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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