ch 5 - active filters 2015 (1)

CHAPTER 5:

ACTIVE FILTERS

EMT 283/3 Analog Electronic II

Outlines

1. Introduction

2. Advantages of Active Filters over Passive Filters

3. Types of filter

4. Filter Response Characteristic

5. Active Low-Pass Filter

6. Active High-Pass Filter

7. Active Band-Pass Filter

8. Active Band-Stop Filter

9. Summary

Introduction

Filters are circuits that are capable of passing signals within a band of frequencies while rejecting or blocking signals of frequencies outside this band.

This property of filters is also called frequency selectivity.

Passive filters - built using components such as resistors, capacitors and inductors.

Active filters - employ transistors or op-amps in addition to resistors and capacitors.

Active filters are mainly used in communication and signal processing circuits.

They are also employed in a wide range of applications such as entertainment, medical electronics, etc.

Advantages of Active Filters over Passive Filters

1. Active filters can be designed to provide required

gain, and hence no attenuation as in the case of

passive filters.

2. No loading problem, because of high input

resistance and low output resistance of op-amp.

3. Active Filters are cost effective as a wide variety

of economical op-amps are available.

Active Filter

There are 4 basic categories of active filters:

Low-Pass filters

High-Pass filters

Band-Pass filters

Band-Stop filters

Each of these filters can be built by using op-amp

or transistor as the active element combined with

RC, RL or RLC circuit as the passive elements.

Active Filter

The passband is the range of frequencies that are allowed to pass through the filter.

The critical frequency, fc is specified at the point where the response drops by -3 dB from the passband response (i.e. to 70.7% of the passband response)

The stopband is the range of frequencies that have the most attenuation

The transition region is the area where the fall-off occurs

Low-Pass Filter

Allows the frequency from

0 Hz to critical frequency,

fH (also known as cutoff

frequency)

Ideally, the response drops

abruptly at the critical

frequency fH

Ideal response

Actual response

1st Order Low-Pass Filter

o

o

i

o

s

CRs

CRsV

VsH

1

1

o

o

ssH

CRo

1

o

o

o

jj

jH

1

1


2

1

1log20

o

dBjH

2

1

1

o

jH


When

And

10

1

1

2

o

jH

0

dB 01log20dB

jH


When

And

707.011

1

1

1

2

o

c

jH

CRoc

1

dB 3707.0log20dB

jH

CRc

1

is known as 3-dB or cutoff frequency (rad/s)


f c

0 dB

-10 dB

-20 dB

-30 dB

-40 dB

-3 dB

PASSBAND

SLOPE

= -20 dB/decade

0.1f c 0.01f c 10f c 100f c

STRAIGHT-LINE APPROX.

2nd Order Low-Pass Filter

R

C

+

V i

-

+

V o

-

R

C

V 1

I 1 I 2 I 3I 4

R

VVsCV

R

VV oi 1

1

1

R

VV

R

sCR

R

V oi

1

2

o

o sCVR

VV

1

(1)

oVsCRV 11 (2)

(1) & (2) R

VVsCR

R

sCR

R

V oo

i

1

2

13

12

sCRsCRV

VsH

i

o

22

2

13

1

CRCRss

CRsH

2nd Order Low-Pass Filter

22

2

3 oo

o

sssH

CRo

1where

f c 0.1f c 0.01f c 10f c 100f c

0 dB

-20 dB

-40 dB

-60 dB

-80 dB

-3 dB SLOPE

= -40 dB/decade

PASSBAND

High-Pass Filter

Allows the frequencies

above the critical

frequency fL. (also known

as the cutoff frequency.

Ideally, the response rises

abruptly at the critical

frequency. 0

1

f L f

Gain,

Pass

band

Stop

band

Ideal response

Actual response

1st Order High-Pass Filter

o

i

o

s

s

CRs

s

sCR

Rs

V

V

1

1

os

ssH

CRo

1Where;


oo jj

jjH

1

1

2

1

1

o

jH

2dB

1

1log20

o

jH


When

When

When

2

1

1

o

jH

0 jH

dB 01jH

oc dB 3707.0 jH


f c 0.1f c 0.01f c 10f c 100f c

0 dB

-10 dB

-20 dB

-30 dB

-40 dB

-3 dB

SLOPE

= 20 dB/decade

PASSBAND

STRAIGHT-LINE APPROX.

2nd Order High-Pass Filter

+

V i

-

+

V o

-R R

V 1C C

22

2

3 ooss

ssH

CRo

1where

2nd Order High-Pass Filter

f c 0.1f c 0.01f c 10f c 100f c 0.001f c

0 dB

-20 dB

-40 dB

-60 dB

-80 dB

-3 dB

SLOPE

= 20 dB/decade4

1st & 2nd Order Filter

Low-Pass Filter (LPF) High-Pass Filter

cs

ssH

c

c

ssH

o

j

jH

1

1

oj

jH

1

1

22

2

3 oo

o

sssH

22

2

3 ooss

ssH

Band-Pass Filter

Allows frequencies

between a lower

cutoff frequency (fL) and an upper cutoff

frequency (fH). 0

1

f L f

Gain,

Pass

band

f H

Ideal response

Actual response

Band-Pass Filter

Bandwidth (BW)

Center frequency

Quality factor (Q) is the

ratio of center frequency fo

to the BW

12 cc ffBW

BW

fQ 0

210 cc fff

Band-Stop Filter

Frequencies below fc1

(fL) and above fc2 (fH) are passed.

0

1

f L f

Gain,

Stop

band

f H

Ideal response

Actual response

Filter Response Characteristics

Identified by the shape

of the response curve

Passband flatness

Attenuation of

frequency outside

the passband

Three types:

1. Butterworth

2. Bessel

3. Chebyshev

Av

f

Chebyshev: rapid roll-off characteristic

Butterworth: flat amplitude response

Bessel: linear phase response


1. Butterworth

Amplitude response is very flat in passband.

The roll-off rate -20 dB per decade (per filter order).

normally used when all frequencies in the passband must

have the same gain.

2. Chebyshev

overshoot or ripples in the passband.

The roll-off rate greater than 20 dB.

can be implemented with fewer poles and less complex

circuitry for a given roll-off rate.

3. Bessel

Linear phase response.

Ideal for filtering pulse waveforms.


Damping Factor

determines the type of

response characteristic

either Butterworth,

Chebyshev, or Bessel.

The output signal is

fed back into the filter

circuit with negative

feedback determined

by the combination of

R1 and R2 2

12R

RDF

The Damping Factor

Order

Roll-off

dB/deca

de

1st stage 2nd stage

Poles DF R1 /R2 Poles DF R1 /R2

1 -20 1 Optional

2 -40 2 1.414 0.586

3 -60 2 1.00 1.00 1 1.00 1.00

4 -80 2 1.848 0.152 2 0.765 1.235

Parameters for Butterworth filters up to four poles are

given in the following table. (See text for larger order

filters).

Notice that the gain is 1 more than this resistor ratio. For example,

the gain implied by this ratio is 1.586 (4.0 dB).

Example

As an example, a two-pole VCVS Butterworth filter

is designed in this and the next two slides. Assume

the fc desired is 1.5 kHz. A basic two-pole low-pass

filter is shown.

+

R1

R2

RBRA

CA

CB

Vin

Vout

4.7 nF

4.7 nF

22 kW 22 kW

Step 1: Choose R and C for the

desired cutoff frequency

based on the equation

By choosing R = 22 kW, then

C = 4.8 nF, which is close to a

standard value of 4.7 nF.

12c

fRC

Example

Order

Roll-off

dB/deca

de

1st stage 2nd stage

Poles DF R1 /R2 Poles DF R1 /R2

1 -20 1 Optional

2 -40 2 1.414 0.586

3 -60 2 1.00 1.00 1 1.00 1.00

4 -80 2 1.848 0.152 2 0.765 1.235

Step 2: Using the table for the Butterworth filter,

note the resistor ratios required.

Step 3: Choose resistors that are as close as practical to the desired

ratio. Through trial and error, if R1 = 33 kW, then R2 = 56 kW.

Example

+

R1

R2

RBRA

CA

CB

Vin

Vout

33 kW

56 kW

22 kW 22 kW

4.7 nF

4.7 nF

Two-pole Low-Pass Butterworth Design


Critical Frequency The critical frequency, fc is

determined by the values of R and C in the frequency-selective RC circuit.

For a single-pole (first-order) filter, the critical frequency is:

The above formula can be used for both low-pass and high-pass filters

RCfc

2

1


Roll-off rate

Greater roll-off rates can be achieved with more poles.

Each RC set of filter components represents a pole.

Cascading of filter circuits also increases the poles which

results in a steeper roll-off.

Each pole represents a 20 dB/decade increase in roll-off


Roll-off depends on number of poles

Active High-Pass Filters

At critical frequency,

Resistance = capacitive

reactance

cXR

CfR

c2

1

CR

c

1

critical frequency:

RCfc

2

1


Roll-off depends on number of poles.


A Single-Pole Filter

RCfc

2

1

2

11R

RAcl


The Sallen-Key

second-order (two-pole) filter

roll-off -40dB per decade

RCfc

2

1

Lets RA = RB = R and CA = CB = C;

BABA

cCCRR

f2

1


C C

R

R

R 1

R 2

+

-V o

V i

V f

V 1

V 2

I 1

I 2

I 3 I 4

0

fVV 2

of VRR

RV

21

2

2

1

2

21

2

1R

R

R

RR

V

V

V

VA

f

ooo

o

o

A

VV 2

321 III

211

1 VVsCR

VVVVsC oi

At node V1:


o

ooi

A

VVsC

R

VVVVsC 1

11 (1)

43 II

R

VVVsC 221

RA

V

A

VVsC

o

o

o

o

1

sCRA

V

A

VV

o

o

o

o 1 (2)

At node V2:

(2) in (1): oo

o

oo

i VA

sC

RV

sCRAACs

RsCV

1112

1


1322

sCRAsCR

sCRA

V

VsH

o

o

i

o

o

o

oi V

sCRA

sCRAsCRsCV

2

213

2

2

2

13

CRs

CR

As

sA

o

o

222

3 ooo

o

sAs

sA

CRo

1where

oA 3

22

2

oo

o

ss

sAsH

= Damping Factor


C C

R

R

R 1

R 2

+

-V o

V i

V f

V 1

V 2

I 1

I 2

I 3 I 4

0oA 3

2

113R

R

2

12R

R


Cascaded HPF Six pole

cascade 3 Sallen-Key two-pole stages

roll-off -120 dB per decade

Active Low-Pass Filters

At critical frequency,

Resistance =

capacitive reactance

cXR

CfR

c2

1

CR

c

1 critical frequency:

RCfc

2

1


A Single-Pole Filter

2

11R

RAcl RC

fc2

1


The Sallen-Key

second-order (two-pole) filter


RCfc

2

1

For RA = RB = R and CA = CB = C

BABA

cCCRR

f2

1


C

C

R R

R 1

R 2

+

-

V i

V o

22

2

oo

oo

ssAsH

CRo

1

2

12R

R&


The 3-dB frequency c is related to o by a factor known as the FREQUENCY CORRECTION FACTOR

(kLP), thus;

Parameter table for Sellen-Key 2nd order LPF

RC

kk LPoLPc

Type of response kLP

Bessel 1.732 0.785

Butterworth 1.414 1

Chebyshev (1 dB) 1.054 1.238

Chebyshev (2 dB) 0.886 1.333

Chebyshev (3 dB) 0.766 1.390

Example

Design a Bessel 2nd order low-pass filter with a 3-

dB frequency of 5 kHz. Use C = 22 nF.

C

C

R R

R 1

R 2

+

-

V i

V o

Solution

From the table:

RC

kk LPoLPc

C

kR

c

LP

732.1

785.0

LPk

Solution

C

C

R R

R 1

R 2

+

-

V i

V o

1.136 kW 1.136 kW

22 nF

22 nF

2.88 kW

10.74 kW

A Bessel 2nd order LPF

Solution

Frequency response curve of a 2nd order Bessel LPF

Frequency

100Hz 1.0KHz 10KHz 100KHzVDB(C5:2)

-50

-25

0

10

fc

Example

1. Determine critical frequency

2. Set the value of R1 for Butterworth response


Cascaded LPF Three-pole

cascade two-pole and single-pole



Cascaded LPF Four pole

cascade two-pole and two-pole


Example

Determine the capacitance values required to

produce a critical frequency of 2680 Hz if all

resistors in RC low pass circuit is 1.8 kW

Solution

RCfc

2

1

F 033.02

1

RfC

c

F 033.02211 BABA CCCC


SELLEN-KEY LPF

The slope (roll-off) of a filter is associated with the its

order the higher the order of a filter, the steeper will

be its slope.

The slope increases by 20 dB for each order, thus;

ORDER SLOPE (dB/decade)

1 20

2 40

3 60

4 80


Active filters can be cascaded to increase its order,

e.g. cascading a 2nd order and a 1st order filter will

produce a 3rd order filter

2nd order

LPF 1st order

LPF Input signal

Output signal

Cascading a 2nd order and a 1st order filter produces a 3rd order filter with a roll-off of 60dB/decade

Active Band-Pass Filters

A cascade of a low-pass and high-pass filter

Band-pass filter formed by cascading a two-pole high-pass and a two-pole low-pass filters


210 cc fff

1111

12

1

BABA

cCCRR

f

2222

22

1

BABA

cCCRR

f


Multiple-Feedback

BPF

The low-pass circuit

consists of R1 and C1.

The high-pass circuit

consists of R2 and C2.

The feedback paths

are through C1 and R2.

Center frequency:

212310

//2

1

CCRRRf


Multiple-Feedback BPF

For C1 = C2 = C, the resistor values can be

obtained using the following formulas:

The maximum gain, Ao occurs at the center

frequency.

ooCAf

QR

21

Cf

QR

o2 )2(2 2

3

oo AQCf

QR


State-Variable BPF

It consists of a summing amplifier and two integrators.

It has outputs for low-pass, high-pass, and band-pass.

The center frequency is set by the integrator RC circuits.

R5 and R6 set the Q (bandwidth).

1

3

1

6

5

R

RQ


The band-pass output peaks sharply the center

frequency giving it a high Q.


Biquad Filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name biquad is

short for biquadratic. It is also sometimes called the 'ring of 3' circuit.

contains an integrator, followed by an inverting amplifier, and then an

integrator.

In a biquad filter, the bandwidth is independent and the Q is dependent on

the critical frequency.

Active Band-Stop Filters

The BSF is opposite of BPF in that it blocks a specific

band of frequencies.

The multiple-feedback design is similar to a BPF

with exception of the placement of R3 and the

addition of R4

Active Band-Stop Filters

State Variable Band-Stop Filter

Summing the low-pass and the high-pass responses of

the state-variable filter with a summing amplifier

creates a state variable band-stop filter

Summary

The bandwidth of a low-pass filter is the same as

the upper critical frequency.

The bandwidth of a high-pass filter extends from

the lower critical frequency up to the inherent limits

of the circuit.

The band-pass passes frequencies between the

lower critical frequency and the upper critical

frequency.

A band-stop filter rejects frequencies within the

upper critical frequency and upper critical

frequency.

Summary

The Butterworth filter response is very flat and has

a roll-off rate of 20 dB.

The Chebyshev filter response has ripples and

overshoot in the passband but can have roll-off

rates greater than 20 dB.

The Bessel response exhibits a linear phase

characteristic, and filters with the Bessel response

are better for filtering pulse waveforms.

Summary

A filter pole consists of one RC circuit. Each pole

doubles the roll-off rate.

The Q of a filter indicates a band-pass filters

selectivity. The higher the Q the narrower the

bandwidth.

The damping factor determines the filter response

characteristic.

Exercises

1. A band-pass filter can be created by cascading a high-pass

filter and a low-pass filter.

2. The bandwidth of a band-pass filter is the sum of the two

cutoff frequencies.

3. A Sallen-Key filter is a second-order filter.

4. Filters with Bessel characteristics are used for filtering pulse

waveforms.

5. The bandwidth of a practical high-pass filter is infinite.

(Answer: TRUE )

(Answer: FALSE )

(Answer: TRUE )

(Answer: TRUE )

(Answer: FALSE )

Exercises

6. In filters, a single RC network is called a pole.

7. A band-pass filter passes all frequencies within a band

between a lower and an upper cutoff frequency.

8. An active filter uses capacitors and inductors in the feedback

network.

9. A second-order filter has a roll-off of 20 dB/decade.

10. Butterworth filters have the characteristic of a very flat

response in the passband and a roll-off of 20 dB/decade.

(Answer: TRUE )

(Answer: TRUE )

(Answer: FALSE )

(Answer: FALSE)

(Answer: TRUE )

Exercises

Identify the frequency response curve for a band-

pass filter.

Answer: A

Exercises

A ________ filter rejects all frequencies within a specified

band and passes all those outside this band.

A. low-pass B. high-pass

C. band-pass D. band-stop

Filters with the ________ characteristic are useful when a

rapid roll-off is required because it provides a roll-off rate

greater than 20/dB/decade/pole.

A. Butterworth B. Chebyshev

C. Bessel

Answer: D

Answer: B

ch 5 - active filters 2015 (1)

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