Decibels

The bel (abbreviated B) is named after Alexander Graham Bell, who did much pioneering work with sound and the way our ears respond to sound. Our ears respond to sounds ranging from an intensity less than 10 -16 W/cm 2 to intensities larger than 10 -4 W/cm 2 (where we begin to experience pain). This is a range of more than 1x1012 times from the softest to the loudest sounds. Logarithms provide a convenient way to represent these values, because they compress this scale into a range of 12, rather than a range of a billion. A bel is defined as the logarithm of a power ratio. It gives us a way to compare power levels with each other and with some reference power.

While the bel was first defined in terms of sound power, to describe sound intensities, in electronics we often use it to compare electrical power levels. The decibel is one-tenth of a bel, and is abbreviated

dB.

It takes 10 decibels to make 1 bel,
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The Equation for calculating decibels is as follows

Where P1 is the reference power and P2 is the power you are comparing to the reference level.
Since the Decibel measurement system is always in relation to a reference power level the reference level should usually be mentioned

Common Reference levels used in radio work are

dBW Decibels in relation to 1 Watt

dBm Decibels in relation to 1 Milli-Watt (0.001 W)

In relation to Antenna Gain the following reference levels are commonly used.

dBi Decibels in relation to an Isotopic radiator.

dBd Decibels in relation to a Dipole antenna.

Values given in dBd can be converted to dBi units by Adding 2.14 this is because a Dipole Antenna has 2.14dB gain over an isotropic radiator, I.e. a Dipole antenna is said to have 2.14 dBi gain.

dB can be used without giving the reference level when talking about system gain or loss.

Some Common Decibel Values and power-ratio Equivalents.
Anytime you double the power, it represents approximately a 3-dB increase in power What is the decibel change when you cut the power in half?
Again, it doesn’t matter if you are going from 1000 W to 500, 100 to 50 or 2 W to 1 W; the power ratio is still 0.5

Suppose you double the power, and then double it again?
The final power is four times the starting power, so you can calculate the decibel increase using the equation given.
You can also calculate the total power change “by inspection” because you know each time you double the power there is a 3-dB increase.
In this example you have a 3-dB increase, plus a second 3-dB increase. If you add these two decibel values, you have a 6-dB total increase. If you double the power again, you have a 9-dB total increase. Doubling the power a fourth time gives a 12-dB total increase.
The same relationship is true of power decreases. Each time you cut the power in half you have a 3-dB decrease. Cutting the power in half and then in half again is a 6-dB decrease, and so on.
The addition and subtraction of decibel values is very important in electronics. Amplification factors, gains and losses of antennas, antenna feed lines and all kinds of circuits can simply be added when they are expressed in decibels.

Logarithims
You will require a basic understanding of logarithms to understand Decibels, if you are unsure about logarithms them please read on.

A logarithm is an exponent. Common logarithms use the number 10 as their base. You have someexperience with “powers of 10” from writing numbers in exponential or scientific notation. A common log, as it is usually called, is the exponent or power to which you must raise 10 to get a certain number. In the examples above, we raised 10 to the third power to get 1000. The log of 1000, then, is 3. The log of 1000000 is 6. In general, we define a common logarithm with two equations. If: N = 10 x , then:
log (N) = x
Sometimes you will see this written as log10 (N) = x. This is simply to ensure that you know the base of the logarithm is 10.
Finding the log of a multiple of 10 is easy, as these examples show. You may wonder to what power you can raise 10 to get a number like 2. That is a good question, and the answer is 0.301.
Logs are usually decimal fractions rather than whole numbers. Logs for numbers smaller than 10 are less than 1; logs for numbers larger than 10 are greater than 1.

From the definition of a log, we can write the expression:

2 = 100.301
The easiest way to find any logarithm is with your calculator. Simply enter the number whose log you want to find, and then push the button labeled “log.” It is easy to find that log (5) = 0.699, for example.
It is interesting to note that log (1) = 0, because anything (including 10) raised to the zero power is1.
The log of 0 is undefined, because there is no power to which you can raise 10 and get 0. The inverse log is called the antilog (often written log –1 ). When we know the log and want to find the original number, we want the antilog. To find an antilog, simply raise 10 to the given power. Your calculator probably has a button labeled “10 x ” or something similar.

What is the antilog of 1.845
101.845 = 70. Don’t try to follow the rules for significant figures when finding logs or antilogs. Do follow the rules with the values you calculate from logs and antilogs, however.

The second base that is frequently used for logarithms is a number usually represented by e. (Some-times
the Greek letter epsilon () is used to represent e although this is an incorrect representation.)

This number is approximately 2.71828. This is not an exact value, because the decimal fraction doesn’t end with this last 8. This value is rounded off, but there is no exact value for e because you can never find the last digit. Mathematicians call such numbers with no exact value, irrational numbers. The number represented by e appears in several electronics calculations, and is called the natural number, because it appears as a constant of nature. You will use e to calculate the voltage on a capacitor as it charges or discharges, for example. Logarithms that use e for their base are called natural logarithms, or Naperian logarithms. This can be written as loge , but to more easily distinguish it from common logs, we usually abbreviate it ln.

We define natural logs the same way we define common logs.

If M = ey ,
then
loge (M) = ln (M) = y
The easiest way to find a natural log is with a scientific calculator. Enter the number whose ln you want to know, then press the “ln” button on the calculator. For example, ln (2) = 0.693 and ln (20) = 2.996.

As you might expect, ln (e) = 1, ln (1) = 0 and ln (0) is undefined.
Inverse natural logs, or antilogs are also easy with a calculator.
Just raise e to that power: e2.996 = 20.
Converting between common logarithms and natural logarithms is easy, however. If you want to find a common log, and know the natural log value, divide that by the natural log of 10. log(x) = ln(x) / ln(10) = ln(x) / 2.3025851
log(x) = 0.4342945 ln(x)
If you know the common log, and want to find the natural log, divide that value by the common log
of e.

ln(x) = log(x) / log(e) = log(x) / 0.4342945
ln(x) = 2.3025851 log(x)