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1054 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
dB
Fig. 23.57(c) we are plotting Av versus the maximum value of Av. On
most data sheets and for the majority of the investigative techniques
commonly employed, the normalized plot of Fig. 23.57(c) is used
because it establishes 0 dB as an asymptote for the dB plot. To ensure
that the impact of using either Fig. 23.57(b) or Fig. 23.57(c) in a frequency
plot is understood, the analysis of the filter of Fig. 23.55 will
include the resulting dB plot for both normalized curves.
For the network of Fig. 23.55:
Vo � �R2� �V R2Vi 1
�� �� i
R1 � R2 � jXC R1 � R2 � jXC Dividing the top and bottom of the equation by R1 � R2 results in
R2 1
Vo ��
——
R1 � R2 XC
�1 � j��� R1 � R2 XC 1
1
but �j� ��j�� ��j��
R1 � R2 q(R1 � R2)C 2pf(R1 � R2)C fc 1
��j� with fc �� and RTh � R1 � R2 f
2pRThC
so that Vo � � �V �� i
If we divide both sides by V i, we obtain
Vo �
Vi
R 2
� R1 � R 2
A v � � � � (23.41)
from which the magnitude plot of Fig. 23.57(b) can be obtained. If we
divide both sides by A vmax � R 2/(R 1 � R 2), we have
(23.42)
from which the magnitude plot of Fig. 23.57(c) can be obtained.
Based on the past section, a dB plot of the magnitude of A′ v �
Av/Avmax is now quite direct using Fig. 23.54(b). The plot appears in Fig.
23.58.
For the gain Av � Vo/Vi, we can apply Eq. (23.5):
20 log10 ab � 20 log10 a � 20 log10 b
where
R2 1
20 log10� ����� R2 1
� ��
R1 � R2 1� j( fc/f )
A� Av
1
v � �� � �
A 1 � j(
fc/f )
�
vmax
1
� 20 log10 � 20 log10 ��
�1� ��(�fc�/f�)
The second term will result in the same plot of Fig. 23.58, but the
first term must be added to the second to obtain the total dB response.
Since R2/(R1 � R2) must always be less than 1, we can rewrite the
first term as
R2 1
R
20 log 1 � R
10 � 20 log 2
R1 � R 10 � 20 log
2 R1 � R 101 � 20 log10 2
R2 0
2
�
R1 � R2 1 � j( fc/f )
R2 �
R1 � R2 �
R 2
1
1� j( fc/f )