Analog Circuits and Systems...Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 22:...

Post on 18-Mar-2021

4 views 0 download

Transcript of Analog Circuits and Systems...Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 22:...

Analog Circuits and Systems Prof. K Radhakrishna Rao

Lecture 22: Passive Filters

1

Review

�  Second Order Filters

2

Review (contd.,)

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

3

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.

4

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.

5

First Order Passive Low Pass Filters

�  A first order RC-network

6

( )

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

7

( )

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

8

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

9

First Order Low Pass R L Filter

10

( )

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

11

o2

i

V 1V 1 sCR s LC

=+ +

42

1

1 X+ ε

Second Order Butterworth Passive Low Pass Filter

12

( )

( )

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

13

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

14

Magnitude Plot

15

Phase Plot

16

Band Pass Filter – Fourth Order

17

Maximally Flat Function

18

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

Chebyschev or Equi-ripple Low Pass Filter

19

( )1

peak 1 12 21

1

If then where

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

2 Q K1-

4

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.,)

20

�  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

21

Elliptic Filter

22

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

Inverse Chebyshev Low Pass Filter (contd.,)

23

( )( )

( )( ) ( )

( )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.,)

24

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.

25

Conclusion

26