## Decibel Arithmetic

Decibels greatly simplify calculations involving RF and microwave power levels

Decibels greatly simplify the multiplication and division of small and large numbers. This is especially useful in RF communications, where very small and very large power levels often feature in the same calculation.

To multiply two numbers together, just add their equivalent decibel values. For example, to find 10 multiplied by 10:

$$10 \times 10 = \log 10+ \log 10 = 10\:\text{dB} + 10\:\text{dB} = 20 \:\text{dB} = 100$$

To divide one number by another, simply subtract their equivalent decibel values. For example, to find 100 divided by 10:

$$100 \div 10 = \log 100\:– \log 10 = 20 \:\text{dB}\: – 10\:\text{dB} = 10 \:\text{dB} = 10$$

Some common decimal number to dB conversions are shown in the following table:

Number dB
0.01 -20
0.1 -10
0.5 -3
1 0
2 3
10 10
100 20

In the above table, decibel values are rounded off. Greater accuracy can be obtained by including more decimal places when using a calculator or spreadsheet. However, for a quick estimate, rounded decibels are very useful.

For example, 0.5 x 10 x 100 = 500. Taking the log of 500 and multiplying the result by 10 gives 27 dB. Or more simply, just add the decibel values of each number: – 3 + 10 + 20 = 27 dB.

Many RF and microwave engineers make a point of memorising the most common decibel equivalents so they can perform very quick power (or voltage) calculations in their heads without needing a calculator!

### Converting from decibels to number ratios

To convert from decibels back to a number ratio, a calculator is generally used, although it’s still a straightforward process. First divide the decibel value by 10 and then use the inverse log function of the calculator to find the number ratio.

For example, starting with 27 dB:

$$27 \div 10 = 2.7$$

Now take the inverse base 10 logarithm of 2.7:

$$10^{2.7} = 501$$

The rounded off number method gives 500 as the answer, which is sufficiently accurate for a first approximation.