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CHAE 3: ACE FE

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INTRODUCTION TO FILTER

Gain GainGainGain

A filter is a circuit that passes certain frequencies and rejects all others. The

passband is the range of frequencies allowed through the filter.

Below show the basic filter response:

f fff

H B B

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BASIC LOW-PASS FILTER

.

.

0 dB3 dB

Gain (normalized to 1)

Actual response of a

-70.7%

f

BW

20 dB

10fc

40 dB

60 dB0.1fc fc0.01fc 100fc 1000fc

Passband

Transition

region

Stopband

region

20dB/decad

e

VoutR

Vs C

C ,

( 1)

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BASIC HIGH-PASS FILTERThe high-pass filter passes all frequencies above a critical frequency and

rejects all others. The simplest high-pass filter is a passive RCcircuit with the

output taken across R.

0 dB3 dB

Gain (normalized to 1)

Actual response

f

20 dB

fc

40 dB

60 dB 0.01fc 0.1fc0.001fc 10fc 100fc

of a single-pole

RCfilter

20

dB/decad

e Vout

RVs

C

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BASIC BAND-PASS FILTERA band-pass filterpasses all frequencies between two critical frequencies.

The bandwidth is defined as the difference between the two critical

frequencies. The simplest band-pass filter is an RLCcircuit.

0.707

1

Vout(normalized to 1)

VoutR

BW

fc1 f0 fc2

f

s

12 cc ffBW =

120 cc fff =W

fQ 0=

Quality Factor, Q

-as indicator in filter selection.

-higher Q, BW small and goodfor f0 selectivity and vice versa.

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BASIC BAND-STOP FILTERA band-stop filterrejects frequencies between two critical frequencies; the

bandwidth is measured between the critical frequencies. The simplest band-stop

filter is anRLC

circuit.

3

Gain (dB)

0

VoutC

L

fc1 f0 fc2f

BW

s

12 cc ffBW =

120 cc fff =

W

fQ 0=

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PASSIVE FILTER VERSUS ACTIVEFILTER

1. Passive filter contain passive components only; resistor, inductance and

capacitor.2. Active filter will combine active device such as transistor or operational amplifier

in designing filter.

3. The active device (op-amp) will provide the needed gain and the passive circuit

provide frequency selectivity.4. Four general types of active filter:

a. Low-Pass active filter.

b. High-Pass active filter.

c. Pass-Band active filter.d. Band-Stop active filter.

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FILTER RESPONSE CHARACTERISTICAv

C: ,

f

B:

F

20B//

B:

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THE DAMPING FACTOR (DF)The damping factor primarily determines if the filter will have a Butterworth, Chebyshev,

or Bessel response.

1

2

2DF=

The term pole has mathematical

+

Frequency-

selective

RCcircuit

Vin

Vout

s gn cance w e g er eve

math used to develop the DFvalues.

For our purposes, a pole is the

number of non-redundant reactive

elements in a filter. For example, a

one-pole filter has one resistor andone capacitor.

R1

R2

Amplifier

Negative feedback circuit

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GENERAL ACTIVE FILTER

+

R1

Frequency-selective

RCcircuit

Vin

Amplifier

Vout

feedback circuitto determine the

gain of filter and

damping ratioR2

Negative feedback circuit

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TRANSFER FUNCTION ACTIVE FILTER F .

:

F

()

0()

0() () 0() ().

=, 0()=H().() .

)(

)()( 0

sV

sVsHi

=

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1ST ORDER LOW-PASS ACTIVE FILTER

0

00)()(

1

11)(

+==

+

+=

s

AsHsA

sRCR

RsA

v

A

Bv

+=

1

1

0

0

sA

RCf

RC

R

RA

CHCH

A

B

2

1@

1

1

0

0

===

+=

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1ST ORDER LOW-PASS ACTIVE FILTERC : DC +

.

AB RRR ||=

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1ST ORDER LOW-PASS ACTIVE FILTERF :

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1ST

ORDER LOW-PASS ACTIVE FILTERE 1:

F 1 .

) F , H.) F B .

) F G, A0 .

) GB ,

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1st ORDER HIGH-PASS ACTIVE FILTER

+

+=sRC

R

R

AA

B

SV 11

1

1)(

+=+==

s

As

s

sHsAv0

0

0

0

1)()(

Cf

C

R

RsA

CLCL

A

B

2

1@

1

1)(

0

0

===

+=

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1st ORDER HIGH-PASS ACTIVE FILTERF :

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1st

ORDER HIGH-PASS ACTIVE FILTERE 2:

F 1

C=20H) F .

) F A .

) F G, A0 (B)

,

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2ND

ORDER LOW-PASS ACTIVE FILTERB 2

C C .

F 1=2=, C1=C2=C

( )

++== 1)3(2

sCRAsCRA

VVA

O

O

i

OV

BO

RA += 1

Cfo

2

1=

A

B

R

R=

2

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2ND

ORDER LOW-PASS ACTIVE FILTER,

B, C B.

C , , ,CH(3B )

RC

kfkf LPoLPCH

2==

B 1.732 0.785

B 1.414 1

C (1B) 1.054 1.238

C (2B) 0.886 1.333

C (3B) 0.776 1.390

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2ND

ORDER LOW-PASS ACTIVE FILTERC

AB

ABAB

RRRRRRR

+==||2

/=40B/

G ,B

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2ND

ORDER LOW-PASS ACTIVE FILTERE: F 2 . B

, :

) .) C , CH

) G, A (B)

)

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BAND-PASS ACTIVE FILTERE 1: CACADG A AD HGHA ACE FE

H

11111122 CR

k

CCRR

kf HPHPCL

==

: H :

22222222 CR

k

CCRR

kf HPHPCH

==

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BAND-PASS ACTIVE FILTER

H

G (B)

F (H)C CH

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BAND-PASS ACTIVE FILTERE 2: E FEEDBAC BADA ACE FE

21231 )||(21

CCRRRfo

=

C , :

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BAND-PASS ACTIVE FILTER

321

31

2

1

RRR

RR

fo+

=

C1= C2= C ( )

Q

CRfo 2

1

=

ooCAf21

Cf

QR

o

=2

)2(2 23

oo AQCfQR

=

2Q

o

==

CR

BW

2

1

=

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BAND-PASS ACTIVE FILTERF :

G

(B)

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BAND-STOP ACTIVE FILTERE FEEDBAC

BADA ACE

FE

3

Gain (dB)

fc1

f0 f

c2

f

0

BW

0.707

1

Vout(normalized to 1)

BW

fc1 f0 fc2

f