Decibels Explained

Friday, February 24, 2017

by Jim ArcarowD8PFK

I have found over the years, in talking to both students and technicians, that there is a great deal of confusion and misunderstanding when it comes to talking about decibels (dB). Decibels are a unit of measurement that makes it simple to “add up” the gains and losses in a system, and determine what the output really is.

If you know how to divide, multiply and use a calculator, figuring out the gain or loss in dB is easy. Decibels are most commonly used in two areas of electronics. The first is audio, where a standard output level, typically 0 dBm, is often specified.

The second area is RF, where transmission line losses are given in dB, and antenna gains are related in dBi or dBd. The two areas share amplifier gain, where a gain of 10 or 20 dB is not uncommon. What does the above all mean, and what are these dB things anyway?

To dB Or Not To dB

We’ll spare the history lesson about Alexander Graham Bell, and how the unit came to be, so let's get started by answering some common questions and explaining the math. You may have read or heard that a gain of 3 dB is a doublingof power, and a loss of 3 dB is a halving of power. How was this figured out? The power formula for calculating dB looks intimidating, but it is quite simple.

The output in dB is equal to 10 times the log of the quantity (Pout divided by Pin). So throw some numbers in. Let’s say Pout equals 100 Watts, and Pin equals 50 W. The ratio is 2 to 1. The seemingly hard part is the log. Logarithms are exponents of numbers. The two systems used are related either to base 10, or to the natural number “e”.

To keep from confusing the two “log” is usually used to denote the base of 10, and in is used to denote the natural log. We will only use log, or base 10 here. Recall from scientific notation that 100 equals 10 times 10, or 10 to the second power. The log of 100 is the exponent, the number of times 10 is used as a factor, or 2. Similarly the log of 1,000 is three, and the log of 10,000 is four - just count the zeros! Can you guess what the log of 10 is?

It's 1, of course. Returning to our example, our power ratio of 2 makes things a bit harder, requiring a calculator (or an old slide rule, if you have one).

Two isn’t a nice number like 100, or 1,000, so we have to calculate what o power often would yield 2 as answer. We can guess that since 10 to the zero is one (by definition), and ten to the first is 10, the answer will be between zero and one.

Practice Makes Perfect

First a bit of practice. Pull out the scientific calculator and verify that when o you enter 100 and then press the “LOG” key the answer is 2. Press CLEAR. Try again by entering 1,000, then press LOG and verify the answer is 3. Got it?

OK, now clear the display and enter 100, DIVIDE, 50, equals, and the result is 2. Without clearing this, press LOG. The answer is 0.30103, but 0.3 is close enough, and between zero and one just as we predicted. So 10 to the 0.3 power equals 2. From the power formula above, the gain in dB is 10 times log (Pout/Pin). So the log of Pout/Pin in our example is 0:3, multiplied by 10 equals 3. That’s the answer, 3. Any example having a power ratio of 2 will be 3 dB, because the log of 2 is 0.3.

The Question and Answer

Let’s try another example. The power out is 120 watts, the input power is 30 watts, what’s the gain in dB? OK, 120 divided by 30 is 4. The log of 4 is 0.60205. Multiply that by ten and your answer is 6 db.

Now, here’s a quick way to check this. We started with 30 watts, double that is 60 watts (3 dB gain), double 60 is 120 watts (another 3 db gain). The total gain is additive, and is 6 dB. You can do the same thing with losses, just use a minus sign in front of your answer to indicate the loss.

Here’s an example: the input to a splitter is 10 watts, each output is 4 watts, calculate the loss in each port. The power ratio is 10 divided by 4, which is equal to 2.5. The log of 2.5 is 0.3979; we'll call it 0.4. Multiply by 10 and the answer is 4 dB of loss, or 4.0dB.

Now here’s something interesting. If you take the same numbers and instead use 4 divided by 10, then take the log of the answer, you get 0.3979, multiply by 10 and you end up with 3.979 dB, or 4.0. Either way, your answer is correct. Next time we’ll work some of these problems backwards knowing the amount of gain and calculating the input, or output required.

Learn more about Decibels with CIE's Decibel Lesson.

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