D i d S h iDesign and Synthesis ofof

Active Filter

R fReferences: H. Lam, “ANALOG AND DIGITAL FILTERS: Design and Realization”W K Chen “Passive and Active Filters: Theory and Implementations”W. K. Chen,  Passive and Active Filters: Theory and ImplementationsV. K. Aatre, “Network Theory And Filter Design”

What is a Filter?What is a Filter?

• An electric filter is a frequency selective• An electric filter is a frequency selectivecircuit that passes a specified band off i d bl k tt tfrequencies and blocks or attenuatessignals of frequencies outside this band.

• Filter modifies signals in frequencyf g f q ydomain – both in amplitude and phase.

•Analog filters are designed to process analog signals.

Di it l filt l i l i di it l t h i•Digital filters process analog signals using digital techniques.

•Elements used in the passive filters : resistors, inductors, capacitors.•Elements used in the active filters : Transistors or Op‐Amps in additiont i t d itto resistors and capacitors

Most commonly used filtersMost commonly used filters 

Most commonly used filtersMost commonly used filters 

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Low pass FilterLow pass Filter

• Passes low frequency components andPasses low frequency components and attenuates high frequency components

Ideal Characteristics of Low pass filter

Low pass FilterLow pass Filter

• Passes low frequency components andPasses low frequency components and attenuates high frequency components

Actual Characteristics of Low pass filter

High pass FilterHigh pass Filter

• Passes high frequency components andPasses high frequency components and attenuates low frequency components

Ideal Characteristics of High pass filter

High pass FilterHigh pass Filter

• Passes high frequency components and asses g eque cy co po e ts a dattenuates low frequency components

Actual Characteristics of High pass filter

Band pass FilterBand pass Filter

• Passes certain band of frequencies and• Passes certain band of frequencies andattenuates frequencies outside this band.

• Has pass‐band between two stop‐bands

• Bandwidth= (ωH ‐ ωL)Bandwidth (ωH ωL)

Band pass FilterBand pass Filter

Id l Ch t i ti f B d filtIdeal Characteristics of Band pass filter

Band pass FilterBand pass Filter

A t l Ch t i ti f B d filtActual Characteristics of Band pass filter

Band stop FilterBand stop Filter

Ideal Characteristics of Band stop filter

Band stop FilterBand stop Filter

Actual Characteristics of Band stop filter

Narrow Band‐pass filter: known as Tuned Filter N B d fil k N h FilNarrow Band‐stop filter: known as Notch Filter 

Tuned Filter  Notch Filter 

Filter Transfer FunctionFilter Transfer Function

m = number of zeros  and       n = number of poles

First Order FilterFirst  Order Filter

For Low-pass Filter:

Pass-band gain = (a0 / b0)Cut-off frequency = b rad/secCut-off frequency b0 rad/sec

First Order FilterFirst  Order FilterFor Low-pass Filter:

P b d i ( / b )Pass-band gain = (a0 / b0)Cut-off frequency = b0 rad/sec

First Order FilterFirst  Order Filter

For High-pass Filter:For High pass Filter:

P b d iPass-band gain = a1Cut-off frequency = b0 rad/sec

Second Order FilterSecond  Order Filter

For Low-pass Filter:For Low pass Filter:

Pass-band gain = a0 / b0Cut-off frequency of the filter = rad/s

Second Order FilterSecond  Order Filter

For High-pass Filter:For High pass Filter:

Pass-band gain = a2Cut-off frequency of the filter = rad/s

Second Order FilterSecond  Order Filter

For Band-pass Filter:For Band pass Filter:

Pass-band gain = a1 / b1Centre frequency of the filter = rad/s

Second Order FilterSecond  Order Filter

For Band-stop Filter:For Band stop Filter:

[a2 = a0 / b0]

Pass-band gain = a2Centre frequency of the filter = rad/s

Schematic of Band-stop Filter:Schematic of Band stop Filter:

Schematic of Band-stop Filter:Schematic of Band stop Filter:

a1 /b1 = 1a1 /b1  1

Pass-band gain = 1Pass band gain 1 Centre frequency of the filter = rad/s

Filter Circuit ComponentsBasic Building Blocks:

Resistors, Capacitors & 0p‐Amps, p p p

Filter Circuit ComponentsSecondary Building Block:

V lt C t ll d V lt S (VCVS)Voltage Controlled Voltage Source (VCVS)

First Order Low‐pass Filter

Pass-band gain = a0/b0 = KPass band gain a0/b0 KCut-off frequency b0 = 1/R1C2

First Order High‐pass Filter 

Pass band gain = a =KPass-band gain = a1=KCut-off frequency b0 = 1 / R2C1

Second Order Filter Circuit

Second Order Filter Circuit

Second Order Low‐pass Filter

P b d i /b K VCVS iPass-band gain = a0/b0 = K=VCVS gain

Cut-off frequency =

Second Order High‐pass Filter

Pass-band gain = a2 =K = VCVS gainPass band gain a2 K VCVS gain

Cut-off frequency = 4321/1 RRCC

Band‐pass Filter Circuit

Pass-band gain = a1/b1 =

Band‐pass Filter Circuit

Center/ cut-off frequency = =