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CHAPTER 14
Introduction to Frequency
Selective Circuits
Electronic Circuits, Tenth Edition
James W. Nilsson | Susan A. Riedel
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CHAPTER CONTENTS
• 14.1 Some Preliminaries
• 14.2 Low-Pass Filters
• 14.3 High-Pass Filters
• 14.4 Bandpass Filters
• 14.5 Bandreject Filters
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James W. Nilsson | Susan A. Riedel
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CHAPTER OBJECTIVES
1. Know the RL and RC circuit configurations that act
as low-pass filters and be able to design RL and RC
circuit component values to meet a specified cutoff
frequency.
2. Know the RL and RC circuit configurations that act
as high-pass filters and be able to design RL and RC
circuit component values to meet a specified cutoff
frequency.
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CHAPTER OBJECTIVES
3. Know the RLC circuit configurations that act as
bandpass filters, understand the definition of and
relationship among the center frequency, cutoff
frequencies, bandwidth, and quality factor of a
bandpass filter, and be able to design RLC circuit
component values to meet design specifications.
4. Know the RLC circuit configurations that act as
bandreject filters, understand the definition of and
relationship among the center frequency, cutoff
frequencies, bandwidth, and quality factor of a
bandreject filter, and be able to design RLC circuit
component values to meet design specification.
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Frequency-selective circuits
• Varying source frequency on circuit voltages and
currents. The result of this analysis is the frequency
response of a circuit.
• Frequency-selective circuits are also called filters,
such as telephones, radios, televisions, and satellites,
employ frequency-selective circuits.
Figure 14.1 The action of a filter on an
input signal results in an output signal.
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14.1 Some Preliminaries
• The signals passed from the input to the output fall
within a band of frequencies called the passband.
Input voltages outside this band have their
magnitudes attenuated by the circuit and are thus
effectively prevented from reaching the output
terminals of the circuit. Frequencies not in a circuit’s
passband are in its stopband. Frequency-selective
circuits are categorized by the location of the
passband.
Figure 14.2 A circuit
with voltage input and
output.
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• One way of identifying the type of frequency-
selective circuit is to examine a frequency response
plot. A frequency response plot shows how a circuit’s
transfer function (both amplitude and phase) changes
as the source frequency changes.
• A frequency response plot has two parts. One is a
graph of |H(jω)|versus frequency ω. This part of the
plot is called the magnitude plot. The other part is a
graph of θ(jω) versus frequency ω. This part is called
the phase angle plot.
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• One passband and one stopband, which are defined
•
•
•
•
by the cutoff frequency.
Low-pass filter, which passes signals at frequencies
lower than the cutoff frequency from the input to the
output.
High-pass filter, which passes signals at frequencies
higher than the cutoff frequency.
Bandpass filter, which passes a source voltage to the
output only when the source frequency is within the
band defined by the two cutoff frequencies.
Bandreject filter, which passes a source voltage to the
output only when the source frequency is outside the
band defined by the two cutoff frequencies.
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Figure 14.3 Ideal frequency response plots of the four
types of filter circuits. (a) An ideal low-pass filter. (b) An ideal
high-pass filter. (c) An ideal bandpass filter. (d) An ideal
bandreject filter.
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Passive filters
• Passive filters, their filtering capabilities depend only
on the elements: resistors, capacitors, and inductors.
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14.2 Low-Pass Filters
• The Series RL Circuit—
Qualitative Analysis
Figure 14.4 (a) A series RL lowpass filter. (b) The equivalent
circuit at ω = 0 and (c) The
equivalent circuit at ω = ∞.
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Figure 14.5 The frequency response plot
for the series RL circuit in Fig. 14.4(a).
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Defining the Cutoff Frequency
• The definition for cutoff frequency widely used by
electrical engineers is the frequency for which the
transfer function magnitude is decreased by the factor
1 2 from its maximum value:
where Hmax is the maximum magnitude of the transfer
function.
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The Power Delivered
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• At the cutoff frequency the average power delivered
by the circuit is one half the maximum average power.
Thus ωc is also called the half-power frequency.
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The Series RL Circuit—Quantitative
Analysis
Figure 14.6 The s-domain equivalent
for the circuit in Fig. 14.4(a).
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• Cutoff frequency for RL filters
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Example 14.1
• Electrocardiology is the study of the electric signals
produced by the heart. These signals maintain the
heart’s rhythmic beat, and they are measured by an
instrument called an electrocardiograph. This
instrument must be capable of detecting periodic
signals whose frequency is about 1 Hz (the normal
heart rate is 72 beats per minute). The instrument
must operate in the presence of sinusoidal noise
consisting of signals from the surrounding electrical
environment, whose fundamental frequency is 60
Hz—the frequency at which electric power is
supplied.
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Example 14.1
• Choose values for R and L in the circuit of Fig. 14.4(a)
such that the resulting circuit could be used in an
electrocardiograph to filter out any noise above 10 Hz
and pass the electric signals from the heart Vo at or
near 1 Hz. Then compute the magnitude of at 1 Hz,
10 Hz, and 60 Hz to see how well the filter performs.
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Example 14.1
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Example 14.1
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Example 14.1
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A Series RC Circuit as a low-pass filter
1.
2.
3.
Zero frequency (ω = 0): The impedance of
the capacitor is infinite, and the capacitor
acts as an open circuit. The input and
output voltages are thus the same.
Frequencies increasing from zero: The
impedance of the capacitor decreases
relative to the impedance of the resistor,
and the source voltage divides between the
resistive impedance and the capacitive
impedance. The output voltage is thus
smaller than the source voltage.
Infinite frequency (ω = ∞): The impedance
of the capacitor is zero, and the capacitor
acts as a short circuit. The output voltage
is thus zero.
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Figure 14.7 A series RC
low-pass filter.
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Example 14.2
• For the series RC circuit in Fig. 14.7:
a) Find the transfer function between the source voltage
and the output voltage.
b) Determine an equation for the cutoff frequency in the
series RC circuit.
c) Choose values for R and C that will yield a lowpass
filter with a cutoff frequency of 3 kHz.
Figure 14.7 A series RC
low-pass filter.
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Example 14.2
Figure 14.8 The sdomain equivalent for the
circuit in Fig. 14.7.
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Example 14.2
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Example 14.2
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Figure 14.9 Two lowpass filters, the series RL
and the series RC,
together with their
transfer functions and
cutoff frequencies.
• Transfer function for a low-pass filter
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Relating the Frequency Domain to the
Time Domain
• This result is a direct consequence of the relationship
between the time response of a circuit and its
frequency response.
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14.3 High-Pass Filters
• The Series RC Circuit—
Qualitative Analysis
Figure 14.10 (a) A series
RC high-pass filter; (b) the
equivalent circuit at ω = 0
and (c) the equivalent
circuit at ω = ∞.
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Figure 14.11 The frequency response plot
for the series RC circuit in Fig. 14.10(a).
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The Series RC Circuit—Quantitative
Analysis
Figure 14.12 The s-domain equivalent of
the circuit in Fig. 14.10(a).
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Example 14.3
• Show that the series RL circuit in Fig. 14.13 also acts
like a high-pass filter:
a) Derive an expression for the circuit’s transfer
function.
b) Use the result from (a) to determine an equation for
the cutoff frequency in the series RL circuit.
c) Choose values for R and L that will yield a highpass
filter with a cutoff frequency of 15 kHz.
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Example 14.3
Figure 14.13
14.3.
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The circuit for Example
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Example 14.3
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Example 14.3
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Example 14.3
Figure 14.14 The sdomain equivalent of the
circuit in Fig. 14.13.
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Example 14.4
• Examine the effect of placing a load resistor in
parallel with the inductor in the RL high-pass filter
shown in Fig. 14.15:
a) Determine the transfer function for the circuit in Fig.
14.15.
b) Sketch the magnitude plot for the loaded RL highpass filter, using the values for R and L from the
circuit in Example 14.3(c) and letting. On the same
graph, sketch the magnitude plot for the unloaded RL
high-pass filter of Example 14.3(c).
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Example 14.4
Figure 14.15 The circuit
for Example 14.4.
Figure 14.16 The sdomain equivalent of the
circuit in Fig. 14.15.
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Example 14.4
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Example 14.4
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Example 14.4
Figure 14.17 The magnitude plots
for the unloaded RL high-pass filter of
Fig 14.13 and the loaded RL highpass filter of Fig. 14.15.
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Transfer function for a high-pass filter
Figure 14.18 Two
high-pass filters, the
series RC and the series
RL, together with their
transfer functions and
cutoff frequencies.
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14.4 Bandpass Filters
• Ideal bandpass filters have two cutoff frequencies
which identify the passband.
• For realistic bandpass filters, cutoff frequencies are
defined as the frequencies for which the magnitude of
the transfer function equals (1/ 2)Hmax .
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Center Frequency, Bandwidth, and
Quality Factor
• Center frequency ωo, defined as the frequency for
•
•
•
•
which a circuit’s transfer function is purely real.
Resonant frequency, is the same name given to the
frequency that characterizes the natural response of
the second-order circuits.
When a circuit is driven at the resonant frequency, we
say that the circuit is in resonance. O  c1c 2
Bandwidth, b, is the width of the passband.
Quality factor, is the ratio of the center frequency to
the bandwidth.
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The Series RLC Circuit—Qualitative
Analysis
Figure 14.19 (a) A series
RLC bandpass filter; (b) the
equivalent circuit for ω = 0;
and (c) the equivalent circuit
for ω = ∞.
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The frequency response plot for the
series RLC bandpass filter circuit
Figure 14.20 The frequency
response plot for the series RLC
bandpass filter circuit in Fig. 14.19.
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The Series RLC Circuit—Quantitative
Analysis
Figure 14.21 The s-domain equivalent for the
circuit in Fig. 14.19(a).
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• Center frequency
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Cutoff frequencies, series RLC filters
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Relationship between center frequency
and cutoff frequencies
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Relationship between bandwidth and
cutoff frequencies
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Quality factor
• The ratio of center frequency to bandwidth.
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Cutoff frequencies
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Example 14.5
• A graphic equalizer is an audio amplifier that allows
you to select different levels of amplification within
different frequency regions. Using the series RLC
circuit in Fig. 14.19(a), choose values for R, L, and C
that yield a bandpass circuit able to select inputs
within the 1–10 kHz frequency band. Such a circuit
might be used in a graphic equalizer to select this
frequency band from the larger audio band (generally
0–20 kHz) prior to amplification.
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Example 14.5
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Example 14.5
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Example 14.5
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Example 14.5
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Example 14.6
a) Show that the RLC circuit in Fig. 14.22 is also a
bandpass filter by deriving an expression for the
transfer function
b) Compute the center frequency,
c) Calculate the cutoff frequencies, and the bandwidth,
and the quality factor, Q.
d) Compute values for R and L to yield a bandpass filter
with a center frequency of 5 kHz and a bandwidth of
200 Hz, using a capacitor.
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Example 14.6
Figure 14.22
The circuit for Example 14.6.
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Example 14.6
Figure 14.23
The s-domain equivalent of the circuit in Fig. 14.22.
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Example 14.6
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Example 14.6
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Example 14.6
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Example 14.6
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Example 14.6
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Example 14.7
• For each of the bandpass filters we have constructed,
we have always assumed an ideal voltage source, that
is, a voltage source with no series resistance. Even
though this assumption is often valid, sometimes it is
not, as in the case where the filter design can be
achieved only with values of R, L, and C whose
equivalent impedance has a magnitude close to the
actual impedance of the voltage source. Examine the
effect of assuming a nonzero source resistance, Ri, on
the characteristics of a series RLC bandpass filter.
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Example 14.7
a) Determine the transfer function for the circuit in Fig.
14.24.
b) Sketch the magnitude plot for the circuit in Fig. 14.24,
using the values for R, L, and C from Example 14.5
and setting Ri = R. On the same graph, sketch the
magnitude plot for the circuit in Example 14.5, where
Ri = 0.
Figure 14.24
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The circuit for Example 14.7.
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Example 14.7
Figure 14.25 The sdomain equivalent of
the circuit in Fig.
14.24.
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Example 14.7
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Example 14.7
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Example 14.7
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Example 14.7
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Example 14.7
Figure 14.26 The magnitude plots
for a series RLC bandpass filter with a
zero source resistance and a nonzero
source resistance.
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Figure 14.27 Two RLC
bandpass filters, together
with equations for the
transfer function, center
frequency, and bandwidth of
each.
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Transfer function for RLC bandpass
filter
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Relating the Frequency Domain to the
Time Domain
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14.5 Bandreject Filters
• Bandreject filters are characterized by the same
parameters as bandpass filters: the two cutoff
frequencies, the center frequency, the bandwidth, and
the quality factor.
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The Series RLC Circuit—Qualitative
Analysis
Figure 14.28 (a) A series
RLC bandreject filter. (b) The
equivalent circuit for ω = 0 (c)
The equivalent circuit for ω =
∞.
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Figure 14.29 The frequency
response plot for the series RLC
bandreject filter circuit in Fig.
14.28(a).
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The Series RLC Circuit—Quantitative
Analysis
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Figure 14.30 The sdomain equivalent of the
circuit in Fig. 14.28(a).
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Example 14.8
• Using the series RLC circuit in Fig. 14.28(a),
compute the component values that yield a bandreject
filter with a bandwidth of 250 Hz and a center
frequency of 750 Hz. Use a 100 nF capacitor.
Compute values for R, L, c1, c2 and Q.
Figure 14.28 (a) A series
RLC bandreject filter.
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Example 14.8
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Example 14.8
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Transfer function for RLC bandreject
filter
Figure 14.31 Two RLC bandreject filters, together with equations for the
transfer function, center frequency, and bandwidth of each.
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Summary
• A frequency selective circuit, or filter, enables
signals at certain frequencies to reach the output, and
it attenuates signals at other frequencies to prevent
them from reaching the output. The passband
contains the frequencies of those signals that are
passed; the stopband contains the frequencies of
those signals that are attenuated.
• The cutoff frequency, c, identifies the location on
the frequency axis that separates the stopband from
the passband. At the cutoff frequency, the magnitude
of the transfer function equals (1/ 2)Hmax .
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Summary
• A low-pass filter passes voltages at frequencies
below and attenuates frequencies above c. Any
circuit with the transfer function
functions as a low-pass filter.
• A high-pass filter passes voltages at frequencies
above and attenuates voltages at frequencies below
Any circuit with the transfer function
functions as a high-pass filter.
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Summary
• Bandpass filters and bandreject filters each have two
cutoff frequencies, c1 and c2.These filters are
further characterized by their center frequency (o),
bandwidth (b), and quality factor (Q). These
quantities are defined as
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Summary
• A bandpass filter passes voltages at frequencies
within the passband, which is between c1 and c2. It
attenuates frequencies outside of the passband.Any
circuit with the transfer function
functions as a bandpass filter.
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Summary
• A bandreject filter attenuates voltages at frequencies
within the stopband, which is between c1 and c2. It
passes frequencies outside of the stopband. Any
circuit with the transfer function
functions as a bandreject filter.
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Summary
• Adding a load to the output of a passive filter changes
its filtering properties by altering the location and
magnitude of the passband. Replacing an ideal
voltage source with one whose source resistance is
nonzero also changes the filtering properties of the
rest of the circuit, again by altering the location and
magnitude of the passband.
Electronic Circuits, Tenth Edition
James W. Nilsson | Susan A. Riedel
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