Ch. 12 Active Filters Part 1 1ECES 352 Winter 2007

Active Filters

* Based on use of amplifiers to achieve filter function

* Frequently use op amps so filter may have some gain as well.

* Alternative to LRC-based filters* Benefits

Provide improved characteristics

Smaller size and weight Monolithic integration in IC Implement without inductors Lower cost More reliable Less power dissipation

* Price Added complexity More design effort

dBin)T(20log)A(

1)A(for )A(function n Attenuatio

or

dBin)T(20log)G(

1)A(for )G(function Gain

:as expressed becan Magnitude

)(

)(

sV

sVsT

i

o

Transfer Function

Vo(s)Vi(s)

Ch. 12 Active Filters Part 1 2ECES 352 Winter 2007

Filter Types* Four major filter types:

Low pass (blocks high frequencies)

High pass (blocks low frequencies)

Bandpass (blocks high and low frequencies except in narrow band)

Bandstop (blocks frequencies in a narrow band)

Low Pass High Pass

Bandpass Bandstop

Ch. 12 Active Filters Part 1 3ECES 352 Winter 2007

Filter Specifications

* Specifications - four parameters needed Example – low pass filter: Amin, Amax, Passband, Stopband

Parameters specify the basic characteristics of filter, e.g. low pass filtering Specify limitations to its ability to filter, e.g. nonuniform transmission in

passband, incomplete blocking of frequencies in stopband

Ch. 12 Active Filters Part 1 4ECES 352 Winter 2007

Filter Transfer Function

* Any filter transfer function T(s) can be written as a ratio of two polynomials in “s”

* Where M < N and N is called the “order” of the filter function Higher N means better filter performance Higher N also means more complex circuit implementation

* Filter transfer function T(s) can be rewritten as

where z’s are “zeros” and p’s are “poles” of T(s) where poles and zeroes can be real or complex

* Form of transfer function is similar to low frequency function FL(s) seen previously for amplifiers where A(s) = AMFL(s)FH(s)

oN

NN

oM

MM

M

bsbs

asasasT

....

....)(

11

11

N

MM

pspsps

zszszsasT

....

....)(

21

21

Ch. 12 Active Filters Part 1 5ECES 352 Winter 2007

First Order Filter Functions

0

11

0

1)(

s

a

asa

s

asasT

o

o

* First order filter functions are of the general form

Low Pass

High Pass

a1 = 0

a0 = 0

Ch. 12 Active Filters Part 1 6ECES 352 Winter 2007

First Order Filter Functions* First order filter functions are of the form

0

11

0

1)(

s

a

asa

s

asasT

o

o

General

All Pass

a1 0,a2 0

Ch. 12 Active Filters Part 1 7ECES 352 Winter 2007

Example of First Order Filter - Passive* Low Pass Filter

00

0

2

2/1

2

001

0

0

0

log20,For

01,For

1log10

1

1log20)(T

0a

thfilter wi pass) (loworder first a of form thehas This

1

1

1

)/1(

11

1

1

11

1

)(

)()(

T

dBT

dBin

a

RCwhere

ss

RCs

sRCsC

R

sCRZ

Z

sV

sVsT

o

o

o

C

C

i

o

0 dB

Ch. 12 Active Filters Part 1 8ECES 352 Winter 2007

20 log (R2/R1)

Example of First Order Filter - Active* Low Pass Filter

01

20

121

20

2

1

2

2/1

21

2

01

201

2

0

0

1

2

0

1

2

2

1

2

21

2

1

2

1

2

11

2

log20log20,For

0log20,For

1log10log20

1

1log20log20)(T

0a

thfilter wi pass) (loworder first a of form thehas This

1

1

1

)/1(

11

1

)1(

)/1(

1

)(

)()(

R

RT

RRfordBR

RT

R

R

R

RdBin

R

Ra

CRwhere

sR

RsR

R

CRsR

R

CsRR

R

R

sCR

R

ZR

RI

ZRI

sV

sVsT

o

o

o

CCo

i

o

V_= 0

Io

I1 = Io

Gain Filter function

Ch. 12 Active Filters Part 1 9ECES 352 Winter 2007

Second-Order Filter Functions

o

o

bsbs

asasasT

1

21

22)(

* Second order filter functions are of the form

which we can rewrite as

where o and Q determine the poles

* There are seven second order filter types:Low pass, high pass, bandpass, notch,Low-pass notch, High-pass notch andAll-pass

20

02

12

2)(

sQ

s

asasasT o

2

0

02121 41

22,, Q

Qj

Qpp PP

js-plane

o

x

x

Qo

2

This looks like the expression for the new poles that we had for a feedback amplifier with two poles.

Ch. 12 Active Filters Part 1 10ECES 352 Winter 2007

Second-Order Filter Functions

Low Pass

High Pass

Bandpass

a1= 0, a2= 0

a0= 0, a1= 0

a0= 0, a2= 0

20

02

12

2)(

sQ

s

asasasT o

Ch. 12 Active Filters Part 1 11ECES 352 Winter 2007

Second-Order Filter Functions

Notch

Low Pass Notch

High Pass Notch

a1= 0, ao = ωo2

a1= 0, ao > ωo2

a1= 0, ao < ωo2

20

02

12

2)(

sQ

s

asasasT o

All-Pass

Ch. 12 Active Filters Part 1 12ECES 352 Winter 2007

Passive Second Order Filter Functions

* Second order filter functions can be implemented with simple RLC circuits

* General form is that of a voltage divider with a transfer function given by

* Seven types of second order filters High pass Low pass Bandpass Notch at ωo

General notch Low pass notch High pass notch

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

Ch. 12 Active Filters Part 1 13ECES 352 Winter 2007

* Low pass filter

Example - Passive Second Order Filter Function

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

General form of transfer function

20012

22

2

22

0a,0a

thfilter wiorder second a of form thehas This

1

11

1

1

1

11

1

1

1)(

111

111

)(

)()(

a

L

CRRCQand

LCwhere

sQ

sLCRC

ss

LC

LCsRLs

sRCR

sLRsRC

sRCR

sL

sRCR

ZRZ

RZsT

sosRC

R

RsC

RZ

RZwhere

ZRZ

RZ

sV

sVsT

oo

oo

o

LC

C

C

C

LC

C

i

o

T(dB)

0

01

0

)(22

2

sas

jsasQ

sas

sQ

ssT o

oo

o

0 dBQ

Ch. 12 Active Filters Part 1 14ECES 352 Winter 2007

Example - Passive Second Order Filter Function* Bandpass filter

01

a,0a

thfilter wiorder second a of form thehas This

1

1

11

1

11

1)(

111

111

)(

)()(

012

222

2

2

2

2

aQRC

L

CRRCQand

LCwhere

sQ

s

sRC

LCRCss

RCs

LCsRsL

sL

LCs

sLR

LCs

sL

RZZ

ZZsT

soLCs

sL

sLsC

ZZ

ZZwhere

RZZ

ZZ

sV

sVsT

o

oo

oo

LC

LC

LC

LC

LC

LC

i

o

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

General form of transfer function

00

1

0

)(

2

2

22

0

sass

jsat

sass

s

sQ

s

Qs

sT

o

o

oo

T(dB)

0

0 dB

-3 dB

Qo

Ch. 12 Active Filters Part 1 15ECES 352 Winter 2007

Single-Amplifier Biquadratic Active Filters* Generate a filter with second order

characteristics using amplifiers, R’s and C’s, but no inductors.

* Use op amps since readily available and inexpensive

* Use feedback amplifier configuration Will allow us to achieve filter-like

characteristics

* Design feedback network of resistors and capacitors to get the desired frequency form for the filter, i.e. type of filter, e.g bandpass.

* Determine sizes of R’s and C’s to get desired frequency characteristics (0 and Q), e.g. center frequency and bandwidth.

* Note: The frequency characteristics for the active filter will be independent of the op amp’s frequency characteristics.

Example - Bandpass Filter

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

General form of transfer function

Ch. 12 Active Filters Part 1 16ECES 352 Winter 2007

Design of the Feedback Network

* General form of the transfer function for feedback network is

* Loop gain for feedback amplifier is

* Gain with feedback for feedback amplifier is

* Poles of feedback amplifier (filter) are found from setting

)(

)()(

sD

sN

V

Vst

b

a

)(

)()(

sD

sNAAstAf

Ast

A

A

AA

ff )(11

0)(

)()(

sincefilter theof poles theion)approximat good a (to

becomecircuit feedback from t(s)of zeros theSo

1since01

)(

0)(1

sD

sNst

AA

st

orAst

Conclusion: Poles of the filter are the same as the zeros of the RC feedback network !

Design Approach: 1. Analyze RC feedback network to find expressions for zeros in terms R’s and C’s.2. From desired 0 and Q for the filter, calculate R’s and C’s. 3. Determine where to inject input signal to get desired form of filter, e.g. bandpass.

Ch. 12 Active Filters Part 1 17ECES 352 Winter 2007

Design of the Feedback Network* Bridged-T networks (2 R’s and 2C’s)

can be used as feedback networks to implement several of the second order filter functions.

* Need to analyze bridged-T network to get transfer function t(s) of the feedback network. We will show that

* Zeros of this t(s) will give the pole frequencies for the active filter..

4321413231

2

4321321

2

1111

1111

)(

RRCCRCRCRCss

RRCCRCCss

V

Vst

b

a

Bridged – T network

b

a

V

Vst )(

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

General form of filter’s transfer function

Ch. 12 Active Filters Part 1 18ECES 352 Winter 2007

Analysis of t(s) for Bridged-T Network

Vb

Va

I3 = (Vb-Va)/R3

I2 = I3

Ia = 0I1

for t(s)result final get the wegRearrangin

11

11111

1

11111

1

11

11

11

1

. and of in terms and find Now

V and V of in terms I and I findingby Begin

)(

2323

2433132341

1211

24333234

3243234241

2432344

124

2323232212

41

332

33

ba32

CsR

V

CsRV

CRsRRsC

V

RCsRRsC

V

VZIV

CRsRRV

RCsRRV

R

VV

CRsR

V

CsRR

VIII

CRsR

V

CsRR

V

R

VI

CsR

V

CsRV

sCR

VVVZIVV

VVII

R

VVIIand

R

VVI

V

Vst

ba

ba

Cb

ba

abba

ba

ba

abaCa

ba

abab

b

a

V12

I4

Analysis for t(s) = Va / Vb

4321413231

2

4321321

2

1111

1111

)(

RRCCRCRCRCss

RRCCRCCss

V

Vst

b

a

Ch. 12 Active Filters Part 1 19ECES 352 Winter 2007

4321413231

2

4321321

2

1111

1111

)(

RRCCRCRCRCss

RRCCRCCss

V

Vst

b

a

Analysis of Bridged-T Network* Setting numerator of t(s) = 0 gives zeroes

of t(s), which are also the poles of filter’s transfer function T(s) since

* Where the general form of filter’s T(s) is

* Then comparing the numerator of t(s) and the denominator of T(s), o and Q are related to the R’s and C’s by

* so

* Given the desired filter characteristics specified by o and Q, the R’s and C’s can now be calculated to build the filter.

20

02

12

2

)(

)()(

sQ

s

asasa

sV

sVsT o

i

o

4321

1

RRCCo

321

111

RCCQo

4

321

2121

213

4321

11

R

RCC

CCCC

CCR

RRCCQQ

o

o

.0)(

1)()(

)(11)(

)(

)(

sAstwhensTASo

Ast

A

A

AsT

sV

sVA

f

fi

of

These have the same form – a quadratic !