Analog Circuits and Systems Prof. K Radhakrishna Rao

Lecture 22: Passive Filters

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Review

�  Second Order Filters

2

Review (contd.,)

�  Higher order wide band filter with stagger tuned narrow band filters of lower order

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What are passive filters?

�  Filters that use only passive components R, L, C and transformer are known as passive filters.

�  Before the commercial availability of Op Amps, all base band filters were mainly passive in nature because of reliability, precision, and low sensitivity to temperature variations and aging.

�  Transformers were mainly used for impedance matching. �  Present day base band filters no longer use discrete

transformers.

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Passive base band filters

�  Passive filters are still used in microwave region �  Interconnect models also are low pass passive filters �  Passive filters are mainly designed as first or second

order filters �  In higher order passive filters the coefficients in the filter

functions can become very complex functions of passive component values.

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First Order Passive Low Pass Filters

�  A first order RC-network

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( )

o

i

o

i2 *

o o o

i i i

2 2

V 1V 1 sCR

V 1V 1 j CR

V V VV V V

1 111 CR

CR

=+

=+ ω

⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

= =+Ω+ ω

Ω = ω

@

For sinusoidal excitation

where

First Order MFM

�  It is similar to a Maximally Flat Magnitude (MFM) (Butterworth) function

�  Response is similar to that of low pass filter.

�  Square of magnitude and the delay are frequency dependent in the pass band.

�  We always consider the filter response in the region W2>0

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( )

1o

i

0 02 20

V tan CR ; delayV

CR 1 1 1;CR11 CR

− ∂φφ = = − ω = φ − τ∂ω

⎡ ⎤∂φ ⎣ ⎦− = τ = = ω τ ω = =∂ω τ+Ω+ ω

@ @

where

Magnitude and Delay Plots

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The normalized magnitude

and delay

plots of this filter are

relevant for >0.

o2

i

0

2

V 1 ,V 1

T

⎛ ⎞⎜ ⎟⎜ ⎟+Ω⎝ ⎠

⎛ ⎞τ =⎜ ⎟τ⎝ ⎠

Ω

@

Magnitude and Delay Plots (contd.,)

�  W = 1 is recognized as the (half-power) bandwidth of the filter. �  Filters with maximally flat magnitude function are called

Butterworth filters �  Filters with maximally flat delay characteristics are called Bessel or

Thompson filters. �  Rate of attenuation at the edge of pass band (W = 1) is -0.5

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First Order Low Pass R L Filter

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( )

o

i

0

0

V 1LV 1 sR

1 RRC L

=+

ω = =

ω ω?

First order RC and RL low pass filters

have a bandwidth of

The magnitude decreases in the

stop band at the rate of

20 dB/decade or 6dB/Octave).

Second Order Butterworth Passive Low Pass Filter

�  The second order Butterworth filter will have a magnitude function similar to

where e2 indicates the deviation from 1 in magnitude at X = 1

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o2

i

V 1V 1 sCR s LC

=+ +

42

1

1 X+ ε

Second Order Butterworth Passive Low Pass Filter

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( )

( )

ω = =ω

= ω =+ ω − ω

= =⎛ ⎞⎛ Ω ⎞ + − Ω + Ω− Ω + ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

ωΩ = ω = = =ω ω

= =+ Ω

22

0 20

o2

i2

o2

2 2 4i 22

00 0

o4

i

1 ss LCLC

V 1s jV 1 j CR LC

V 1 1V 11 21 QQ

1 1 L 1QCR C RLC

QV1 1QV2 1

Define

Substitute

where where and

is known as quality factor.

If then

Phase of the second-order filter

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o2

i

V 1V 1 j CR LC

=+ ω −ω

o2i

V 1V 1 j

Q

= Ω−Ω +

( )-1o2

i

Ω QVPhaseof =Φ=-tanV 1-Ω

∂ΦΤ = −∂ΩDelay

2

2 42

1 1Q 11 ( 2 )

Q

⎡ ⎤⎢ ⎥

+Ω⎢ ⎥Τ = ⎢ ⎥⎛ ⎞⎢ ⎥+ − + Ω +Ω⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Tmax = 2Q at W = 1

Second Order Low Pass RLC Filter

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Magnitude Plot

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Phase Plot

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Band Pass Filter – Fourth Order

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Maximally Flat Function

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For delay (Q = 1/sqrt(3)) Thomson’s/Bessel’s filter

Chebyschev or Equi-ripple Low Pass Filter

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( )1

peak 1 12 21

1

If then where

K 1 1= where K = 2- and the peak 1+ =

2 Q K1-

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for =0.1

o1 22 4

i 1

1

21 1

V1 1 1Q K 2V Q2 1 K

2K 0.83112

> = = −− Ω + Ω

Ω ε

= = ε+ε + ε

Chebyschev or Equi-ripple Low Pass Filter (contd.,)

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�  Filter with gives a better performance at the pass band edge (faster rate of attenuation). Achieved at the cost of deviation from flatness (ripple) in the pass band

�  Functions of the type are known as second order Chebyshev functions.

1Q2

>

( )2 41

1

1 K− Ω +Ω

Inverse Chebyshev Low Pass Filter

�  Addition of a zero to a Chebyshev function improves the response at the pass band edge

�  It is known as inverse Chebyshev function

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Elliptic Filter

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R=40, L2 = 0.9m, L1=0.1m, C=0.1micro

Inverse Chebyshev Low Pass Filter (contd.,)

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( )( )

( )( ) ( )

( )and

221o o1

2 2 22i i1 21 2

22 1 2

12pz

1 L CV V1 s L C ;V V1 s L L C sCR 1 L L C CR

L L1 1K QCR C R

− ω+= =+ + + − ω + + ω

+ω = Ω = =ωω

( )( ) ( )

where

becomes zero when

22 2

z 1 2 p21 1 2p

21o o

i i 12 42

1 1; L L C ;L C L L C

1 NV V 1;V V N11 2

Q

ωω = ω + = = Ω ω =+ω

− Ω= Ω =

⎛ ⎞+ Ω − + + Ω⎜ ⎟⎝ ⎠

Inverse Chebyshev Low Pass Filter (contd.,)

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If is selected to be less than 1, zero occurs outside the pass band

becomes zero for

is of the type

For this function to become a maximally flat

1

o

i

2 2 2 4o 1 1 1

2 42 4i 11

1 1

N

V 1V

V 1 N X 1 2N X N XV 1 K X X1 K X X

2N K .

Ω >

− − +=

− +− +

= For

the zero will occur at For the response will peak in

the pass band and will have higher rate of attenuation at the edge of the pass band.

1 1

1 11

2N K 0.5

1X 2. K 2NN

= =

= = >

Elliptic Low Pass filter

�  Filter with a zero(s) in stop band and peak(s) in the pass band

�  For N1=0.25 and K1=0.7 the response of the Elliptic filter in comparison with the inverse Chebyshev and second order Butterworth filters.

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Conclusion

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