>Path: gonix!uunet!newsgate.melpar.esys.com!melpar!phb

>From: phb@syseng1.melpar.esys.com (Paul H. Bock)

>Subject: TUTORIAL: dB & dBm

>Sender: news@melpar.esys.com (Melpar News Administrator)

>Organization: E-Systems, Melpar Division

>Date: Tue, 11 Oct 1994 17:23:05 GMT

>Message-ID: phb.781896185@,melpar

>Lines: 247

USING AND UNDERSTANDING DECIBELS

By

Paul H. Bock, Jr. K4MSG

Author's Note: This tutorial was originally written for the use of non-RF/analog engineers (digital, software) and non-engineers who needed an easy-to-follow reference on the general use of the decibel. I hope that some amateur operators may find it useful as well.

While the historical accuracy of the comments relating to the telephone company and telephone company engineers may be open to question (the information as supplied to me was anecdotal), the technical points made should be valid regardless of the exact turn of history.

*General*

The decibel, or dB, is a means of expressing either the gain of an active device (such as an amplifier) or the loss in a passive device (such as an attenuator or length of cable). The decibel was developed by the telephone company to conveniently express the gain or loss in telephone transmission systems. The decibel is best understood by first discussing the rationale for its development.

If we have two cascaded amplifiers as shown below, with power gain factors Al and A2 as indicted, the total gain is the product of the individual gains, or A1 x A2.

Input >-------- Amp #1 --------- Amp #Z ------- > output

A1 = 275 A2 = 55

In the example, the total gain factor At = 275 x 55 = 15,125.

Now, imagine for a moment what it would be like to calculate the total gain of a string of amplifers. It would be a cumbersome task at best, and especially so if there were portions of the cascade which were lossy and reduced the total gain, thereby requiring division as well as multiplication.

It was for the reason stated above that Bell Telephone developed the decibel. Thinking back to the rules for logarithms, we recall that rather than multiplying two numbers we can add their logarithms and then take the antilogarithms of this sum to find the product we would have gotten had we multiplied the two numbers. Mathematically,

log (A x B) = log A + tog B

If we want to divide one number into another, we subtract the logarithm of the divisor from the logarithm of the dividend, or in other words

log (A/B) = log A - log B

The telephone company decided that it might be convenient to handle gains and losses this way, so they invented a unit of gain measurement called a "Bel," named after Alexander Graham Bell. They defined the Bel as Gain in Bels = log A

where A = Power amplification factor

Going back to our example, we find that log 275 = 2.439 and log 55 = 1.740, so the total gain in our cascade is

2.439 + 1.74 = 4.179 Bels

It quickly occurred to the telephone company engineers that using Bels meant they would be working to at least two decimal places. They couldn't just round off to one decimal place, since 4.179 bels is a power gain of 15, 101 while 4.2 bels is a power gain of 15,849, yielding an error of about 5%. At that point it was decided to express power gain in units which were equal to one-tenth of a Bel, or in deci -Bels. This simply meant that the gain in Bels would be multiplied by 10, since there would be ton times more decibels than Bets. This changes the formula to

Gain in decibels (dB) = 10 log A (Eq. 1)

Again using our example, the gain in the cascade is now

24.39 + 17.40 = 41.79 decibels

The answer above is accurate, convenient to work with, and can be rounded off to the first decimal place will little loss in accuracy; 41.79 dB is a power gain of 15,101, while 41.8 dB is a power gain of 15,136, so the error is only 0.23%.

What if the power gain factor is less than one, indicating an actual power loss? The calculation is performed as shown above using Equation 1, but the, result will be different. Suppose we have a device whose power gain factor is 0.25, which means that it only outputs one-forth of the power fed into it? Using Equation 1, we find

G = 10 log (0.25)

G = 10 (-0.60)

G = -6.0 dB

The minus sign occurs because the logarithm of any number less than 1 is always negative. This is convenient, since a power loss expressed in dB will always be negative.

There are two common methods of using the decibel. The first is to express a known power gain factor in dB, as just described. The second is to determine the power gain factor and convert it to dB, which can all be done in one calculation. The formula for this operation is as follows:

Po

G= 10 log ---- (E. 2)

Pi

where G = Gain in dB

Po = Power output from the device

Pi = Power input to the device

Both Po and Pi should be in the same units; i.e., watts, milliwatts, etc. Note- that Equation 2 deals with power, not voltage or current; these are handled differently when converted to dB, and are not relevant to this discussion. Below are two examples of the correct application of Equation 2:

Ex. 1: An amplifier supplies 3.5 watts of output with an input of 20 milliwalts. What is the gain in dB?

3.5 watts

G= 10 log ----

0.02 wafts

G = 10 log (175)

G = 10 (2.24)

G = 22.4 dB

Ex. 2: A length of coaxial transmission line is being fed with 15 0 watts from a transmitter, but the power measured at the output end of the line, is only 112 watts. What is the line loss in dB?

112 watts

G= 10 log ---

150 watts

G = 10 log 0.747

G = 10 (-0. 127)

G = -1.27 dB

*Non-relative, (Absolute) Uses of the Decibel*

The most common non-relative, or absolute, use of the decibel is the dBm, or decibel relative to one milliwatt. It is different

from the dB because it represents, in physical terms, an absolute amount of power which can be measured.

The difference between "relative" and "absolute" can be understood easily by considering temperature. For example, if I say that it is "20 degrees colder now than it was this morning,"

it's a relative measurement; unless the listener knows how cold it was this morning, it doesn't mean anything in absolute terms. If, however, I say, "It was 20 degrees C this morning, but it's 20 degrees colder now," then the listener knows exactly what is meant; it is now 0 degrees C. This can be measured on a thermometer and is referenced to an absolute temperature scale.

So it is with dB and dBm. A dB is merely a relative measurement, while a dBm is referenced to an absolute quantity: the milliwalt (1/1000 of a watt). We can apply this concept to Equation I as follows:

dBm = 10 log (P) (1000 mW/watt)

where dBm = Power in dB referenced to 1 milliwatt

P = Power in watts

For example, take the case where we have a power level of I milliwatt:

dBm = 10 log (0.001 watt) (1000 mW/watt)

dbm = 10 log (1)

dBm = 10 (0)

dBm = 0

Thus, we see that a power level of 1 milliwatt is 0 dBm. This makes sense intuitively, since our reference, power level is also 1 milliwatt. If the power level was I watt, however, we find that

dbm = 10 log (I watt) (1000 iiiW/watt)

dBm = 10 (3) dBm = 30

The dBm can also be negative, just like the dB; if our power level is 1 microwatt, we find that

dBm = 10 log (I x IOE-6 watt) (1000 mW/watt)

dBm = -30 dBm

Since the dBm is an absolute amount of power, it can be converted back to watts if desired. Since it is in logarithmic

form it may also be conveniently combined with other dB terms, making system analysis easier. For example, suppose we have a signal source with an output power of -70 dBm, which we wish to connect to an amplifier having 22 dB gain through a cable having 8.5 dB loss. What is the output level from the amplifier? To find the answer, we just add the gains and losses as follows:

Output = -70 dBm + 22 dB + (-8.5 dB)

Output = -70 dbm + 22 dB - 8.5 dB

Output = -56.5 dBm

As a final note, power level may be referenced to other quantities and expressed in dB form. Below are some examples:

dBW = Power level referenced to I watt

dBk = Power level referenced to 1 kilowatt (1000 watts)

One other common usage is dBc,, which is a relative term like dB alone. It means "dB referenced to a carrier level" and is most commonly seen in receiver specifications regarding spurious signals or images. For example, "Spurious signals shall not exceed -50 dBc" means that spurious signals will always be at least 50 dB less than sot-ne specified carrier level present (which could mean "50 dB less than the desired signal").

Paul H. Bock, Jr. K4MSG

Principal Systems Engineer

E-System/Melpar Div.

7700 Arlington Blvd.

Falls Church, VA 22046

"Imagination is more important than knowledge." - Albert Einstt@in