Active Filters

1
Active Filters

• Based on use of amplifiers to achieve
  filter function
• Frequently use op amps so filter may have
  some gain as well.
• Alternative to LRC-based filters
• Benefits
• Provide improved characteristics
• Smaller size and weight
• Monolithic integration in IC
• Implement without inductors
• Lower cost
• More reliable
• Less power dissipation
• Price
• Added complexity
• More design effort

Vo(s)
Vi(s)
Transfer Function
2
Filter Types

• Four major filter types
• Low pass (blocks high frequencies)
• High pass (blocks low frequencies)
• Bandpass (blocks high and low
  frequencies except in narrow band)
• Bandstop (blocks frequencies in a
  narrow band)

Low Pass
High Pass
Bandpass
Bandstop
3
Filter Specifications

• Specifications - four parameters needed
• Example low pass filter Amin, Amax, Passband,
  Stopband
• Parameters specify the basic characteristics
  of filter, e.g. low pass filtering
• Specify limitations to its ability to
  filter, e.g. nonuniform transmission in
  passband, incomplete blocking of frequencies
  in stopband

4
Filter Transfer Function

• Any filter transfer function T(s) can be
  written as a ratio of two polynomials in
  s
• Where M lt N and N is called the order
  of the filter function
• Higher N means better filter performance
• Higher N also means more complex circuit
  implementation
• Filter transfer function T(s) can be
  rewritten as
• where zs are zeros and ps are poles
  of T(s)
• where poles and zeroes can be real or
  complex
• Form of transfer function is similar to
  low frequency function FL(s) seen previously
  for amplifiers where A(s) AMFL(s)FH(s)

5
First Order Filter Functions
First order filter functions are of the
general form
Low Pass
a1 0
High Pass
a0 0
6
First Order Filter Functions
First order filter functions are of the
form
General
a1 ? 0, a2 ? 0
All Pass
7
Example of First Order Filter - Passive

• Low Pass Filter

0 dB
8
Example of First Order Filter - Active

• Low Pass Filter

I1 Io
Io
V_ 0
Gain
Filter function
20 log (R2/R1)
9
Second-Order Filter Functions
j?
Second order filter functions are of the
form which we can rewrite as where
?o and Q determine the poles There
are seven second order filter types Low
pass, high pass, bandpass, notch, Low-pass
notch, High-pass notch and All-pass
s-plane
x
?o
?
x
This looks like the expression for the new
poles that we had for a feedback
amplifier with two poles.
10
Second-Order Filter Functions
Low Pass
a1 0, a2 0
High Pass
a0 0, a1 0
Bandpass
a0 0, a2 0
11
Second-Order Filter Functions
Notch
a1 0, ao ?o2
Low Pass Notch
a1 0, ao gt ?o2
High Pass Notch
a1 0, ao lt ?o2
All-Pass
12
Passive Second Order Filter Functions

• Second order filter functions can be
  implemented with simple RLC circuits
• General form is that of a voltage divider
  with a transfer function given by
• Seven types of second order filters
• High pass
• Low pass
• Bandpass
• Notch at ?o
• General notch
• Low pass notch
• High pass notch

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Example - Passive Second Order Filter Function

• Low pass filter

T(dB)
Q
0 dB
General form of transfer function
?
?0
14
Example - Passive Second Order Filter Function

• Bandpass filter

T(dB)
0 dB
General form of transfer function
-3 dB
?
?0
15
Single-Amplifier Biquadratic Active Filters

• Generate a filter with second order
  characteristics using amplifiers, Rs and
  Cs, but no inductors.
• Use op amps since readily available and
  inexpensive
• Use feedback amplifier configuration
• Will allow us to achieve filter-like
  characteristics
• Design feedback network of resistors and
  capacitors to get the desired frequency form
  for the filter, i.e. type of filter, e.g
  bandpass.
• Determine sizes of Rs and Cs to get
  desired frequency characteristics
  (?0 and Q), e.g. center frequency and
  bandwidth.
• Note The frequency characteristics for the
  active filter will be independent of the
  op amps frequency characteristics.

Example - Bandpass Filter
General form of transfer function
16
Design of the Feedback Network

• General form of the transfer function for
  feedback network is
• Loop gain for feedback amplifier is
• Gain with feedback for feedback amplifier
  is
• Poles of feedback amplifier (filter) are
  found from setting

Conclusion Poles of the filter are the
same as the zeros of the RC feedback
network ! Design Approach 1. Analyze RC
feedback network to find expressions
for zeros in terms Rs and Cs. 2. From
desired ?0 and Q for the filter, calculate
Rs and Cs. 3. Determine where to inject
input signal to get desired form of
filter, e.g. bandpass.
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Design of the Feedback Network

• Bridged-T networks (2 Rs and 2Cs) can be
  used as feedback networks to implement
  several of the second order filter
  functions.
• Need to analyze bridged-T network to get
  transfer function t(s) of the feedback
  network. We will show that
• Zeros of this t(s) will give the pole
  frequencies for the active filter..

Bridged T network
General form of filters transfer function
18
Analysis of t(s) for Bridged-T Network
Analysis for t(s) Va / Vb
I3 (Vb-Va)/R3
I2 I3
I1
Ia 0
V12
Va
Vb
I4
19
Analysis of Bridged-T Network

• Setting numerator of t(s) 0 gives zeroes
  of t(s), which are also the poles of
  filters transfer function T(s) since
• Where the general form of filters T(s) is
  
• Then comparing the numerator of t(s) and
  the denominator of T(s), ?o and Q are
  related to the Rs and Cs by
• so
• Given the desired filter characteristics
  specified by ?o and Q, the Rs and Cs can
  now be calculated to build the filter.

These have the same form a quadratic !