Transcript Lecture Slide 8

Active Filters: concepts
• All input signals are composed of sinusoidal components of various
frequencies, amplitudes and phases.
• If we are interested in a certain range of frequencies, we can
design filters to eliminate frequency components outside the range
• Filters are usually categorized into four types: low-pass filter, highpass filter, band-pass filter and band-reject filter.
• Low-pass filter passes components with frequencies from DC up to
its cutoff frequency and rejects components above the cutoff
frequency.
• Low-pass filter composed of OpAmp are called active filter (as
opposed to lumped passive filter with resistor, capacitor and
inductor)
• Active filters are desired to have the following characteristics:
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Contain few components
Insensitive to component variation
Not-too-hard-to-meet specifications on OpAmp
Easy reconfiguration to support different requirements (like cutoff freq)
Require a small spread of component values
Applications of Analog Filters
• Analog filters can be found in almost every electronic circuit.
• Audio systems use them for pre-amplification, equalization, and tone
control.
• In communication systems, filters are used for tuning in specific
frequencies and eliminating others (for example, to filter out noise).
• Digital signal processing systems use filters to prevent the aliasing of outof-band noise and interference.
Butterworth low-pass filter
• Many low-pass filter are designed to have a Butterworth transfer
function with magnitude response as follows:
| H ( f ) |
H0
1  ( f / fb )
2n
, where n is the order of the filter and
f b is the 3db cutoff frequency, H 0 is the gain magnitude at DC.
Graphs from Prentice Hall
Low-pass filter: Sallen-Key Circuits
• Active low-pass Butterworth filter can be implemented by
cascading modified Sallen-Key circuits.
• The Sallen-Key circuit itself is a 2nd order filter. To obtain an nth
order filter, n/2 SK circuits should be cascaded
H( f ) 
Vo ( s )
K

, the 3db cutoff frequency is f b  1 /( 2RC )
2 2 2
Vin ( s ) 1  (3  K ) RCs  R C s
• During design, capacitance
can be selected first and then
resistor values.
• As K increase from 0 to 3,
the transfer function displays
more and more peaking.
• It turns out that if K>3, then
the circuit is not stable.
• Empirical values have been
found for filters of different
orders
Example of a 4th-order Lowpass filter
by cascading two 2nd-order SK filters
Comparison of gain versus frequency for the stages of the fourth-order Butterworth low-pass filter.
Butterworth high-pass filter
• By a change, the lowpass Butterworth transfer function can be
transformed to a high-pass function.
| H ( f ) |
H0
1  ( fb / f )
2n
, where n is the order of the filter and
f b is the 3db cutoff frequency, H 0 is the gain magnitude at DC.
Butterworth high-pass filter: Sallen-Key
• By a change, the lowpass Butterworth transfer function can be
transformed to a high-pass function.
• With real OpAmp, the Sallen-Key is not truly a high-pass filter,
because the gain of the OpAmp eventually falls off. However, the
frequencies at which the OpAmp gain is fairly high, the circuit
behaves as a high-pass filter.
• Since the high-pass Sallen
Key circuit is equivalent the
same as the low-pass one,
the empirical values for K
would be still valid in this
case also.
Band-pass filter: Sallen-Key Circuits
• If we need to design a band-pass filter in which the lower cutoff
frequency is much less than the upper cutoff frequency, we can
cascade a low-pass filter with a high-pass filter.
• The below band-pass filter uses the first stage as a low-pass filter
which passes frequency less than 10KHz and the second stage as
a high-pass filter that passes only frequency above 100Hz. Thus,
frequency components in-between is passed to the output.
Graphs from Prentice Hall
Figure 11.11 Bode plots of gain magnitude for the active filter of Example 11.2.
A summary
Application: Equalizer (EQ)