Lecture 7
Active Filters
Active Filters
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Agenda
 Introduction to Active Filters

Highlighting differences with respect to passive filters

First-order lowpass active filter

First-order highpass active filter

Active bandpass filter

Active bandreject filter

Example
Active Filters
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Introduction
 Some limitations of the passive filters
1.
cannot generate gain greater than one
• passive elements cannot add energy to the network
2.
may require bulky and expensive circuit elements
in case of using inductors
3.
may have source loading
4.
no isolation between the input and the output
Active Filters
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Introduction
 Active filters consist of combinations of
resistors, capacitors, and Op Amps
 They offer some advantages over passive RLC
filters

often smaller and less expensive

can provide amplifier gain in addition to providing
the same frequency response as RLC filters

can be combined with buffer amplifiers (voltage
followers) to isolate each stage of the filter from
source and load impedance effects
• realizing the desired transfer function
Active Filters
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First-Order Active Filter
•
•
•
The components selected for 𝑍𝑖 and 𝑍𝑓 determine whether the
filter is lowpass or highpass, but one of the components must be
reactive
First order filters have one pole due to the usage of one
capacitor, reactive element, besides resistors in the filtering
circuit
Active filters comprise active circuit components such Op Amps
Active Filters
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Active First-Order Lowpass Filter
 The transfer function
𝐻 𝜔 =
𝑉𝑜
𝑉𝑖
=−
𝑖𝑍𝑓
𝑖𝑅𝑖
i
 In case of passive filters  no gain
 In case of DC analysis inverting amplifier
𝐻 𝜔 =−
𝑅𝑓
𝑅𝑖
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Active First-Order Lowpass Filter
 Knowing the transfer function
 The maximum gain |𝐻 𝜔 | =
𝑅𝑓
𝑅𝑖
in case DC
(dc gain)
 How to find the 3dB cutoff or the corner
frequency 𝜔𝑐 ?


It is the frequency when the maximum |𝐻 𝜔 |
1
dropped by 2
Then, 𝜔𝑐 = 𝑅
1
𝑓 𝐶𝑓
Active Filters
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Active First-Order Highpass Filter
 The transfer function
 In case of high frequencies inverting amplifier
𝐻 𝜔 =−
𝑅𝑓
𝑅𝑖
 the 3dB cutoff or the corner frequency 𝜔𝑐
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Active Bandpass Filter
Unity gain
Unity gain
𝜔2 > 𝜔1
Active Filters
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Active Bandpass Filter
 Analysis (cascaded Op Amp circuit)
Active Filters
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Active Bandpass Filter
 Analysis (cascaded Op Amp
circuit)
 The lowpass section sets the
upper corner frequency as
 while the highpass section sets
the lower corner frequency as
Active Filters
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Active Bandpass Filter
 Analysis (cascaded Op Amp circuit)
 With the known values of 𝜔1 and 𝜔2 , the
center frequency𝜔0 , bandwidth 𝐵, and quality
factor 𝑄

the passband gain 𝐾:
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Active Bandreject Filter
Active Filters
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Active Bandreject Filter
 Analysis
Active Filters
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Active Bandreject Filter
 Analysis
 The transfer function


In terms of corner frequencies
1
1
𝜔1 =
,𝜔 =
𝐶1 𝑅 2 𝐶2 𝑅
In the two passbands ( 𝜔 → 0
and 𝜔 → ∞) the gain is
Active Filters
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Example
 Design a first-order lowpass active filter with
a dc gain of 4 and a corner frequency of 500
Hz (in other words, get values for 𝑅𝑖 𝑅𝑓 𝑎𝑛𝑑 𝐶𝑓 )

Solution
• The dc gain
• We can assume a value for 𝐶𝑓
• 𝐶𝑓 = 0.2 𝜇𝐹
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Lecture-related Question
 Design a first-order highpass active filter with a high-
frequency gain of 5 and a corner frequency of 2 kHz. Use
a 0.1 𝜇𝐹 capacitor in your design.
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Lecture Summary
Covered material
 Introduction to Active Filters






Highlighting differences with respect to passive filters
First-order lowpass active filter
First-order highpass active filter
Active bandpass filter
Active bandreject filter
Example
Next Week: Mid-term Exam
Good Luck
Active Filters
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