October 30, 2012
This month we continue our discussion of power. A logarithm is an exponent. Two types of logarithms that we use in our industry are the common logarithm (log) and the natural logarithm (ln). The common logarithm is the exponent or power that you must raise 10 to get a certain number. Example: If you raise 10 to the second power you get 100. Therefore the log of 100 is 2 . The equation is written:
N = 10x or log (N) = x
To raise 10 to a power to achieve a number less than 10 requires that x be less than one.
Example: 2 = 10x x = .301
Note: It is not possible to raise 10 to a power equal to zero. The log of zero is undefined.
The natural logarithm (ln) is the exponent or power that you must raise the base e to get a certain number. The number e is called a natural number because it appears as a constant of nature. It is an irrational number ( it is like pi in that it has no end ) that I will round off to 2.72 . The equation is similiar to common logarithms and is written:
M = ey or loge ( M) = y or ln (M) = y
Example: ln 2 = 0.693 e raised to 0.693 = 2, ln 20 = 2.996 e raised to 2.996 = 20.
The background above on logarithms prepares us for the explanation of decibels. The bel (abbreviated B) is named after Alexander Grahm Bell who was a pioneer in the field of sound. Our ears hear logarithmically from intensities of less than 10-16 W/cm2 to 10-4 W/cm2. This range exceeds 1012 ! Logarithms compress this number ( one trillion ) to 12.
The bel is defined as the logarithm of a power ratio. It gives us a way to compare power levels with each other and with a reference power.
bel = log P1/P0 where P0 = reference power , P1 is power you are comparing to the reference.
Because the bel is such a large ratio, we commonly use one tenth of a bel or the decibel (dB) in our work.
dB = 10 log P1/P0
Example: What is the power gain in dB if an amplifier boosts a signal from 1w to 50w ?
dB = 10 Log 50W/1W = 10 log (50) = 10 ( 1.699) = 16.99 dB
When we know the current or the voltage in a circuit we can also calculate the power as long as we know the impedance. From last months discussion of power: P=E2 /R, P=I2R.
We can take a shortcut comparing signals when the impedance is the same in both circuits or the impedance doesn’t change when the levels change. Under these conditions we can use E2 or I2 in place of power in the decibel equation getting:
dB = 10 log E21 / E20 dB = 20 log E1/ E0 and
dB= 10 log I21/ I2 0 dB = 20 log I1/I0.
Again remember that the impedances must be the same to use the above equations.
Logarithms allow you to take the square term 2 and move it out as a multiple if 10. When calculating ratios of power or squared terms use 10 as the mutiplier, when calculating power using the measured voltage/current (unsquared) use 20 as the multiplier.
Power ratios that we commonly use are the doubling and halving of power. Doubling the power dB = 10 log 2 dB = 10 (.301) = 3.01
For half the power dB = 10 log (0.5) = 10 (-.301) = -3.01
So, to double the power is about 3dB, to halve the power is about -3 dB. To quadruple the power ( 4 times) is 3dB plus 3 dB or 6 dB. Same with reducing the power to one fourth its original value, -3dB plus -3dB is -6dB. Below is a table of common ratios to dBs:
It is often convenient to compare a power level to a reference power. A common reference used in electronics is the milliwatt (mW). A decibel value of a signal compared to one milliwatt is specified in dBm. Therefore 1 milliwatt is 0 dBm.
Example: The signal from an antenna going into a receiver is 2 x 10-13 mW. The signal strength in dBm is:
dBm = 10 log 2 x 10-13 mW /1 mW = 10 log (2 x10-13) = 10 x -12.7 = -127
Many signal generators have attenuators with dual scales that show voltage and power (in dBm). Examples are the IFR FM/AM 1200S, FM/AM 500 and NAV 750B.
There are other reference powers used, one being the watt (W). Its notation is dBW. Generally when you see a letter after dB you know that a reference power is being compared to some other power.
Next month: We go to the airplane and onto the bench to examine the applications of logarithms and dBs in our work.