Converting mW to dB

Date: 06/14/2001 at 21:45:44
From: Jeff Carson
Subject: Conversion of mW to dB

I this equation during a radio receiver discussion.

4x10-12mW = -114dBm

How does it equate? What is the math that performs this conversion?

Date: 06/15/2001 at 12:04:22
From: Doctor Peterson
Subject: Re: Conversion of mW to dB

Hi, Jeff.

According to my favorite source on units, How Many? A Dictionary of 
Units of Measurement, by Russ Rowlett:

    http://www.unc.edu/~rowlett/units/   

    dB m, dB W 
    logarithmic units of power used in electronics. These units
    measure power in decibels above the reference level of 1
    milliwatt in the case of dB m and 1 watt in the case of dB W.
    A power of n watts equals 10 log n dB W; conversely, a power
    of p dB W equals 10^(p/10) watts. The same formulas link dB m
    to milliwatts. An increase of 10 dB m or 10 dB W represents a
    10-fold increase in power. Since 1 watt = 1000 milliwatts, 0
    dB W = 30 dB m. 

(Look up decibel and bel there, too.)

Another source I found, a column by Ron Hranac in the April 2000 issue 
of Communications Technology:

    Broadband DBmV: Power in Terms of Voltage
    http://www.cabletoday.com/ct2/archives/0400/0400col1.htm   

says that dBm is read as "decimal milliwatt," and gives the same 
formula:

    dBm = 10log(P/1 mW), where P is a power level in milliwatts

Note that a bel is the base-ten logarithm of any ratio; in this case 
the ratio is that of the given power to a 1 mW output. A decibel is 
1/10 of a bel, accounting for the factor of ten.

So in your example, with P = 4x10^-12 mW, we get

    10 log(4x10^-12) = 10 (log(4) - 12) = -113.98

which agrees with what you read, rounded appropriately.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/