1. Lecture 11
Chapter 15
ACTIVE FILTERS
BASIC FILTER RESPONSES
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2. INTRODUCTION
• The frequency response of a filter is the graph of its
voltage gain versus frequency.
• A filter is a circuit that passes certain frequencies and
attenuates or rejects all other frequencies.
• The passband of a filter is the range of frequencies
that are allowed to pass through the filter with
minimum attenuation. (usually defined as less than
– 3 dB of attenuation).
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3. • The critical frequency, fc ,(also called the cutoff
frequency) is defined as the end of the passband
and is normally specified at the point where the
response drops – 3 dB (70.7%) from the
passband response.
• Following the passband is a region called the
transition region that leads into a region called
the stopband.
• There is no precise point between the transition
region and the stopband.
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4. Low-Pass Filter Response
• A low-pass filter is one that passes frequencies
from dc to fc , and significantly attenuates all
other frequencies.
• The passband of the ideal low-pass filter is shown
in the blue shaded area of Figure 15–1(a);
• The bandwidth of an ideal low-pass filter is equal
to fc
BW = fc
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5. A decade of frequency change is a ten-times change
(increase or decrease).
15–1(a);
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7. Poles
• A single RC circuit is called a pole.(In filter terminology).
• Actual filter responses depend on the number of poles.
• The -20 dB/decade roll-off rate for the gain of a basic RC
filter means that at a frequency of 10fc,the output will be
-20 dB (10%) of the input.
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8. • The critical frequency of a low-pass RC filter
occurs when XC = R, where
fc =1/2πRC
• The output at the critical frequency is 70.7%
of the input.
• This response is equivalent to an attenuation
of -3 dB
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10. Active filters
• Filters that include one or more op-amps in the
design are called active filters.
• by combining an op-amp with frequency-selective
feedback circuits, these filters can optimize:
1. the roll-off rate or
2. phase response
• In general, the more poles the filter uses, the
steeper its transition region will be
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11. Order of Filter
• The order of a passive filter (symbolized by n)
equals the number of inductors and capacitors in
the filter.
• If a passive filter has two inductors and two
capacitors, n=4.
• The order of an active filter depends on the
number of RC circuits (called poles) it contains.
n ≈ the number of capacitors
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12. High-Pass Filter Response
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14. Band-Pass Filter Response
• A band-pass filter passes all signals lying within a band
between a lower-frequency limit and an upper-
frequency limit and essentially rejects all other
frequencies that are outside this specified band.
Bandwidth BW = fc2 - fc1
center frequency
F0 =
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15. Quality Factor (Q)
• The frequency about which the passband is
centered is called the center frequency,
f0, defined as the geometric mean of the
critical frequencies.
• Quality Factor (Q) of a band-pass filter is the
ratio of the center frequency to the bandwidth.
• Q=f0/BW
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16. • The value of Q is an indication of the selectivity of
a band-pass filter.
• The higher the value of Q, the narrower the
bandwidth and the better the selectivity for a
given value of f0.
• Band-pass filters are sometimes classified as
narrow-band (Q > 10)or wide-band(Q < 10).
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17. EXAMPLE 15–1
• A certain band-pass filter has a center frequency of 15
kHz and a bandwidth of 1 kHz. Determine Q and classify
the filter as narrow-band or wide-band
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18. Band-Stop Filter Response
also known as notch, band-reject, or band-
elimination filter.
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19. All-Pass Filter
• The all-pass filter is useful when we want to produce a
certain amount of phase shift for the signal being filtered
without changing its amplitude.
• The phase response of a filter is defined as the graph of
phase shift versus frequency.
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21. Approximate Responses
• Attenuation
• Attenuation refers to a loss of signal.
• With a constant input voltage, attenuation is
defined as the output voltage at any frequency
divided by the output voltage in the midband:
Attenuation =vout/vout(mid)
• if the output voltage is 1 V and the output voltage in the midband
is 2 V, then:
• Attenuation = 1 V/ 2 V = 0.5
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22. • Attenuation is normally expressed in decibels using
this equation:
• Decibel attenuation = -20 log attenuation
• For an attenuation of 0.5, the decibel attenuation is:
Decibel attenuation = -20 log 0.5 = 6 dB
• We will use the term attenuation to mean decibel
attenuation.
an attenuation of 3 dB means that the output voltage
is 0.707 of its midband value.
An attenuation of 6 dB means that the output voltage
is 0.5 of its midband value.
An attenuation of 12 dB means that the output
voltage is 0.25 of its midband value.
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23. Passband and Stopband Attenuation
• In filter analysis and design, the low-pass filter is a
prototype,
• Typically, any filter problem is converted into an
equivalent low-pass filter problem and solved as a
low-pass filter problem;
• the solution is converted back to the original filter
type.
• For this reason, our discussion will focus on the
low-pass filter and extend the discussion to other
filters.
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24. • Zero attenuation in the passband, infinite
attenuation in the stopband, and a vertical
transition are unrealistic.
• To build a practical low-pass filter, the three regions
are approximated as shown in Fig. 19-6.
• The passband is the set of frequencies between 0
and fc.
• The stopband is all the frequencies above fs.
• The transition region is between fc and fs.
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25. Five approximations
• The five approximations we are about to discuss are trade-
offs between the characteristics of the passband,
stopband, and transition region.
• The approximations may optimize the flatness of the
passband, or the roll-off rate, or the phase shift
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26. ??????????
• In some fi lters, the attenuation
• at the edge frequency is less than 3 dB. For
this reason, we will use f3dB for
• the frequency when the attenuation is down 3
dB and fc for the edge frequency,
• which may have a different attenuation.
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27. Order of Filter
• The order of a passive filter (symbolized by n)
equals the number of inductors and capacitors in
the filter.
• If a passive filter has two inductors and two
capacitors, n=4.
• The order of an active filter depends on the
number of RC circuits (called poles) it contains.
n ≈ the number of capacitors
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28. 1- Butterworth Approximation
• Also called the maximally flat approximation
because the passband attenuation is zero through
most of the passband and decreases gradually to Ap
at the edge of the passband.
• Well above the edge frequency, the response rolls
off at a rate of approximately 20n dB per decade,
where n is the order of the filter:
• Roll-off = 20n dB/decade
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29. • For example, a first-order Butterworth filter rolls
off at a rate of 20 dB decade,
• a fourth-order filter rolls off at a rate of 80 dB per
decade,
• The major advantage of a Butterworth filter is the
flatness of the passband response.
• The major disadvantage is the relatively slow roll-
off rate compared with the other approximations
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31. 2-Chebyshev Approximation
• It rolls off faster in the transition region than a
Butterworth filter.
• The price paid for this faster roll-off is that
ripples appear in the passband of the frequency
response.
• # Ripples =n/2
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