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 = 10^{x} 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 = 10^{x } 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 = e^{y } or log_{e} ( 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/cm^{2} to 10^{-4} W/cm^{2}. This range exceeds 10^{12} ! 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 P_{1}/P_{0} where P_{0} = reference power , P_{1} 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 P_{1}/P_{0}

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=E^{2 }/R, P=I^{2}R.

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 E^{2} or I^{2} in place of power in the decibel equation getting:

dB = 10 log E^{2}_{1} /_{ }E^{2}_{0} dB = 20 log E_{1}/ E_{0} and

dB= 10 log I^{2}_{1}/ I^{2} _{0} dB = 20 log I_{1}/I_{0.}

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:

Ratio | dB | Ratio | dB |

100:1 | 20 | .01:1 | -20 |

10:1 | 10 | .1:1 | -10 |

4:1 | 6 | .25:1 | -6 |

2:1 | 3 | .5:1 | -3 |

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.