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dB
P
dBm � 10 log10 �
1mW
�� 600 �
(23.11)
Note the subscript m to denote that the decibel level is determined with
a reference level of 1 mW.
In particular, for P � 40 mW,
dB m � 10 log 10
whereas for P � 4 W,
dB m � 10 log 10
� 10 log 10 40 � 10(1.6) � 16 dB m
� 10 log 10 4000 � 10(3.6) � 36 dB m
Even though the power level has increased by a factor of 4000 mW/
40 mW � 100, the dB m increase is limited to 20 dB m. In time, the significance
of dB m levels of 16 dB m and 36 dB m will generate an immediate
appreciation regarding the power levels involved. An increase of
20 dB m will also be associated with a significant gain in power levels.
Voltage Gain
Decibels are also used to provide a comparison between voltage levels.
Substituting the basic power equations P 2 � V 2 2/R 2 and P 1 � V 2 1/R 1 into
Eq. (23.10) will result in
dB � 10 log 10
� 10 log 10
40 mW
� 1 mW
4000 mW
��
1 mW
P 2
� P1
V 2 2/V 2 1
� R2/R 1
� 10 log 10
V 2 2/R 2
� V 2 1/R 1
� 10 log10� � 2 V2 R2 � � 10 log10�� V1
R1�
V2 �
V1
and dB � 20 log 10 � 10 log 10
For the situation where R 2 � R 1, a condition normally assumed
when comparing voltage levels on a decibel basis, the second term of
the preceding equation will drop out (log 10 1 � 0), and
2
dBv � 20 log10 �� V1
Note the subscript v to define the decibel level obtained.
(dB) (23.12)
EXAMPLE 23.3 Find the voltage gain in dB of a system where the
applied signal is 2 mV and the output voltage is 1.2 V.
Solution:
dBv � 20 log10 � Vo
1. 2V
� � 20 log10 �� � 20 log10 600 � 55.56 dB
Vi
2 mV
for a voltage gain Av � Vo/Vi of 600.
EXAMPLE 23.4 If a system has a voltage gain of 36 dB, find the
applied voltage if the output voltage is 6.8 V.
V
R 2
� R1
DECIBELS ⏐⏐⏐ 1023