Those Pesky Decibels!

Newbie: What’s the difference between a 3-element Yagi and a dipole antenna?
Elmer: About 6 dB.

In the Technician License Course you learn two rules of thumb about decibels that get you through the exam and provide a rudimentary understanding of their use:

  1. A doubling or halving of power equals a change of approximately 3 dB.
  2. A 10X change of power equals a change of 10 dB.

Those two rules applied semi-brainlessly tend to cover about 90% of the typical application scenarios for most new-to-intermediate hams. But let’s see if we can get at a slightly deeper understanding of decibels, their utility in amateur radio and electronics, and how to use them with a bit more sophistication. After all, we’re hams! We’re all about learning!

Apples to Apples:  Be sure to keep one thing in mind as you proceed with decibels:  Decibels are used to make comparisons between two measures. The most common measure for comparison in amateur radio for which decibels are used is power, or signal strength. That’s what the Elmer above was getting at in the response to Newbie. A typical 3-element Yagi antenna will provide about a 6 dB increase in signal strength (gain) as compared to a dipole antenna’s signal strength. See, it’s a comparison, with the dipole antenna being the standard against which the Yagi is compared. No standard for comparison means no use of decibels.

“But why,” you now ask, “use this confusing decibel thing to make comparisons instead of just plain numbers?” The answer is really “for convenience,” although you may not consider decibels to be so handy if you haven’t worked with them for a while. Consider the following…

Let’s say you use a 100 watt transmitter to send a signal several thousand miles over the horizon on the 10m band. Your signal is received by a station in Australia generating a voltage on the receiving antenna of about 10 µV (10 microvolts, or 0.000010 volts). Going into the Aussie’s 50 ohm antenna impedance this generates a signal power of about 2 pW (2 picoWatts, or 0.000000000002 Watts). This is quite a large range of power values to consider, from 100 W to 2 pW. If we wanted to compare the strength of the transmitted signal with the received signal strength the comparison looks something like this:

(100 W /  0.000000000002 W) = 50,000,000,000,000

That’s a big bunch of zeros following the 5, and they tell us that the transmitted signal power is 50 trillion (5 x 1013) times greater than the received signal power. See how convenient those conventional comparisons can be when we get into large differences of values? And this is not even an extreme example, so you get the point – we sometimes need to compare values across a very large range of potential magnitudes, and conventional decimal comparisons are just a bit clunky for such comparisons because of the necessity to track all those zeros.

Decibels offer a more elegant solution for comparing values across a broad range of magnitudes, and they are used extensively in RF engineering, in electrical applications, and simply in every-day ham radio operations. Decibels are the standard way that hams compare signal strength or characterize antenna gain, regardless of the number of zeros involved.

The Power of Ten:  The decibel is a logarithmic unit. (No, don’t run! It’s not that scary, really!) A logarithm is the inverse of an exponent, and you may think of a logarithm as telling you “how many 10s have been multiplied together” in a number. Take a look at the table below.

log_table_hr101_decibels
Pierce, Rod. (12 Jan 2014).
“Introduction to Logarithms”. Math Is Fun. Retrieved 4 May 2014 from http://www.mathsisfun.com/algebra/logarithms.html

The decimal value of numbers is in the left-most column. The middle column represents the same number as multiples of the number 10. For instance, the number 100 is the same as 10 x 10. The number 1,000 is the same as 10 x 10 x 10, and so on. (The digit 1 is always included up front for consistency, especially for fractional values as follows.)

For decimal values less than one the same concept applies, only the “10” representation is by dividing some quantity of 10s repeatedly. So, 0.01 (one one-hundredth) is 1 ÷ 10 ÷ 10.

Notice that the right-most column indicates how many 10s have been multiplied or divided to obtain the number in the left-most column. For the number 100 the logarithm is 2, meaning that two 10s are multiplied to obtain 100. For 1000 three 10s are multiplied. For fractional values the number of divided 10s is depicted as a negative number. This right column number of “10 multiples” is the logarithm of the left column number. (Specifically, the base 10 logarithm, but we will not muddy the waters with the base in this article.)

So, what is the logarithm of the power of our 100 W transmitted signal? That’s easy, the log of 100 is 2. How about the 2 pW received signal? The weak signal’s logarithm is roughly -12. (Using a scientific calculator you can take the log of 0.000000000002 and find the value is a -11.7, rounded to the nearest 0.1.)

The base 10 logarithm of a number is also known as a bel, after Alexander Graham Bell, when compared to unity (1). The bel is a unit that is seldom used, but if you slice it into 10 parts you have units of decibels. Ten decibels comprise one bel, or one multiple of 10. Below are two different depictions of decibels, a graph and a table.

dB_graph_hr101_decibels
Courtesy PhysClips: http://www.animations.physics.unsw.edu.au/jw/dB.htm

The table equates decibel values with equivalent power dB_table_hr101_decibelsratios. Notice the following in the table:

  • 3 dB equates to about a 2:1 power ratio
  • 10 dB (aka 1 bel, or 1×10) equates to a 10:1 power ratio
  • 20 dB (aka 2 bel, or 1x10x10) equates to a 100:1 power ratio
  • 30 dB (aka 3 bel, or 1x10x10x10) equates to a 1000:1 power ratio
  • … and so on

The graph relates decibels on the vertical axis to power ratio on the horizontal. The resulting function, or curve of the graph, is called a logarithmic function. Although not depicted here, the curve will pass through a point to the right where the power ratio is 100 and the dB value is 20, just as in the table. Similarly, the other table values will be points along the logarithmic curve of the graph. A base 10 logarithmic function is computed as

y = bx

where, in this graph, y is in bel, x is the power ratio, and b is the base value of 10. In our graph bel has been subdivided into decibels to illustrate the common rule-of-thumb relationships mentioned earlier in this article.

Power Ratios:  We can now realize the power of decibels! <ahem>  Our crusty old Elmer told Newbie that the 3-element Yagi provided signal gain over a dipole antenna of about 6 dB. What does that mean in terms of a power ratio? We see from the chart or the graph that 6 dB equates to about a 4X increase in power. The directional signal of the Yagi antenna will be about 4X greater than the dipole signal in the direction of the Yagi’s main lobe, or its pointing direction.

What if you purchase coaxial cable that has a loss characteristic at 28 MHz of 2 dB per 100 feet of cable, and you use 100 feet of it to reach your 10m band antenna? How much of your transmitted power will reach your antenna? From the graph or table we can estimate that -2 dB results in a power ratio of approximately two-thirds, or roughly 0.67. So, about 67% of your transmitted power will reach your antenna, the remainder lost to heat in the cable. Use only half as much coax to reach the antenna and you lose only half as much power, or -1 dB, resulting in a power ratio of about 80%.

Final Calculations:  Finally, we should consider an equation by which a precise value of power gain in decibels may be determined from the power ratio, P2/P1. The equation is:

Gain (dB) = 10 x log(P2/P1)

where P2/P1 is the power ratio of the comparison.

So, using the example above for the Yagi to dipole comparison, we have a gain calculation as follows:

Gain (dB) = 10 x log(4/1)

Gain (dB) = 10 x log(4)

Gain (dB) = 10 x 0.602

Gain (dB) = 6.02

Or, going back to our Australian contact example, we determined the power ratio P2/P1 to be 50,000,000,000,000. Plugging that into the old scientific calculator, taking the logarithm, and multiplying by 10 yields a gain comparison of 137 dB! That is, the transmitted signal of 100 W is 137 dB stronger than the received signal. That’s quite a leap, but see how much more readable “137 dB” is than 50,000,000,000,000? Convenient, huh?

We can also invert this equation, solving for the power ratio to obtain

Power Ratio = 10G/10

where the power ratio is P2/P1, and G is the gain in dB.

Using our coaxial cable problem above with a 2 dB loss (-2 dB) we find

Power Ratio = 10-2/10

Power Ratio = 10-0.2

Power Ratio = 0.63 or 63%.

So, we find that our estimate from the table and graph was close, but not exact, and the power reaching the antenna through 100 feet of coaxial cable is actually about 63% of the transmitter’s output.

Bonus Example: Let’s try putting several of these decibel factors together in an example to illustrate just how useful these things really can be. Let’s try these factors:

  • 100 W transmitter power
  • Transmitter feedline loss of 1.8 dB/100 feet, feedline length of 75 feet
  • 3-element Yagi transmitting antenna gain of 6.1 dB
  • Assume the same -137 dB transmission path reduction from before
  • 2-element quad receiving antenna gain of 5.8 dB
  • Receiver feedline loss of 2.2 dB/100 feet, feedline length of 50 feet

What is the resulting power at the receiver? Let’s sum up the decibel losses along the way:

  • Transmitter feedline loss = -1.8 dB x 0.75 = -1.35 dB
  • Transmitting antenna gain = +6.1 dB
  • Transmission path loss = -137 dB
  • Receiving antenna gain = +5.8 dB
  • Receiving feedline loss = -2.2 dB x 0.5 = -1.1 dB

Summing:  -1.35 + 6.1 – 137 + 5.8 – 1.1 = -127.5 dB

Finally, we compute the power ratio associated with -127.5 dB:

Power Ratio = 10-127.5/10 = 1.78 x 10-13 or 0.000000000000178

So, only a tiny percentage of our 100 W of transmitter power is received:

 (100W x 0.000000000000178) = 0.0000000000178 Watts

at the receiving station, otherwise termed 17.8 pW (picoWatts). A very weak signal, indeed!

That’s a Wrap:  See, decibels are not really so pesky after all. They grow on you, much like a fungus or a bad rash, once you’ve had sufficient contact with them. You learn to love them! Good luck with your studies, and KEEP LEARNING!