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Chapter 21
Decibels, Filters,
and Bode Plots
Introductory Circuit Analysis, 12/e
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OBJECTIVES
• Develop confidence in the use of logarithms and
decibels in the description of power and voltage
levels.
• Become familiar with the frequency response of
high- and low-pass filters. Learn to calculate the
cutoff frequency and describe the phase response.
• Be able to calculate the cutoff frequencies and
sketch the frequency response of a pass-band or
stop-band filter.
• Develop skills in interpreting and establishing the
Bode response of any filter.
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INTRODUCTION
• The unit decibel (dB), defined by a
logarithmic expression, is used
throughout the industry to define
levels of audio, voltage gain, energy,
field strength, and so on.
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INTRODUCTION
Logarithms
• Basic Relationships
– Let us first examine the relationship between
the variables of the logarithmic function.
– The mathematical expression:
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INTRODUCTION
Logarithms
• Some Areas of Application
– The following are some of the most common
applications of the logarithmic function:
• 1. The response of a system can be plotted for a
range of values that may otherwise be impossible or
unwieldy with a linear scale.
• 2. Levels of power, voltage, and the like can be
compared without dealing with very large or very
small numbers that often cloud the true impact of the
difference in magnitudes.
• 3. A number of systems respond to outside stimuli in
a nonlinear logarithmic manner.
• 4. The response of a cascaded or compound system
can be rapidly determined using logarithms if the gain
of each stage is known on a logarithmic basis.
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PROPERTIES OF LOGARITHMS
• There are a few characteristics of logarithms that
should be emphasized:
– The common or natural logarithm of the number 1 is
0
– The log of any number less than 1 is a negative
number
– The log of the product of two numbers is the sum of
the logs of the numbers
– The log of the quotient of two numbers is the log of
the numerator minus the log of the denominator
– The log of a number taken to a power is equal to the
product of the power and the log of the number
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DECIBELS
• Power Gain
• Voltage Gain
• Human Auditory Response
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DECIBELS
TABLE 21.1
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DECIBELS
Pout Pin
dB  10log10 ( Pout Pin )
0.1
-10
1
0
2
3
10
10
20
13
100
20
1000
20
10000
40
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DECIBELS
TABLE 21.2 Typical sound levels and their decibel levels.
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DECIBELS
FIG. 21.5 LRAD (Long Range Acoustic Device) 1000X.
(Courtesy of the American Technology Corporation.)
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FILTERS
• Any combination of passive (R, L, and C)
and/or active (transistors or operational
amplifiers) elements designed to select or
reject a band of frequencies is called a
filter.
• In communication systems, filters are used
to pass those frequencies containing the
desired information and to reject the
remaining frequencies.
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FILTERS
• In general, there are two
classifications of filters:
– Passive filters
– Active filters
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FILTERS
FIG. 21.7 Defining the four broad
categories of filters.
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R-C LOW-PASS FILTER
FIG. 21.8 Low-pass filter.
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FIG. 21.9 R-C low-pass
filter at low frequencies.
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R-C LOW-PASS FILTER
FIG. 21.10 R-C low-pass filter
at high frequencies.
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FIG. 21.11 Vo versus frequency for a lowpass R-C filter.
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R-C LOW-PASS FILTER
FIG. 21.12 Normalized plot of Fig. 21.11.
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R-C LOW-PASS FILTER
FIG. 21.13 Angle by which Vo leads Vi.
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R-C LOW-PASS FILTER
FIG. 21.14 Angle by which Vo lags Vi.
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R-C LOW-PASS FILTER
FIG. 21.15 Low-pass R-L filter.
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FIG. 21.16 Example 21.5.
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R-C LOW-PASS FILTER
FIG. 21.17 Frequency response for the low-pass R-C network in Fig.
21.16.
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R-C LOW-PASS FILTER
FIG. 21.18 Normalized plot of Fig. 21.17.
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R-C HIGH-PASS FILTER
FIG. 21.19 High-pass filter.
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R-C HIGH-PASS FILTER
FIG. 21.20 R-C high-pass filter at very
high frequencies.
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FIG. 21.21 R-C high-pass filter
at f = 0 Hz.
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R-C HIGH-PASS FILTER
FIG. 21.22 Vo versus frequency for a high-pass R-C filter.
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R-C HIGH-PASS FILTER
FIG. 21.23 Normalized plot of Fig. 21.22.
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R-C HIGH-PASS FILTER
FIG. 21.24 Phase-angle response for the high-pass R-C filter.
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R-C HIGH-PASS FILTER
FIG. 21.25 High-pass R-L filter.
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R-C HIGH-PASS FILTER
FIG. 21.26 Normalized plots for a low-pass and a high-pass filter using the same elements.
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R-C HIGH-PASS FILTER
FIG. 21.27 Phase plots for a low-pass and a high-pass filter using the same elements.
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PASS-BAND FILTERS
FIG. 21.28 Series resonant pass-band filter.
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PASS-BAND FILTERS
FIG. 21.29 Parallel resonant pass-band filter.
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PASS-BAND FILTERS
FIG. 21.30 Series resonant pass-band filter for Example
21.7.
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PASS-BAND FILTERS
FIG. 21.31 Pass-band response for the network.
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PASS-BAND FILTERS
FIG. 21.32 Normalized plots for the pass-band filter in Fig. 21.30.
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PASS-BAND FILTERS
FIG. 21.33 Pass-band filter.
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PASS-BAND FILTERS
FIG. 21.34 Pass-band characteristics.
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PASS-BAND FILTERS
FIG. 21.35 Pass-band filter.
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FIG. 21.36 Pass-band characteristics for the
filter in Fig. 21.35.
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PASS-BAND FILTERS
FIG. 21.37 Network of Fig. 21.35 at f = 994.72 kHz.
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BAND-REJECT FILTERS
• Since the characteristics of a bandreject filter (also called stop-band or
notch filter) are the inverse of the
pattern obtained for the band-pass
filter, a band-reject filter can be
designed by simply applying
Kirchhoff’s voltage law to each circuit.
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BAND-REJECT FILTERS
FIG. 21.38 Demonstrating how an applied signal of fixed magnitude can be broken
down into a pass-band and band-reject response curve.
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BAND-REJECT FILTERS
FIG. 21.39 Band-reject filter using a series resonant circuit.
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BAND-REJECT FILTERS
FIG. 21.40 Band-reject filter using a parallel resonant network.
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BAND-REJECT FILTERS
FIG. 21.41 Band-reject filter.
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BAND-REJECT FILTERS
FIG. 21.42 Band-reject characteristics.
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