ELG4139: Op Amp-based Active Filters

• Advantages:

– Reduced size and weight, and therefore parasitics.

– Increased reliability and improved performance.

– Simpler design than for passive filters and can realize a wider range of

functions as well as providing voltage gain.

– In large quantities, the cost of an IC is less than its passive counterpart.

• Disadvantages:

– Limited bandwidth of active devices limits the highest attainable pole

frequency and therefore applications above 100 kHz (passive RLC

filters can be used up to 500 MHz).

– The achievable quality factor is also limited.

– Require power supplies (unlike passive filters).

– Increased sensitivity to variations in circuit parameters caused by

environmental changes compared to passive filters.

• For applications, particularly in voice and data communications, the

economic and performance advantages of active RC filters far outweigh

their disadvantages.

First-Order Low-Pass Filter

ffB

Bi

f

ffi

f

i

f

ff

f

f

f

ff

f

f

ff

i

f

i

o

CRf

ffjR

R

CfRjR

R

Z

ZfH

CfRj

RZ

R

CfRj

R

fCj

RZ

Z

Z

V

VfH

2

1

)/(1

1

21

1)(

21

21

2

1

111

)(

A low-pass filter with a dc gain of –Rf /Ri

dttvRC

tv

t

o in

0

1

First-Order High-Pass Filter

iiB

B

B

i

f

ii

if

i

i

ii

if

ii

f

i

f

ffi

ii

i

f

i

o

CRf

ffj

ffj

R

R

CRfj

CRfj

R

R

CRfj

CRfj

CfjR

R

Z

ZfH

RZCfj

RZ

Z

Z

v

vfH

2

1

)/(1

)/(

21

2

21

2

2

1)(

2

1

)(

A high-pass filter with a high frequency gain of –Rf /Ri

Higher Order Filters

n

B

n

i

fn

n

ffjR

R

fHfHfHfH

)/(1

1)1(

)()()()( 21

Single-Pole Active Filter Designs

High Pass Low Pass

)/1()/1(

1

1

11

1

RCs

s

RCsRC

sRC

sCR

sRC

sCR

v

v

in

out

RCs

RC

v

v

in

out

/1

/1

7

Two-Pole (Sallen-Key) Filters

-

+

+V

-V

R1

Rf1

Rf2

C1

vin

vout

C2

R2

-

+

+V

-V

R1

Rf1

Rf2

C2

vin

vout

R2

C1

Low Pass Filter High Pass Filter

8

Three-Pole Low-Pass Filter

-

+

+V

-V

R1

Rf1

Rf2

C1

vin

C2

R2

-

+

+V

-V

R3

Rf3

Rf4

C3

vout

Stage 1 Stage 2

9

Two-Stage Band-Pass Filter

R2

R1

vin

C1

C2

Rf1

Rf2

C4

C3

R3

R4

+V

-V

vout

Rf3

Rf4

+

-

+

-

+V

-V

Stage 1

Two-pole low-pass

Stage 2

Two-pole high-pass

BW

f1

f2

f

Av

Stage 2

response

Stage 1

response

fo

BW = f2 – f1

Q = f0 / BW

10

Multiple-Feedback Band-Pass Filter

R1

R2

C1

C2

vin

Rf

+V

-V

-

+v

out

11

Transfer function H(j)

Transfer

Function

)( jHV

oV

i

)(

)()(

jV

jVjH

i

o

)Im()Re( HjHH

22 )Im()Re( HHH

)Re(

)Im(tan 1

H

HH 0)Re( H

)Re(

)Im(tan180 1

H

HH o

0)Re( H

12

Frequency Transfer Function of Filters

H(j)

HL

HL

o

o

o

o

ffffjH

fffjH

ffjH

ffjH

ffjH

ffjH

and 0)(

1)(

Filter Pass-Band (III)

1)(

0)(

Filter Pass-High (II)

0)(

1)(

Filter Pass-Low (I)

response phase specific a has

allfor 1)(

Filter shift)-phase(or Pass-All (V)

and 1)(

0)(

Filter (Notch) Stop-Band (IV)

fjH

ffffjH

fffjH

HL

HL

Bode Plot

To understand Bode plots, you need to use Laplace transforms!

The transfer function of the circuit is:

1

1

/1

/1

)(

)(

sRCsCR

sC

sV

sVA

in

o

v

R

Vin(

s)

b

v

f

fj

RCfjRCjfA

1

1

21

1

1

1)(

where fc is called the break frequency, or corner

frequency, and is given by: RCf

c

2

1

14

Bode Plot (Single Pole)

o

jCRj

jH

1

1

1

1)(

2

1

1)(

o

jH

2

101011log20)(log20)(

o

dBjHjH

o

dBjH

10log20)(

For >>o

R

C VoV

i

Single pole low-pass filter

15

dBjH )(

(log)x

x

2x

10

6d

B2

0d

B slope

-6dB/octave

-20dB/decade

o

jH

10log20)(

For octave apart,

1

2

o

dBjH 6)(

For decade apart, 1

10

o

dBjH 20)(

16

Bode Plot (Two-Pole)

R1

R2

C1

C2

vi v

o

21 oo

2

2

2

1

10111log20)(

oo

jH

Corner Frequency

• The significance of the break frequency is that it represents the frequency where

Av(f) = 070.7-45

• This is where the output of the transfer function has an amplitude 3-dB below the input amplitude, and the output phase is shifted by -45 relative to the input.

• Therefore, fc is also known as the 3-dB frequency or the corner frequency.

Bode plots use a logarithmic scale for frequency, where a decade is

defined as a range of frequencies where the highest and lowest

frequencies differ by a factor of 10.

Magnitude of the Transfer Function in dB

2

/1

1)(

b

v

fffA

b

bb

bdBv

ff

ffff

fffA

/log20

/1log10/1log20

/1log201log20)(

22

2

• See how the above expression changes with frequency:

– at low frequencies f << fb, |Av|dB = 0 dB

• low frequency asymptote

– at high frequencies f >>fb,

|Av(f)|dB = -20 log f/ fb

• high frequency asymptote

Real Filters

• Butterworth Filters – Flat Pass-band.

– 20n dB per decade roll-off.

• Chebyshev Filters – Pass-band ripple.

– Sharper cut-off than Butterworth.

• Elliptic Filters – Pass-band and stop-band ripple.

– Even sharper cut-off.

• Bessel Filters – Linear phase response – i.e. no signal distortion in pass-band.

Filter Response Characteristics

The magnitude response of a Butterworth filter.

Butterworth Filters

Magnitude response for Butterworth filters of various

order with = 1. Note that as the order increases, the

response approaches the ideal brickwall type

transmission.

Sketches of the transmission characteristics of a representative even- and odd-

order Chebyshev filters.

Chebyshev Filters

First-Order Filter Functions

First-Order Filter Functions

Second-Order Filter Functions

Second-Order Filter Functions

Second-Order Filter Functions

Second-Order LCR Resonator

The Antoniou inductance-simulation circuit. (b) Analysis of the circuit assuming ideal op

amps. The order of the analysis steps is indicated by the circled numbers.

Second-Order Active Filter: Inductor Replacement

The Antoniou inductance-simulation circuit. Analysis of the circuit assuming ideal op amps.

The order of the analysis steps is indicated by the circled numbers.

Second-Order Active Filter: Inductor Replacement

Realizations for the various second-order filter functions using the op amp-RC resonator of

Fig. 11.21 (b). (a) LP; (b) HP; (c) BP, (d) notch at 0;

Second-Order Active Filter: Inductor Replacement

The Second-Order Active Filter: Inductor Replacement

Derivation of an alternative two-integrator-loop biquad in which all op amps are used in a

single-ended fashion. The resulting circuit in (b) is known as the Tow-Thomas biquad.

Second-Order Active Filter: Two-Integrator-Loop

Low-Pass Active Filter Design

Design a fourth-order low-pass Butterworth filter having a frequency cut-off of 100 Hz

k

HzxCfR

B

92.15

)100)(101.0(2

1

2

1

F0.1C Choose

6

Low-Pass Active Filter Design

Low-Pass Active Filter Design

Infinite-Gain Multiple-Feedback (IGMF)

Negative Feedback Active Filter

Vi

Z1

Vx

Z3

Z2

Z4 Z5

Vo

)2( ,at KCLBy

)1( 0 0

4321

x

5

3

3

5

Z

VV

Z

V

Z

V

Z

VVV

VZ

ZVV

Z

ZVvv

oxxxxi

oxxoii

Substitute (1) into (2) gives

V

Z

Z

Z ZV

Z

Z ZV

V

Z

Z

Z ZV

V

Zi

o o

o

o

o

1

3

1 5

3

5 2 5

3

4 5 4

(3)

rearranging equation (3), it gives,

HV

V

Z Z

Z Z Z Z Z Z Z

o

i

1

1 1 1 1 1 1

1 3

5 1 2 3 4 3 4

H

V

V

YY

Y Y Y Y Y Y Y

o

i

1 3

5 1 2 3 4 3 4

Or in admittance form:

Z1 Z2 Z3 Z4 Z5

LP R1 C2 R3 R4 C5

HP C1 R2 C3 C4 R5

BP R1 R2 C3 C4 R5

Filter Value

IGMF Band-Pass Filter

H s Ks

s as b( )

2Band-pass:

To obtain the band-pass response, we let

55

44

4

33

32211 11

11

RZsCCj

ZsCCj

ZRZRZ

R1

C3R2

C4 R5H s

sC

R

s C C sC C

R R R R

( )

3

1

2

3 4

3 4

5 5 1 2

1 1 1

This filter prototype has a very low

sensitivity to component tolerance when

compared with other prototypes.

Simplified Design (IGMF Filter)

R1

C

C R522

551

1

21)(

CsR

Cs

RR

R

sC

sH

Comparing with the band-pass response

22

)(

P

P

P sQ

s

sKsH

It gives,

2

1

5

51

2 2

1

1QH

R

RQ

RRCppp

Example: IGMF Band Pass Filter

To design a band-pass filter with and 10 Hz512 QfO

)512(21

51

HzRRC

P

2

51 741,662,9 100 RRnFC

102

1

1

5 R

RQ

P

170,62 4.155 51 RR vin v

o

+

-

100nF

4.155

170,62

100nF

With similar analysis, we can choose the following values:

700,621 and 554,1 10 51 RRnFC

Butterworth Response (Maximally Flat)

n

o

jH2

1

1

where n is the order

Normalize to o = 1rad/s

162.1162.01

124.324.524.524.3

185.1177.0

161.241.361.2

11

122

12

1

22

2345

5

22

234

4

2

23

3

2

2

1

sssss

ssssssB

ssss

sssssB

sss

ssssB

sssB

ssB

Butterworth polynomials

Butterworth polynomials:

)(

1)(ˆ

jBjH

n

njH

21

1)(ˆ

Second Order Butterworth Response

Started from the low-pass biquadratic function 22

1)(

P

P

P sQ

s

KsH

2

111 QKp

njH

jH

jH

jH

jjH

sssH

222

4

242

222

2

2

1

1

1

1)(

1

1)(

221

1)(

21

1)(

12

1)(

)polynomial butterwothorder (second12

1)(

For

Second order Butterworth Filter

224321

2

31214

'

11)(

P

P

P sQ

s

K

CCRRsKCsRRRsC

KsH

42

31

31

42

32

41 1

1

CR

CRK

CR

CR

CR

CRQ

P

Setting R1= R2 and C1 = C2

KKKQ

P

3

1

12

1

1111

1

Now K = 1 + RB/ RA

vin

C4

vo

+

-

R1

R2

C3

RB

RA

A

B

A

B

P

R

R

R

RKQ

2

1

13

1

3

1

For Butterworth response:

2

1

PQ

A

B

P

R

RQ

2

1

2

1

We have 414.122 A

B

R

R

We define Damping Factor (DF) as:

414.121

A

B

P R

R

QDF