Active Filters

1
Active Filters

• Conventional passive filters consist of LCR
  networks.
• Inductors are undesirable components
• They are particularly non-ideal (lossy)
• They are bulky and expensive
• Active filters replace inductors using op-amp
  based equivalent circuits.

2
Active Filter Designs

• Three active filter design techniques will be
  covered
• Synthesis by Sections
• Cascade of second order sections.
• Component Simulation
• Replace inductors with op-amp inductor
  simulations.
• Operational Simulation
• Simulate all currents and voltages in the LCR
  ladder using an analogue computer.

3
Analogue Filter Responses
H(f)
H(f)
0
0
f
f
fc
fc
Ideal brick wall filter
Practical filter
4
Standard Transfer Functions

• Butterworth
• Flat Pass-band.
• 20n dB per decade roll-off.
• Chebyshev
• Pass-band ripple.
• Sharper cut-off than Butterworth.
• Elliptic
• Pass-band and stop-band ripple.
• Even sharper cut-off.
• Bessel
• Linear phase response i.e. no signal distortion
  in pass-band.

5
(No Transcript)
6
Analogue Transfer Functions
The transfer function of any analogue filter
(active or passive) can be expressed as the ratio
of two polynomials
7
Poles and Zeros

• Poles
• Complex values of s where the transfer function
  is infinite.
• i.e. the denominator of the transfer function is
  zero.
• Zeros
• Complex values of s where the transfer function
  is zero.
• An N-th order filter will have N poles and up to
  N zeros.
• Some poles may be in the same place (as may some
  zeros).

8
Example Two Pole Bessel Filter
Low pass, cut-off frequency 1 rad/s, from
tables
9
Operational Amplifiers

• All the active filters we shall study are based
  on operational amplifiers (op-amps).
• Analysis of linear op-amp circuits is usually
  based on simplifying assumptions
• The difference between the non-inverting and
  inverting inputs is zero.
• The input current is zero.
• The output voltage and current is arbitrary.

10
Op-Amp Assumptions
I
V

Iout
Vout
I-
-
V-
11
Inverting Amplifier
Z2
Z1
VIN
-
VOUT

0 V
12
Non-Inverting Amplifier

VIN
VOUT
-
Z1
Z2
0 V
13
Buffer Amplifier

VIN
VOUT
-

• Output voltage Input voltage
• Input impedance is infinite
• Output impedance is zero

14
Single-Pole Passive Filter
R
C
vin
vout

• First order low pass filter
• Cut-off frequency 1/CR rad/s
• Problem Any load (or source) impedance will
  change frequency response.

15
Single-Pole Active Filter
R
C
vin
vout

• Same frequency response as passive filter.
• Buffer amplifier does not load RC network.
• Output impedance is now zero.

16
Low-Pass and High-Pass Designs
High Pass
Low Pass
17
Higher Order Filters

• You might think we could make higher order
  filters by simply cascading N first order filters
• This doesnt work
• The single pole of a first order filter must be
  purely real (no imaginary part)
• The poles of a higher order filter usually need
  to be complex
• Solution Use second order sections, each one
  synthesising a conjugate pair of complex poles

18
Summary

• Active filter designs aim to replace the
  inductors in passive filters.
• Design techniques
• Synthesis by sections
• Component simulation
• Operational simulation
• All based on op-amps understanding of basic
  op-amp circuits is essential.