# Circuit Theory/Decibel

dB power ratio voltage or current ratio
100   10 000 000 000 100 000
90 1 000 000 000 31 623
80 100 000 000 10 000
70 10 000 000 3 162
60 1 000 000 1 000
50 100 000 316 .2
40 10 000 100
30 1 000 31 .62
20 100 10
10 10 3 .162
6 3 .981 1 .995 (~2)
3 1 .995 (~2) 1 .413
1 1 .259 1 .122
0 1 1
-3 0 .501 (~1/2) 0 .708
-6 0 .251 0 .501 (~1/2)
-10 0 .1 0 .316 2
-20 0 .01 0 .1
-30 0 .001 0 .031 62
-40 0 .000 1 0 .01
-50 0 .000 01 0 .003 162
-60 0 .000 001 0 .001
-70 0 .000 000 1 0 .000 316 2
-80 0 .000 000 01 0 .000 1
-90 0 .000 000 001 0 .000 031 62
-100 0 .000 000 000 1 0 .000 01
An example scale showing power ratios x and amplitude ratios √x and dB equivalents 10 log10 x. It is easier to grasp and compare 2- or 3-digit numbers than to compare up to 10 digits.

## Decibel

Very large and very small numbers are hard to keep track of. Filters/amplifiers and test equipment all reduce or magnify signals. Even rounding to a factor 10 is hard to keep track of. It is easier to add and subtract powers of 10 rather than multiplying/dividing. Decibels have evolved as a way of doing this math in your head. Looking over a circuit is easier to see where inserting test equipment causes the signal to drop 3db rather than trying to remember whether one is dealing with power (and thus should translate this into one half) or voltage (which should be ... some uncomfortable number) at the moment.

In addition, the goal is magnify the very small and reduce the very large so that the most information can fit on a graph. Decibels also serve this purpose.

A decibel (dB) is one tenth of a bel (B). The bel represents a ratio between two power quantities of 10:1, and a ratio between two field quantities of √10:1. A field quantity is a quantity such as voltage, current, sound pressure, electric field strength, velocity and charge density, the square of which in linear systems is proportional to power. A power quantity is a power or a quantity directly proportional to power, e.g., energy density, acoustic intensity and luminous intensity.

The calculation of the ratio in decibels varies depending on whether the quantity being measured is a power quantity or a field quantity.

1 dB means a power ratio of 1.25892 (or ${\displaystyle 10^{\frac {1}{10}}\,}$ ) and a voltage/current ratio of 1.12202 (or ${\displaystyle {\sqrt {10}}^{\frac {1}{10}}\,}$ ).

The problem is that logs are taken of unit-less ratios. How is a number like 5.7 Watt converted to dB? What is it to be divided by? For the purposes of this course, use P0 = reference power = 1 Watt, V0 = reference voltage = 1 volt, and I0 = 1 amp. Decibels are then computed like this:

${\displaystyle dB=10\log _{10}{\frac {P}{P_{0}}}}$
${\displaystyle dB=20\log _{10}{\frac {V}{V_{0}}}}$
${\displaystyle dB=20\log _{10}{\frac {I}{I_{0}}}}$

Power goes as the square of the voltage, therefore the 10 log becomes 20 log.