Analog Filters:Basics of OP AMP-RC Circuits

Stefano GregoriThe University of Texas at Dallas

Stefano Gregori Basics of OP AMP-RC Circuits 2

Introduction

So far we have considered the theory and basic methods of realizing filters that use passive elements (inductors and capacitors)

Another type of filters, the active filters, are in very common use

They were originally motivated by the desire to realize inductorless filters, because of the three passive RLC elements the inductor is the most non-ideal one (especially for low-frequency applications of filters in which inductors are too costly or bulky)

When low-cost, low-voltage solid-state devices became available, active filters became applicable over a much wider frequency range and competitive with passive ones

Now both types of filters have their appropriate applications

Stefano Gregori Basics of OP AMP-RC Circuits 3

Active-RC filters

Active filters are usually designed without

regard to the load or source impedance; the terminating impedance may not affect the performance of the filter

it is possible to interconnect simple standard blocks to form complicated filters

are noisy, have limited dynamic ranges and are prone to instability

can be fabricated by integrated circuits

In this lesson we concentrate on active-RC filters. They make use of active devices as well as RC components.

Passive filters the terminating impedance is an

integral part of the filter: this is a restriction on the synthesis procedure and reduces the number of possible circuits

are less sensitive to element value variations

are generally produced in discrete or hybrid form

Stefano Gregori Basics of OP AMP-RC Circuits 4

Operational Amplifier

In an ideal op-amp we assume:

input resistance Ri approaches infinity, thus i1 = 0

output resistance Ro approaches zero

amplifier gain A approaches infinity

Ri

Ro

A(e+-e-)

e+

e-

e2

i1

equivalent circuitsymbol

e+

e-e2

Stefano Gregori Basics of OP AMP-RC Circuits 5

Inverting voltage amplifier

R1

R2

vinvout

i

i1

in )()(

R

tvti

)()()( in1

22out tv

R

RtiRtv

Example:

ftπVtv 2sin)( 0in

given

we have

R1 = 1 kΩR2 = 2 kΩV0 = 1 Vf = 1 MHz

ftπVR

Rtv 2sin)( 0

1

2out

vin(t)

vout(t)

Stefano Gregori Basics of OP AMP-RC Circuits 6

Weighted summer

k

kk R

tvti

)()(

)()( )( 1out titiRtv nf

R1

Rf

v1voutR2

v2

Rn

vn

n

k k

kf R

tvR

1

)(

n

nf R

tv

R

tvR

)()(

1

1

Stefano Gregori Basics of OP AMP-RC Circuits 7

Noninverting voltage amplifier

R1

R2

vin

vouti

i1

in )()(

R

tvti

)( 1)( )( in1

221out tv

R

RtiRRtv

Example:

ftπVtv 2sin)( 0in

given

we have

R1 = 1 kΩR2 = 1 kΩV0 = 1 Vf = 1 MHz

ftπVR

Rtv 2sin1)( 0

1

2out

vin(t)

vout(t)

Stefano Gregori Basics of OP AMP-RC Circuits 8

Buffer amplifier

vin

vout

)()( inout tvtv

Stefano Gregori Basics of OP AMP-RC Circuits 9

Inverting or Miller integrator

R

tvti

)()( in

)(1

)0()(0

inoutout t

dttvRC

vtv

R

C

vinvout

i dt

tdvCti

)()( out

Example:

0

0

2sin

0)(

0in

t

t

ftπVtv

0)0(out v

12cos2

)( 0out πft

fRCπ

Vtv

given

we have

vin(t)

vout(t)

R = 1 kΩC = 1 nFV0 = 1 Vf = 1 MHz

RCs

VV

in

out

Stefano Gregori Basics of OP AMP-RC Circuits 10

Inverting differentiator (1)

dt

tdvRCtRitv

)()()( in

out

dt

tdvCti

)()( in

Example:

0

0

2sin

0)(

0in

t

t

ftπVtv

given

we have

R = 1 kΩC = 100 pFV0 = 1 Vf = 1 MHz

C

R

vinvout

i

vin(t)

vout(t)

0

0

2cos2

0)(

0out

t

t

ftπfRCVπtv

Cs

inout VRCsV

Stefano Gregori Basics of OP AMP-RC Circuits 11

Inverting differentiator (2)

vout(t) is a square waveform with:- vout max 2,068 V- vout min -2,068 V- frequency 500 kHz

R = 22 kΩC = 47 pFvin(t) is a triangular waveform with:

- vin max 2 V- vin min 0 V- frequency 500 kHz

vin(t)

vout(t)

C

R

vinvout

Cs Cs

Stefano Gregori Basics of OP AMP-RC Circuits 12

Inverting lossy integrator

R1

R2

vinvout

C

in

21

out1

1V

RsCR

V

Stefano Gregori Basics of OP AMP-RC Circuits 13

Inverting weighted summing integrator

R1

v1voutR2

v2

Rn

vn

C

n

k k

kout R

V

sC V

1

1

Stefano Gregori Basics of OP AMP-RC Circuits 14

Subtractor

2

321

1031

1

0 VRRR

RRRV

R

R Vout

R1

R0

v1vout

R2

v2

R3

Stefano Gregori Basics of OP AMP-RC Circuits 15

Integrator and differentiator

integrator

differentiator

frequency behavior

R

C

vinvout

C

R

vinvout

integrator differentiator

R = 1 kΩC = 1 nF

2

1V fRCπ

A

fRCπA 2V vin(t) is a sinewave with frequency f.

Figure shows how circuit gain AV changes with the frequency f

AV is the ratio between the amplitude of the output sinewave vout(t) and the amplitude of the input sinewave vin(t)

Stefano Gregori Basics of OP AMP-RC Circuits 16

Low-pass and high-pass circuits

low-passvoutlpvin

R

C

C

vin vouthp

R

low-pass circuit

high-pass circuit

frequency behavior

high-pass

R = 1 kΩC = 1 nF

Stefano Gregori Basics of OP AMP-RC Circuits 17

Inverting first-order section

v1v2

R1

C1

R2

C2

Z1

Z2

v1v2

2

1

1

2

1

2 Y

Y

Z

Z

V

V

1

1

22

11

1

2

1

2

CsR

CsR

R

R

V

V

inverting lossing integrator

Stefano Gregori Basics of OP AMP-RC Circuits 18

Noninverting first-order section

2

1

1

2

1

2 1 1Y

Y

Z

Z

V

V

1

11

22

11

1

2

1

2

CsR

CsR

R

R

V

V

noninverting lossing integrator

Z1

Z2

v1

v2

v1

v2

R1

C1

R2

C2

Stefano Gregori Basics of OP AMP-RC Circuits 19

Finite-gain single op-amp configuration

V1

RCthreeport

V2

V3

i2

i3i1

Many second-order or biquadratic filter circuits use a combination of a grounded RC threeport and an op-amp

3332321313

3232221212

3132121111

VyVyVyI

VyVyVyI

VyVyVyI

μEE

I

/

0

23

3

μ

yy

y

E

E

3332

31

1

2

Stefano Gregori Basics of OP AMP-RC Circuits 20

Infinite-gain single op-amp configuration

32

31

1

2

y

y

E

E

V1

RCthreeport

V2

V3

Stefano Gregori Basics of OP AMP-RC Circuits 21

Gain reduction

V1

N

V2

Z

V1

N

V2

Z1

Z2

V1'

To reduce the gain to α times its original value (α < 1) we make

21

2

1

1

ZZ

Zα

V

V

Z

ZZ

ZZ

21

21and

solving for Z1 and Z2, we get

α

ZZ 1 Z

αZ

1

12and

Stefano Gregori Basics of OP AMP-RC Circuits 22

Gain enhancement

A simple scheme is to increase the amplifier gain and decrease the feedback of the same amount

V1

RCthreeport

V2

V3 K

1/KV2/K

μ

yy

Ky

E

E

3332

31

1

2

Stefano Gregori Basics of OP AMP-RC Circuits 23

RC-CR transformation (1)

is applicable to a network N that contains resistors, capacitors, and dimensionless controlled sources

conductance of Gi [S] → capacitance of Gi [F]

capacitance of Cj [F] → conductance of Cj [S]

the corresponding network functions with the dimension of the impedance must satisfy

sZ

ssZ

11)(

the corresponding network functions with the dimension of the admittance must satisfy

ssYsY

1)(

the corresponding network functions that are dimensionless must satisfy

sHsH

1)(

Stefano Gregori Basics of OP AMP-RC Circuits 24

RC-CR transformation (2)

672

12)(

21

2

ssV

VsH

1 F

v1' v2'2

1/2 F

3

2

v1 v2

12

2

1/3 F

1/2 F

)12(

672)(

2

1

111

ss

ss

I

VsZ

276

12)(

2

2

1

2

ss

s

V

VsH

)2(

276)(

2

1

111

ss

ss

I

VsZ

N N’

Stefano Gregori Basics of OP AMP-RC Circuits 25

Sallen-Key filters

lowpass filter

highpass filter

frequency behavior

C

vin

vout

C

R

R

C

vin

vout

C

RR

lowpass highpass

bandpass

R = 1 kΩC = 1 nF

Stefano Gregori Basics of OP AMP-RC Circuits 26

2 4 6 8 10

0.2

0.4

0.6

0.8

1

2 4 6 8 10

0.2

0.4

0.6

0.8

1

2 4 6 8 10

0.2

0.4

0.6

0.8

1

2 4 6 8 10

0.2

0.4

0.6

0.8

1

2 4 6 8 10

0.2

0.4

0.6

0.8

1

Types of biquadratic filters

012

0

bsbs

Gb

012

2

bsbs

Gs

012

1

bsbs

sGb

012

02

2

bsbs

asa

01

201

2

bsbs

bsbsG

lowpass highpass bandpass bandreject allpass