Decibel Calculator: Power Ratios, dBm, dBi and S Units

Decibels are the language every radio calculation is written in, and the reason is practical rather than mathematical: gains and losses in a chain add rather than multiply. Once you can move between decibels and power ratios in your head, most amateur radio arguments about whether an upgrade is worth it answer themselves in about ten seconds.

The two rules that do most of the work

Memorise two numbers and you can do decibel arithmetic mentally for the rest of your time in the hobby.

• 3 dB doubles the power. Minus 3 dB halves it.
• 10 dB multiplies power by ten. Minus 10 dB divides it by ten.

Everything else is combination. 13 dB is 10 plus 3, so twenty times. 20 dB is 10 plus 10, so a hundred times. 7 dB is 10 minus 3, so five times. 6 dB is 3 plus 3, so four times. That covers essentially every figure that turns up in amateur radio, and it turns questions like "is a 600 watt amplifier worth it over 100 watts" into arithmetic you can do while someone is still explaining the question.

Why decibels rather than ratios

A signal path is a chain of multiplications. The transmitter puts out some power, the feedline divides it, the antenna multiplies it in one direction, the path divides it by an enormous factor, the receiving antenna multiplies it again. Written as ratios that is a long product with numbers spanning fifteen orders of magnitude.

Taking logarithms turns every multiplication into an addition. A 100 watt transmitter into 1.5 dB of coax loss into a 9 dBi antenna is plus 9 minus 1.5, so 7.5 dB relative to isotropic, and you did that in your head. That is the whole reason the unit exists, and it is why every link budget, every antenna specification and every receiver sensitivity figure in radio is written this way.

What a decibel is worth on the air

This is the part that matters for spending decisions, and the honest summary is that small decibel figures are not worth money.

The pattern is that antenna and feedline improvements work on receive as well as transmit, and power improvements do not. A station that is 6 dB better because of its antenna hears 6 dB better as well, which frequently matters more, because being unable to hear a station is a harder problem than being unable to reach one. The full argument is on QRP versus 100 watts.

dBi, dBd and the marketing gap

Antenna gain is always relative to something, and there are two references in common use.

• dBi compares the antenna to an isotropic radiator, a theoretical point that radiates equally in every direction. It does not exist and cannot be built, which is precisely why it is a clean reference.
• dBd compares the antenna to a half-wave dipole in free space. A dipole is a real antenna with a real pattern, and it has 2.15 dBi of gain of its own.

So dBi equals dBd plus 2.15, always. Manufacturers quote whichever number is larger, which is dBi, and a vertical advertised at 7.2 dBi is a 5.05 dBd antenna. When comparing two antennas, check which reference each specification uses before concluding one is better, because the 2.15 difference is larger than most of the differences being argued about.

Be sceptical of large gain claims on small antennas. Gain is not created; it is redistributed, by taking energy from directions you do not want and putting it where you do. A 9 dBi omnidirectional vertical achieves that by squashing its pattern flat, which works beautifully on level ground and can point the main lobe over the top of a station in a valley below you. The Diamond X50A dual-band base antenna$122.99 quotes 4.5 dBi on 2 metres and 7.2 dBi on 70 centimetres, which are believable figures for its length; anything claiming much more from the same physical size is measuring optimistically.

dBm, and reading a receiver specification

dBm is an absolute level rather than a ratio: decibels relative to one milliwatt. It is how receiver sensitivity, signal generator outputs and spectrum analyser readings are expressed.

The S9 landmark at minus 73 dBm is the one worth remembering, because it anchors the whole S meter scale. S9 is defined as 50 microvolts across 50 ohms on HF, and each S unit below it is nominally 6 dB. Above S9 the scale switches to plain decibels, which is why reports are given as "20 over 9" rather than as S numbers.

S meters and why they lie

The standard says one S unit is 6 dB. Actual amateur transceiver S meters commonly show anywhere from 3 to 6 dB per unit, are non-linear across the scale, read differently with the preamplifier on, and vary between two examples of the same model. They are useful for comparing two signals in the same minute on the same radio and close to useless as absolute measurements.

That is worth knowing when someone reports your amplifier made no difference. A 7.8 dB increase should move a properly calibrated meter by one and a third units, and on a typical amateur meter it might move two units or one, depending on where on the scale you started. The signal report conventions and what the numbers are meant to represent are on the RST reference.

Where the arithmetic actually gets used

• Deciding on feedline. Loss in decibels per 100 feet, scaled by your run length, compared against the price difference. Done on the coax loss calculator.
• RF exposure evaluation. Antenna gain in dBi and feedline loss in dB both feed directly into the compliance distance, which is required of every station. See the RF exposure calculator.
• Effective radiated power. Transmitter power, minus feedline loss, plus antenna gain. Some bands and some jurisdictions regulate ERP rather than transmitter output. See ERP and EIRP.
• The Amateur Extra exam. Decibel arithmetic appears throughout Element 4, and it is one of the areas where knowing the two rules cold converts several questions from calculation into recognition. See the Extra class page.