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1046 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
For frequencies where f K f c or ( f c/f ) 2 k 1,
2
2
1 �� � �� �
and AvdB ��10 log10� � 2 fc but log 10 x 2 � 2 log 10 x
resulting in A vdB ��20 log 10
However, logarithms are such that
�log 10 b ��log 10
and substituting b � f c/f, we have
f c
� f
AvdB ��20 log10 �
fc
f K f c
(23.35)
First note the similarities between Eq. (23.35) and the basic equation
for gain in decibels: G dB � 20 log 10 V o/V i. The comments regarding
changes in decibel levels due to changes in V o/V i can therefore be
applied here also, except now a change in frequency by a 2�1 ratio will
result in a 6-dB change in gain. A change in frequency by a 10�1 ratio
will result in a 20-dB change in gain.
Two frequencies separated by a 2:1 ratio are said to be an octave
apart.
For Bode plots, a change in frequency by one octave will result in
a 6-dB change in gain.
Two frequencies separated by a 10:1 ratio are said to be a decade
apart.
For Bode plots, a change in frequency by one decade will result in
a 20-dB change in gain.
One may wonder about all the mathematical development to obtain
an equation that initially appears confusing and of limited value. As
specified, Equation (23.35) is accurate only for frequency levels much
less than fc. First, realize that the mathematical development of Eq. (23.35) will
not have to be repeated for each configuration encountered. Second, the
equation itself is seldom applied but simply used in a manner to be
described to define a straight line on a log plot that permits a sketch of
the frequency response of a system with a minimum of effort and a high
degree of accuracy.
To plot Eq. (23.35), consider the following levels of increasing frequency:
For f � fc/10, f/f c � 0.1 and �20 log10 0.1 ��20 dB
For f � fc/4, f/f c � 0.25 and �20 log10 0.25 ��12 dB
For f � fc/2, f/f c � 0.51 and �20 log10 0.5 ��6 dB
For f � fc, f/f c � 1 and �20 log10 1 � 0 dB
f
f c
� f
� f
f c
� f
1
� b
dB