DECIBEL (dB) MEASURE AND ORDERS-OF-MAGNITUDE
Within any engineering context an order-of-magnitude reference is necessary. For the electrical sciences
the measure is in terms of the decibel (dB).
Aside from the history and a more extensive overlook (https://en.wikipedia.org/wiki/Decibel) the measure
relates to orders of magnitude of power. A factor of 10 change in power corresponds to 10dB, i.e.
p

10 dB  10  log  2  10 
(base 10 logarithms)
 p1

Similarly a change by a factor of 100 will be 20 dB since
p

20 dB  10  log  2  100 
 p1

etc.
So dB measure is the (exponent of 10 to a power) × 10. The logarithm is (by definition) the exponent.
Take note that 0 dB corresponds to a factor of 1.0 since anything to the 0th power = 1.
Of special interest are factors of 2, for which
p

10  log  2  2  = 3dB
 p1

A factor of ½ (the half-power level)
1
10  log  
2
10 log (22) = 2 × 10 log (2) = 6dB
10 log (2-2) = -2 × 10 log (2) = -6dB
10 log (23) = 3 × 10 log (2) = 9dB
Take note that:
= -3dB
etc.
It is possible to construct any integer measure in dB from these simple relationships.
For example a factor of
5
2.5
= 10dB – 3dB = 7dB
= 7dB – 3dB = 4dB
= 10/2
= 5/2
So for integer values of dB there is no need for a calculator. From the above, five of the factor
equivalents are now known. The others are
2 dB
1 dB
5 dB
8dB
= 12dB – 10dB
= 7dB – 6dB
= 9dB – 4db
= 4dB + 4dB
= 16/10
= 2.5/2
= 8/2.5
= 2.5 × 2.5
= 1.6
= 1.25
= 3.2
= 6.25
The same order-of-magnitude factor applies to relative measures. The most common is the power level
relative to 1 watt ≡ dBW,
For example:
100W corresponds to 20 dBW since 10 log [(100W)/(1.0W)] = 20
It is also not uncommon to apply the relative measure of power relative to a milli-Watt ≡ dBm.
For example 0.1W = 100 mW corresponds to 20 dBm . It also corresponds to -10dBW.
Most transfer functions originate as transfer ratios for voltage or current. Since power relates to these
quantities as the square an increase in a factor of 10 in the voltage corresponds to an increase by a factor
of 100 in power
So they will have the prefix of 20 instead of 10 concerning dB measure, i.e.
v

20 dB = 20  log  2  10 
 v1

v

40 dB = 20  log  2  100 
 v1

etc.
So the same context for dB usage as a measure of factor of 10 is kept, except the mathematics is
multiplied by a factor of 2.
Of special interest is the factor √2. This factor corresponds to 3dB since
20 log √2
= 20 × (1/2) log 2
= 10 log 2 = 3dB
The roll-off corner for which |v2/v1| = 1/√2 then corresponds to -3dB, and is so designated. This roll-off
corner also corresponds to the half-power (roll-off) point.