April 25, 2024

### Decibels and dbm

Decibels are dimensionless numbers used to represent the ratio of two quantities of power: db = 10 log P1/P0 . This is a base 10 logarithm. For the common unit of "dbm" the reference power P0 is defined as 1 milliwatt : dbm = 10 log P1/1mw. Easy, right?

This equation can be inverted to gives power in terms of a reference power: P1= P010db/10. For the common unit of dbm the equation is P1= 1mw·10dbm/10

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Why? It is not unusual in a radio, radar, or microwave system to have 100,000 watts of transmit power, 1 microwatt of receive power, and various gains and losses from antennas, distance, and absorbing media. Adding and subtracting db is more convenient than multiplying, particularly if you are using a slide rule and trying to keep track of the decimal point.

For signals trapped on a wire it is much easier to measure volts or amps with a voltmeter or current probe, but dbm can still be used. Since Watts are proportional to Amps squared or Volts squared, db (in power) = 20 log V1/V0 or db = 20 log I1/I0 . This is only useful in cases where the voltage or current is measured at the same impedance or resistance, which allows the ratio of two powers I2R or V2/R to cancel out the R.

Fun facts: the "bell" in Decibel is named after Alexander Graham Bell, the telephone inventor. The concept was developed at Bell Labs, but for some reason of economy or humor when they named it they left out the second "L" in his name. Originally it was called the TU, or "transmission unit." The term "logit" was also bounced around for a while.

 dbm milliwatts -10 0.100 -9 0.126 -8 0.158 -7 0.200 -6 0.251 -5 0.316 -4 0.398 -3 0.501 -2 0.631 -1 0.794 0 1.000 1 1.259 2 1.585 3 1.995 4 2.512 5 3.162 6 3.981 7 5.012 8 6.310 9 7.943 10 10.00 11 12.59 12 15.85 13 19.95 14 25.12 15 31.62 16 39.81 17 50.12 18 63.10 19 79.43 20 100.00

 dbm Watts 20 0.1 21 0.125 22 0.158 23 0.200 24 0.251 25 0.316 26 0.398 27 0.501 28 0.630 29 0.794 30 1.00 31 1.259 32 1.585 33 1.995 34 2.511 35 3.162 36 3.981 37 5.012 38 6.310 39 7.943 40 10.00 41 12.69 42 15.85 43 19.95 44 25.12 45 31.62 46 39.81 47 50.12 48 63.10 49 79.43 50 100.0

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