analog circuits and systems...analog circuits and systems prof. k radhakrishna rao lecture 22:...
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Analog Circuits and Systems Prof. K Radhakrishna Rao
Lecture 22: Passive Filters
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Review
� Second Order Filters
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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
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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
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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
@
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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).
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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
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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
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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
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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
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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-
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
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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
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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
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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⎜ ⎟⎝ ⎠
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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
= =
= = >
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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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