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1044 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
American (Madison, WI;
Summit, NJ;
Cambridge, MA)
(1905–81)
V.P. at Bell
Laboratories
Professor of Systems
Engineering,
Harvard University
FIG. 23.44
Hendrik Wade Bode
Courtesy of AT&T Archives
In his early years at Bell Laboratories, Hendrik Bode
was involved with electric filter and equalizer design.
He then transferred to the Mathematics Research
Group, where he specialized in research pertaining
to electrical networks theory and its
application to long-distance communication facilities.
In 1946 he was awarded the Presidential Certificate
of Merit for his work in electronic fire control
devices. In addition to the publication of the book
Network Analysis and Feedback Amplifier Design in
1945, which is considered a classic in its field, he
has been granted 25 patents in electrical engineering
and systems design. Upon retirement, Bode was
elected Gordon McKay Professor of Systems Engineering
at Harvard University. He was a fellow of
the IEEE and American Academy of Arts and Sciences.
the series resonant circuit is capacitive. The inductance L p is chosen to
establish parallel resonance at the resonant stop-band frequency. The
high impedance of the parallel resonant circuit will result in a very low
load voltage V L.
For rejected frequencies below the pass-band, the networks should
appear as shown in Fig. 23.43. For the reverse situation, L s in Fig.
23.43(a) and L p in Fig. 23.43(b) are replaced by capacitors.
EXAMPLE 23.9 For the network of Fig. 23.43(b), determine L s and
L p for a capacitance C of 500 pF if a frequency of 200 kHz is to be
rejected and a frequency of 600 kHz accepted.
Solution: For series resonance, we have
f s �
1
and Ls � ���� �140.7 mH
4p
At 200 kHz,
XLs � qL � 2pfsLs � (2p)(200 kHz)(140.7 mH) � 176.8 �
2 (600 kHz) 2 1
�2 2
4p f sC
(500 pF)
1
1
and XC ������ �1591.5 �
qC (2p)(200 kHz)(500 pF)
For the series elements,
j (XLs � XC) � j (176.8 ��1591.5 �) ��j 1414.7 ���j X′ C
At parallel resonance (Ql ≥ 10 assumed),
XLp � X′ C
X Lp
1414.7 �
and Lp ����� �1.13 mH
q (2p)(200 kHz)
The frequency response for the preceding network appears as one of the
examples of PSpice in the last section of the chapter.
23.10 BODE PLOTS
1
� 2p�L�C�
There is a technique for sketching the frequency response of such factors
as filters, amplifiers, and systems on a decibel scale that can save a
great deal of time and effort and provide an excellent way to compare
decibel levels at different frequencies.
The curves obtained for the magnitude and/or phase angle versus
frequency are called Bode plots (Fig. 23.44). Through the use of
straight-line segments called idealized Bode plots, the frequency
response of a system can be found efficiently and accurately.
dB
To ensure that the derivation of the method is correctly and clearly
understood, the first network to be analyzed will be examined in some
detail. The second network will be treated in a shorthand manner, and
finally a method for quickly determining the response will be introduced.