Filters and Tuned Amplifiers
Introduction
Filter Transmission, Types and
Specification
Butterworth and Chebyshev Filters
First-Order and Second-Order
Filter Functions
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Introduction
• Electronic Filter, I.e. active filter
– Active RC filters, switched capacitor ckts
Advantages:
No inductors!
Inductors are large and physically bulky for
low frequency applications (such as those
used in passive LC filters)
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Filter Transmission & Types
• Filter, a two port device
• Transfer function T(s) T ( s ) = V0 ( s ) / Vi ( s )
2
Filter Transmission
Let s=jω
T ( jω ) = T ( j ω ) e
jφ (ω )
G (ω ) = 20 log T ( jω ) dB
A(ω ) = −20 log T ( jω ) dB
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Transmission specifications
for a bandpass filter
•Monotonically
decreasing
transmission in
the passband
on both sides
of the peak
frequency.
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Filter Transfer Function
aM s M + aM −1s M −1 + L + a0
T ( s) =
s N + bN −1s N −1 + L + b0
aM ( s − z1 )( s − z 2 ) L ( s − z m )
T ( s) =
( s − p1 )( s − p2 ) L ( s − pm )
N: filter order
For filter to be stable M ≤ N
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Bandpass Filter
Transmission
zeros at
s = ± jω l1 , s = ± jω l 2 , s = 0, s = ∞
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T ( jω ) =
Butterworth Filter
1
⎛ω ⎞
⎟⎟
1 + ε 2 ⎜⎜
ω
⎝ P⎠
2N
• Frequently used
to approximate
transmission
characteristic of
low pass filters
with closed form
expression
• At ω=ωp T ( jω ) =
p
1
1+ ε 2
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Magnitude response
1. Increasing the order
makes response
very flat near ω=0.
2. Converges to brick
wall
3. At the edge of
stopband ω=ωp
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All Pass Filter
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Second Order Filtering Func.
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Notch Filters
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All Pass Filter
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Second-order filter functions
using the LCR resonator
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Opamp RC Resonator
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Low Pass Filter
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High Pass Filter
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Low Pass Notch
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Transforming a first-order lowpass filter
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