www.ku.edu.npDepartment of Electrical & Electronics Engineering www.ku.edu.np/ee

EEEG 213: Network AnalysisLecture-22: Filters

SHAILENDRA KUMAR JHADepartment of Electrical & Electronics EngineeringKathmandu University, Dhulikhel

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Outline• Definition of Filters• Frequency filters• Types of frequency filters• Active filters design

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References:

1. Boylestad R. L., Introductory Circuit Analysis, PHI, 1999

2. Gayakwad R. A., “Op-Amps and Linear Integrated Circuits”, PHI,1998

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Definition

A network designed to either block orpass energy

– Amplitude filters– Frequency filters

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Frequency filtersA network designed to select or reject aband of frequencies

Types:– Analog and digital– Passive or active– Low pass, high pass, band pass and band

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• Analog filters are designed to processanalog signals

• Digital filters process analog signals usingdigital techniques

• Passive filters are designed usingresistors, inductors and capacitors

• Active filters employ transistors or op-amps in addition with resistors,inductors and capacitors

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Low Pass filters

• Allows only low frequency to pass• has constant gain from 0Hz to a high

cutt-off frequency fH

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Pass band stop band

Ideal response

fHfrequency

Gain (Vo/Vi)

1

0

real response

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High Pass filters

• Allows only high frequency to pass• has constant gain after cutt-off

frequency fL

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Stop band pass band

Ideal response

fLfrequency

Gain (Vo/Vi)

1

0

real response

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Band Pass filters• Designed to pass certain band of frequencies while

rejecting all frequencies below and above this range.

• has constant gain between two cut-off frequencies fLand fH

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Ideal response

fLfrequency

Gain (Vo/Vi)

1

0 fH

Passband

stop bandStopband

real response

fC

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Band Stop filters• Designed to reject certain band of frequencies while

pass all frequencies below and above this range.

• has constant gain from 0Hz to low cut of frequencyfL and after high cut-off frequencies fH

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Ideal response

stopband

pass bandpassband

fLfrequency

Gain (Vo/Vi)

1

0 fH

real response

fC

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Real filter designs• Butterworth filters

– Flat pass band as well as stop band– Roll of 20dB/decade/pole– Used when all frequencies in pass band must have

same gain

• Chebyshev filters– Ripple pass band and flat stop band– Sharper cut-off than Butterworth– Roll of greater than 20dB/decade/pole– Use when rapid roll of required

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Real filter designs

• Cauer filters– Ripple pass band and ripple stop band– Gives the best stop band response– Even sharper cut-off than Chebyshev

• Bessel filters– Linear response i.e no distortion in pass band– Roll off rate less than 20dB/decade/pole– Used for filtering pulse waveform without

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Stop Band

Real filter designs

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frequency

GainButterworthChebyshevCauerBessel

Pass Band

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Low Pass RC filter

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At any intermediate frequency, the output voltage Vo isgiven as

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For frequency less than fc will result in an output voltage Vo that isat least 70.7% of the maximum.

For any frequency above fc, the output is less than 70.7% of theapplied signal.

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High Pass RC filter

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At any intermediate frequency, the output voltage Vo isgiven as

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Band Pass Filters

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Band Stop Filters

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Filters and roll of ratesThe number of poles determines the roll-off rate

First order - one pole - roll-off of -20dB/decade

Second order- two poles-roll-off of -40dB/decade

Third order - three poles-roll-off of -60dB/decade

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First Order Low Pass Butterworth Filter

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R1 Rf

R

C RL

VO

Vi

+15V

-15V

V1

= −−According to voltage divider rule,

, = 1 +And output voltage = (1 + )

That is = (1 + ) 1 + = 1 + ( )

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Where, == 1 + =f = frequency of the input signal= 12 =The gain magnitude and phase angle equations of thelow-pass filter can be written as= 1 + ( )And ∅ = − ( )

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At very low frequency, that is f < fH=At f = fH,

At f > fH

Thus the low-pass filter has a constant gain AF from 0Hz to -the high cutoff frequency fH.

At fH the gain is 0.707Af and after fH it decreases at aconstant rate with an increase in frequency.

The frequency f = fH is called the cutoff frequency becausethe gain of the filter at this frequency is down by 3 dB (= 20log 0.707) from 0 Hz.

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= 2 = 0.707<

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A low-pass filter can be designed by implementingthe following steps:

1. Choose a value of high cutoff frequency fH

2. Select a value of C less than or equal to 1F.Mylar or tantalum capacitors arerecommended for better performance.

3. Calculate the value of R using

4. Finally select values of R, and RF dependent onthe desired pass band gain AF using

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= 12= 1 +

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Frequency ScalingThe procedure used to convert an original cutofffrequency fH to a new cutoff frequency f’H is calledfrequency scaling.

• To change a high cutoff frequency multiply R or C,but not both, by the ratio of the original cutofffrequency to the new cutoff frequency.

• In filter design the needed values of R and C areoften not standard. Besides, a variable capacitor C isnot commonly used.

• Therefore, choose a standard value of capacitor,and then calculate the value of resistor for a desiredcutoff frequency. This is because for a nonstandardvalue of resistor a potentiometer can be used.

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… problems …

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Second Order Low Pass Butterworth Filter

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R2

R1 Rf

R3

C3RL

VO

Vi

+15V

-15VC2

The voltage gain magnitude is = ( )A = 1 + RR = pass band gain of the filterf = frequency of the input signal= 1/2π R R C C= high cutoff frequency of the filter

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Except for having twice the roll-off rate in the stop band, thefrequency response of the second-order low-pass filter is identicalto that of the first-order type.

The design steps of the second-order filter are:

1. Choose a value of high cutoff frequency fH

2. To simplify the design calculations, set R2=R3=R and C2=C3=C.then choose a value of C less than or equal to 1F.

3. Calculate the value of R using

4.Because of the equal resistor R2=R3 and capacitorC2=C3, the pass band voltage gain AF of the second-order low- pass filter has to be equal to 1.586.• That is, RF = 0.586R1.• This gain is required for Butterworth response.Hence choose a value of R1 100k and calculate thevalue of RF.

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= 12

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… problems …

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First Order High Pass Butterworth FilterR1 Rf

C

RRL

VO

Vi

+15V

-15V

For a first order high pass filter, the output voltage is= (1 + ) 1 += [ ]

Where, = 1 + =f = frequency of the input signal= 12 =

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The magnitude of the voltage gain is

Since high-pass filters are formed fromlow-pass filters simply by interchangingR's and C's, the design and frequencyscaling procedures of the low-pass filtersare also applicable to the high-passfilters.

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= ( )1 + ( )

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… problems …

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Second Order High Pass Butterworth Filter

C2

R1 Rf

C3

R3

RL

VO

Vi

+15V

-15VR2

The magnitude of the voltage gain is= 1 + ( )Where Af = 1.586 for second order Butterworth reponse

f = frequency of the input signal (Hz)fL = low cutoff frequency (Hz)

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…problems…

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Band Pass Filter

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R’1 R’f

R’

C’ RL

VO

+15V

-15V

R1 Rf

C

RVi

+15V

-15V

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Band Reject Filter

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R’1 R’f

R’

C’

+15V

-15V

R1 Rf

C

R

Vi

+15V

-15V

R2 R4

R3

ROM

RL

VO

+15V

-15V

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Notch filter

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