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College of Southern Nevada
Decibels and dBm’s
Professor Joe Miller
THE DECIBEL SYSTEM
The decibel (dB) is a logarithmic unit that is used in electronics, acoustics, optics, video & digital
imaging, and telecommunications industries to indicate a ratio. For example, power measurements for
rf circuits are usually indicated in terms of decibels (dB), electronic amplifier gain in terms of power or
voltage often is expressed in terms of dBs, initial and final transmitter power when there is an upgrade
can be expressed as a power gain in dBs, and other examples can be found in the fields of acoustics and
optics.
⁄
Equation dB-1
The basic unit of measurement in the system is not the decibel; it is the bel. The bel is a unit that
expresses the logarithmic ratio between the input and the output of any given component, circuit, or
system. It may be ratio of voltages, currents, powers, sound pressure level in µPascals, or the intensity of
the radiation (light). In many cases, it is used to show the ratio between input and output power
otherwise known as gain.
REPRESENTING POWER GAIN IN DECIBELS
You can express the power gain of the amplifier (N) in bels by dividing the output (P1) by the input
(P2) and taking the base 10 logarithm of the resulting quotient. For a review of logarithms see the linked
WISC-Online module Log-Review (note the first equation in the module lacks superscript format on the
“y” )
The formula for determining this gain is:
Equation dB-2
If an amplifier doubles the input power, the quotient of POut to PIn will be 2. If you consult a
logarithm table, you will find that the base 10 logarithm of 2 is 0.3, making the power gain of the
amplifier 0.3 bel. Experience has shown that because the bel is a rather large unit, it is difficult to apply.
A more practical unit, and one that can be used more easily, is the decibel (1/10 bel). You can convert
any figure expressed in bels to decibels by multiplying that figure by 10 or simply by moving the decimal
point one place to the right. Applying this rule, we find that the above ratio of 0.3 bel is equal to 3
decibels.
Equation dB-3
The decibel (dB) cannot be used to represent actual power; only the ratio of one power compared to
another. To say that an amplifier has a 3 dB gain means that the output power is twice the input power.
This gives no indication of the actual power represented. You must be able to state the input power for it
to be meaningful.
Example DB.1
Given: An amplifier has input power level of 50 mW and a output power level of
2.75 W
Find: AP and APdB
AP= PO/PI=2.75W/50 mW = 55
APdB= 10 Log AP = 17.4 dB
The power gain of any amplifier can be calculated in dB using the same steps shown
above.
Example DB.2
Given: An amplifier has output power level of 680 mW and is reconfigured to
have an output power level of 1.2 W
Find: Find the change in power level in decibels
AP= Pfin/Pint=1.2W/680 mW = 1.76
AP dB= 10 Log AP = 2.47 dB
Reverse the calculation. If you are given the gain in dBs you can find the gain as a ration (AP).
Start with the following simple equation for APdB.
AP dB= 10 Log AP = 17.4 dB
Divide both sides of the equation by 10.
AP dB/10 = Log AP
Take the antilog of both sides.
10 AP dB/10 = AP
Equation dB-5
After you have derived AP, you can find either the input or output powers if the other is
known.
Example DB.3
Given: An amplifier has a power gain of 26 dB and an input power of 70 uW
Find: AP and PO
AP = 10 APdB/10 = 10 26/10 = 398
AP = 398 = PO/70uW
PO=398 x 70uW = 27.9 mW
ABSOLUTE POWER GAIN EXPRESSED USING DECIBELS
If a standard reference value is used for PIn , AP dB and even AP represent POut as a ratio to
the standard reference and a specific power. If the standard reference value is one watt
then the gain in dB becomes AP dBm or if the standard reference for telecommunications, 1
mW, is used then the gain in dB becomes AP dBm. In acoustics pref is used and it is the
sound pressure of 20 micro-pascals., specifically - dB SPL (sound pressure level) – for
sound in air and other gases (Note: 20 micro-pascals (µPa) = 2×10-5 Pa, approximately
the quietest sound a human can hear).
The standard reference that is used in electronics and telecommunications the dBm. The
dBm is an abbreviation used to represent power levels above or below 1 milliwatt.
Negative dBm (−dBm) represents power levels below 1 milliwatt, and positive dBm
(+dBm) represents power levels above 1 milliwatt. In other words, a dBm value is a
specific amount of power; 0 dBm is equal to 1 milliwatt. Briefly stated, the amount of power
in a given value of dBm is the power which results if 1 milliwatt is amplified or attenuated
by that dB value. For example, 40 dBm represents an actual power level (watts or
milliwatts) that is 40 dB above 1 milliwatt, whereas −10 dBm represents a power level that
is 10 dB below 1 milliwatt. The formula for finding dBm is a variation of the dB power
formula:
Equation dB-6
You do not need to use the formula in many applications. The following shows
conversions of dBm to mW:
a. 20dBm = 100mW
f. 3dBm = 2mW
b. 10dBm = 10mW g. 0dBm = 1mW
c. 7dBm = 5mW
h. -3dBm = .5mW
d. 6dBm = 4mW
i. -10dBm = .1mW
e. 4dBm = 2.5mW
For a +10 dBm level, start with the 1 milliwatt reference and move the
decimal point one place to the right (+10 dBm = 10 mW). Another 10 dB
increment brings the power level to +20 dBm, thereby moving the decimal
point another place to the right (+20 dBm
100 mW) For a −10 dBm
level, again start with 1 milliwatt, but this time move the decimal point one
place to the left (−10 dBm = .1 mW). An additional 10 dB decrease results
in another decimal point shift to the left (−20 dBm
01 mW)
For a 3 dB increase, you double the power. For a 3 dB decrease, you reduce
the power by one-half (+3 dBm
2 mW and −3 dBm
5 mW) A +6 dBm
level is an additional 3 dB change from +3 dBm. In this case, you just double
the power level of the +3 dBm (+6 dBm = 4 mW).
The dB change can be made in either direction. For example, +7 dBm is a
decrease from +10 dBm. Reducing the +10 dBm power by one-half, we
have +7 dBm, or 5 mW. A +4 dBm power level is a 3 dB decrease from +7
dBm (+4 dBm − 2 5 mW) By using this simple method, you can quickly
find any power level that corresponds to a given dBm.
Example DB.4
Given: A 2.4 W signal
Find: The expression of the signal as APdBm
By manipulating equation dB-6 the way dB-3 was manipulated to yield equation dB5 you
can develop the follow equation for finding a power level in mW that is originally
expressed in dBm’s
0.001W{10 AP dB/10} = Power
Equation dB-7
Example DB.5
Given: A 10 Mhz test tone measures -16 dBm on a spectrum analyzer.
Find: The power level of the signal
0.001W{10 AP dB/10} = 0.001W{10-1.6}
25 1 μW
RELATIVE VOLTAGE GAIN
Many times it is preferable to analyze a circuit in terms of voltage or current and if we use
the following with equation dB-3.
PO=VO2 /RO
APdB = 10 Log { VO2 /RO/ VI2/RI }
PI = VI2/RI
… Then we assume that RO = RI
APdB = 10 Log { VO2 / VI2 } = 10 Log { VO / VI }2 = 20 Log { VO / VI }
Thus
AVdB= 20 Log { VO / VI } Equation dB-8
and in the case of the circuit above we substitute in the values for the above circuit
APdB = 20 Log 3.0/0.3 = 20dB
Equation DB-8 can be used to calculate voltage gain even if the input and output resistance
are not equal, but the result wouldn’t represent the gain in power
Example DB.6
Given: The output of a telecommunications amplifier is 2.00 VRMS for an input of
144 mVRMS
Find: AV and the gain in dBs
AV =VO/VI = 2.00V/144mV = 13.89
AVdB= 20 Log 13.89 = 22.85 dB
Example DB.7
Given: The voltage gain of a network amplifier is 52 dB
Find: Voltage gain as a ratio, VO if VI = 35mV, the drop of the output signal in dBs
if the input decreases from 35mV to 5 mV.
a. AV =VO/VI =10AVdB/20 =1052/20 =398
b. AV =VO/VI => 398 = VO/35mV => VO= 13.9V
c. AVdB = 20 Log Vfin/ VInit = 20 Log 5mV/35mV = -16.9 dB
For a review of decibels and a few more examples please review the linked animation
module from WISC-Online Decibel-Review.
Links:
1. Log-Review – http://www.wisc-online.com/Objects/ViewObject.aspx?ID=SSE2503
2. Decibel-Review - http://www.wisc-online.com/Objects/ViewObject.aspx?ID=SSE2603
Note: You may need to cut and paste the links into a separate browser window.