the low-pass filter shown above:

HzxRC

fB 1000)101(10002

12

16

=⎟⎞

⎜⎛

==−ππ


x )101(2

2 ⎟⎠

⎜⎝ π

π

Exercise 6.11

1( )( )210001

1

ffH

+= ( ) ⎟

⎠⎞

⎜⎝⎛−=∠1000

arctan ffH


First-Order High-Pass Filterg

fRCjfRCj

jjR

Rinininout π

π12

211

11 +

=−

=−

= VVVV

fwhereffjfRC

jfC

jR

BBout

ππ1)/(

21

2

==V


fRCfwhe e

ffj BBin π2)/(1+V


( ) ( )ffff 90∠ oV( ) ( )( )

( )( ) ( )BB

B

B

B

ffff

ffffj

ffjfHarctan1

901 2in

out

∠+

∠=

+==

o

VV

( ) ( )( )B

B

ff

fffH1 2+

=( )

( ) ( ) ( )BB

ffff

fH arctan90arctan

90−=

∠∠

=∠ oo

( )Bff


First-Order High-Pass Filter

( ) ( )Bff ( ) ( )fffH 90∠ o

g

( ) ( )( )21 B

B

ff

fffH+

= ( ) ( )BfffH arctan90 −=∠ o



( ) BfffH =( )( )

( ) ( )

21

B

B

ff

fff

⎟⎞

⎜⎛

+

( ) ( )( )21

log20log20B

BdB

ff

fffHfH

⎞⎛

⎟⎟⎟

⎠⎜⎜⎜

⎝ +==

( )

( )( ) 2/12

2

1l20)l (20

1log20)log(20 BB

ffff

ffff ⎟⎠⎞⎜

⎝⎛ +−=

( )( )

( )( )2

2

1l10)l (20

1log20)log(20 BB

ffff

ffff +−=


( )( )21log10)log(20 BB ffff +−=

Asymptotic Behavior of First-Order High-Pass Filter

( ) ( )( )1l10)l (20 2fffffH ( ) ( )( )( ) )log(20

1log10)log(20 2

≈<<

+−=

fffHffFor

fffffH

BB

BB

( ) ( ) 0log10)log(20 2 =−≈>> fffffHffFor BBB

( ) o arctan90 ⎟⎟⎠

⎞⎜⎜⎝

⎛−=∠

fffHB

( )( ) o

o

0

90

≈∠>>

≈∠<<

fHffFor

fHffFor

B

B


( ) 0∠>> fHffFor B

Bode Plots for the First-Order High-Pass Filter

( ) ( )( )1log10)log(20 2+−= fffffH BB

( )( ) ( ) 0log10)log(20

)log(202 =−≈>>

≈<<

BBB

BB

fffffHffFor

fffHffFor

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

( ) o arctan90

0log10)log(20

)log(20

1log10)log(20

2

⎟⎟⎞

⎜⎜⎛

−=∠

=−≈>>

≈<<

+=

ffH

fffffHffFor

fffHffFor

fffffH

BBB

BB

BB

( )

( )( ) o

o

0

90

arctan90

≈∠>>

≈∠<<

⎟⎟⎠

⎜⎜⎝

∠

fHffFor

fHffFor

ffH

B

B

B

⎞⎛ f( ) ( )( )1log10)log(20 2+ fffffH ( )

( ) o

o

90

arctan90

≈∠<<

⎟⎟⎠

⎞⎜⎜⎝

⎛−=∠

fHffFor

fffH

B

B

( ) ( )( )

( ) )log(20

1log10)log(20

≈

<<

+−=

B

B

BB

fffHffFor

fffffH


( ) o0≈∠>> fHffFor B( ) ( ) 0log10)log(20 2 =−≈

>>

BB

B

fffffH

ffFor

Exercise 6.13

Find the transfer function for this circuit:Find the transfer function for this circuit:

2/)/(1

)/(/21

/22

2Bin

B

Bininout LRfwhere

ffjffj

RfLjRfLj

fLjRfLj

=+

=+

=+

= VVVV ππ

ππ

π

)/(1)/()( Bout

ffjffjfH

+==

VV


)/(1 Bin ffj+V

Exercise 6.14

R=1kΩ

Find the half power frequency and the capacitance for a high-Find the half power frequency and the capacitance for a highpass filter that reduces the amplitude of a frequency component at 1 kHz by 50dB


Exercise 6.14

dB50( ) decadesdecadedBdBdecadesdBdecadesdecadedB 5.2

/2050#50)(#/20 ==→=

What half-power frequency fB is 2.5 decades above 1 kHz?What half power frequency fB is 2.5 decades above 1 kHz?

fflog decades ofNumber 1

2⎟⎟⎠

⎞⎜⎜⎝

⎛=


kHzHzxHzfHz

f 3161031610)1000(1000

log5.2 35.22

2 ===→⎟⎠

⎞⎜⎝

⎛=

Exercise 6.14

HkHf 103163161 3 HzxkHzRC

fB

11

103163162

1 3===π

HzxRfC

B )10316)(1000(21

21

3Ω==

ππ

pFFx 50310503 12 == −


Computer-Generated Bode Plot


Computer-Generated Bode Plot

( ) ( )[ ]LjRCjRRRfH

ωω ++++=

231

3

11 ( )[ ]LjRCjRR ωω ++++ 231 11

Check the low and high frequency behavior to make


sure the computer generated Bode plot is correct

Low Frequency Behaviory

At low frequency the inductance is a short circuit

Th it i i itThe capacitance is an open circuit


Low Frequency Behaviory

( ) 500 3 ==RH ( ) 5.00

321

=++

=RRR

H

( ) ( ) dB60.520log0dB −==H ( ) ( )gdB


High Frequency Behaviorg y

At high frequency the inductance is an open circuit

Th it i h t i itThe capacitance is a short circuit

( ) 526303 ==∞RH


( ) 5263.031

=+

=∞RR

H

High Frequency Behaviorg y

( ) ( ) dB 575.55263.0log20dB −==∞H


MATLAB Code


MATLAB Output


Resonance

Resonance is a phenomenon that can beResonance is a phenomenon that can be observed in mechanical systems and electrical circuitselectrical circuits.


Series Resonance

ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.http://www.intuitor.com/resonance/circuits.html

Series Resonance

fjRfLjfZs π 12)( −+=

resonanceForfC

jfjfs π

11:

2)(

2

For resonance the reactance of the inductor and the capacitor cancel:

f

LCf

CfLf

πππ

1)2(1

212 2

20

00 =→=


LCf

π20 =

Series ResonanceQuality factor QS

Resistance resonance at inductance of ReactanceQs ≡

RLf

Resistance02π

=

LCffrom

CfLSubstitute

R

020

2 21

)()2(1

ππ==

CRfQ

LCCf

s

0

21

2)()2(

π

ππ

=


CRf02π

Series Resonance−+=

fCjfLjRfZs 2

12)(π

π

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

fRCRfLjR

2121

ππ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

⎥⎦⎢⎣ ⎠⎝

LCffff

RLf

jR

f

20

)2(12

1π

==

⎥⎦⎢⎣⎟⎠

⎜⎝

RLf

QandfSubstitute

LCfffR

s0

0

02

0

21

)2(

π

π

⎥⎤

⎢⎡

⎟⎟⎞

⎜⎜⎛

−+=ffjQRfZ

RQ

LCf s

0

0

1)(

2π


⎥⎥⎦⎢

⎢⎣

⎟⎟⎠

⎜⎜⎝

+ff

jQRfZ ss0

1)(

Series Resonance

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=∠⎟⎟

⎠

⎞⎜⎜⎝

⎛−+=

ff

ffQZ

ff

ffQ

RZ

sss0

202 arctan1


⎥⎦⎢⎣⎟⎠

⎜⎝

⎟⎠

⎜⎝ ffffR 00

lecture 19lecture 19 bode plot,,g , high pass filter ... 19.pdf · lecture 19lecture 19 bode...

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