Dbs the Easy Way!

09/22/2012

(no Slide-Rules, Log-Tables, or Calculators needed!)

Many folks that would like to understand the relationship of db gain, and how it translates to what happens on antennas and transmission lines find dbs confusing, let alone how to apply them. We are told that dbs are "simply logarithmic"... Simply stated, logarithms increase their value "exponentially", like increasing their value by doubling, quadrupling, etc., or by factors of 10 (called "factor" increase, where 2 factors = x102 = 100).

Dbs for Power Gain or Power Loss (Attenuation)

Dbs for Voltage or Current Gains or Loss (Attenuation)

Dbs for comparison with the theoretical "Isotropic Antenna" ("dbi")

Dbs for comparison with a Common Dipole Antenna ("dbd")

A special note about the common abbreviation for "Decibel":
Prior to 1973 (or 1972), the proper abbreviation for decibel was "db", except for when it was the leading word in a sentence. At that instance, proper english rules indicated that the first letter should be capitalized, as "Db". At the same time that "Hertz" was designated as the common usage for "cps" (cycles per second), it was deemed to also change the common description of the term "db" to be shown as "dB", to honor the name "Bell". For those of us who were schooled in the days of the "Radio Engineers' Handbook" by Terman (and others), it's still not easy to accept these new "accepted variations". Even the ARRL Handbooks, up through 1971, used the earlier abbreviation for decibel as "db", and as "Db" when it is the first letter of a sentence. Current convention would indicate using the latter abbreviation adopted as the current standard by IEEE & ISO, but there are times when I think we may lose sight of the more important aspect of the clarity of the technical side of the material, which is trying to understand the value, simplicity, and application of this tool of measurement.

For those who might prefer the newer convention of the abbreviation for this presentation, click on Easy_dB. (vs Easy_Db)

### Dbs for Common Power Gain or Power Loss:

Here is a simple procedure for laying out a method to illustrate Power db gain (or loss), and is easy to perform and easy to remember (both of which are key points). While doing this we can see a little bit of what a "logarithm" is. Simply stated for this instance, we find that the "multiplication factor", as it relates to the "db-gain". i.e. Where the db values increase in a "straight-line", we find that the related "multiplication factor" increases radically (actually "logarithmically"), by basically doubling every 3 db. If we lay out a simple table of just 2 columns, identified first as db gain, and secondly as "Factor (multiplier)", we can begin this simple and understandable task.

The way we do this, is to first establish that 0 db has a "Gain Factor" of 1 (which we will later explain why). ..Next, we need to double the "Gain Factor" for every increase of 3 db. This "doubling" is in essence is one example of a "logarithmic increase in value"... A key point here is that these db gains and their respective "Gain Factors" are actually a continuous circle, where the 10 db restarts at the bottom with the same 0 db, but the "Gain Factor" has increased by 10 times.

Let's start out and see how easy this really works: ... The doubling of the "Gain Factor" from x1 to x8, from 0 db to 9 db in 3 db steps, is pretty straightforward... Adding 3 db to the 9 db would give us 12 db, which when "rolled over" on the chart (where 10 db is represented as 0 db at the bottom), 12 db would be on the chart as 2 db (like 10 db + 2 db).

What happens is that the dbs roll over by simply adding, so that 11 db (which is simply 1 db more than 10 db) is shown as 1 db above 0 db. .. The next "db rollover" would result in 19 db + 1, or ... 20 db!! .. To illustrate this, note that where going from 9 db (where we had a "Gain Factor" of x8), 3 db more takes us to the 2 db point (i.e. 9 db + 3 db = 12 db, as 2 db above the 0 db)... Since doubling 8 would give us 16, but rolling over on the chart increases our "Gain Factor" by x10, we can illustrate the 16 as 1.6 instead...

Those relationships are simply accomplished by scientific notation, where a "Gain Factor" of x16 can be expressed as 1.6x101, and a "Gain Factor" of x21.25 would be expressed in scientific notation as 2.125x101.

The simplest process for converting db gain to a "Gain Factor", is to strip off the numbers that precede the "basic 1 figure db", use that basic 1 figure db as from the chart, and the number(s) preceding that "basic 1 figure db" as the power of 10.

This means that an antenna which has a 14 db gain would mean (taking the basic 4 db = x2.5 from the chart) that our antenna has a power gain of 2.5x10+1, ... i.e. 25 times... The thing to remember is that where the dbs roll over by adding, the "Gain Factor" however increases by 10 times, each time that "rollover" occurs.

Think of it like an Odometer (like xxxx.n) where the ".n" represents the "db gain" in each of the Tables below. In that Odometer, when the ".n" goes from ".0" to ".9", and then 0.1 or 0.3 more, we would expect the next number in the adjacent column to pump up by "0.1" or "0.3" above the "0.9". So... 0.0 increasing to 0.9, and then "0.3" more would cause the "0.9" to become "1.2"...... Keep all that in mind as we present these 3 Tables.

 Table #1
 Db Gain Factor (x) . . 9 8 8 . 7 . 6 4 5 . 4 . 3 2 2 . 1 . 0 1
 Table #2
 Db Gain Factor (x) . . 9 8 8 6.4 7 . 6 4 5 3.2 4 . 3 2 2 1.6 1 . 0 1
 Table #3
 Db Gain Factor (x) 10 10 9 8 8 6.4 7 5.0 6 4 5 3.2 4 2.50 3 2 2 1.6 1 1.25 0 1

The reason for the 0 db = 1 , is that any number to the "0" power is equal to "1". .. Think of it this way, everyone knows that 22 = 4, and 23=8. .. But also note that as this process is reversed, we find that 21=2, and 20=1 !!

Examples:

.... An antenna with a given Power Gain of 5 db and driving it with 15W, means that it would have a "Power Gain Factor" of x3.2, resulting with effectively the same as driving with 48W.

Special Note #1: ... This Process is actually the result of numerical descriptions of the resulting values of powers of 10, ranging from 0.0 to 1.0 in 0.1 steps, where 0 db is actually 100.0 = 1. Continuing this process, we find that 100.1 = 1.25 ..... 100.3 = 2.0 .... 100.5 = 3.16 ... ....100.7 = 5.0 ... [ note that 7 db correlates to x5 as a "Gain Factor" ]

Special Note #2: ... In the opposite way of looking at this, we would say that the "log of 5.0 = 0.7", as the necessary power of 10 required to result in the 5.0.

.... A Linear Power Amplifier that has a 550W output if driven with 15W, has a "Power Gain Factor" of 36.7. Using scientific notation, this is expressed as 3.67x101. We find this approximately at 5.6 db on the chart, and by adding the x101) to the db we get 15.6 db as the Power Gain (i.e. 5.6 db + 10 db).

When folks use a calculator, it's easy to get confused as to whether to find the "log" of the number in question, or the "anti-log" or "inverse-log". .. By using this chart method (with scientific notation), it's hard to go wrong. But... "Practice helps a lot!".

### Dbs for Voltage Gains & dbs for Current Gains:

Technically, the proper formula for Power Gain, Voltage Gain, and Current Gain (based on the ratios) in db is:

Db(pwr)=10Log (Po/Pi), ..... where "Po" represents Power Output, and "Pi" represents Power Input

Db(voltage)=20Log(Eo/Ei), ..where "Eo" represents Voltage Output, and "Ei" represents Voltage Input

Db(current)=20Log(Io/Ii), ....where "Io" represents Current Output, and "Ii" represents Current Input

 Table #1
 Power db Gain Factor (x) 10 10 9 8 8 6.4 7 5.0 6 4 5 3.2 4 2.5 3 2 2 1.6 1 1.25 0 1
 Table #2
 Voltage db Gain Factor (x) 20 10 18 8 16 6.4 14 5 12 4 10 3.2 8 2.5 6 2 4 1.6 2 1.25 0 1
 Table #3
 Current db Gain Factor (x) 20 10 18 8 16 6.4 14 5 12 4 10 3.2 8 2.5 6 2 4 1.6 2 1.25 0 1

Please note that the only thing that changes on our charts, is that the db representations for the Voltage & Current dbs are simply illustrated as twice that of the Power dbs! Also note that the Voltage and Current db Charts are exactly the same. Knowing all this makes it really easy for us, in that we can make our chart for the Power db, and then make another (or two) to represent the Voltage or Current chart(s). If we get a good enough handle on this, we might be able to get by on just the Power db Chart.

We need to make a couple of very important observations here! Remember that on the Power db Chart that each time we "rolled up and over" the top to start up from the bottom again, our "Gain Factor" was simply multiplied by 10. That's why a factor of 12.5 would show up as the same place on the chart as 1.25. Now, in the Current db Chart and Voltage db Chart, we still have our "Gain Factor" was simply multiplied by 10, each time we "roll up and over" the top to start up from the bottom again, it takes twice as many dbs to do that. The mathematicians would tell us that this is because power is the product of voltage & current.

Now, let's do an example or two of these new charts.

Example #1: Suppose we have a transistor circuit that when we drive it with a 15microamp signal, and we get out a 3.2ma signal, that would represent a current gain of 213. First convert the 213 to scientific notation to get 2.13x102. Looking on the Current db Gain Chart, we find that 2.13 is a little over the 6db mark (probably not as much as 7db, and more likely about 6.5db). Now, we double that power of 10 that was in the scientific notation and add it to the to the front end of the 6.5db to make it 26.5db as the current gain of that transistor circuit. Remember that in Power dbs we used the x10n as a direct add to the front of the basic db, but in Current Gain we take that x10n and double it (everything in the Current dbs column is doubled).

Example #2: Consider that same transistor circuit, where the signal input voltage is 0.11V, and the output signal is found to be 6.2V. This would represent a voltage gain of ~56.4. Converting this to scientific notation would give us 5.64x101. Now, we look on the Voltage db Gain Chart and find that 5.64 gives us about 15db. Now double the power of 10 that was in the scientific notation and add it to the front end of the 15db to make it about 35db as the voltage gain of that transistor circuit. Remember that in Power dbs we used the x10n as a direct add to the front of the basic db, but in Voltage Gain we take that x10n and double it (everything in the Voltage dbs column is doubled).

Situation: When values are converted to Logarithms, and the original values are multiplied to give a "product" of the originals, and then we find the Log of that product, we will find that adding the Logarithms of each value to each other will give the same number as that Log of the product. We know that power is a product of current and voltage. We also know that the power gain of a circuit can be found by the product of the voltage gain and the current gain.

### Dbs for Power Gains over an "Isotropic Antenna":

Actually, this is the same concept as the first db Power Gain (as it relates to antennas) that was presented at the beginning, with only one exception... It starts with a theoretical assumption of an antenna in free space, radiating from a "point" with no actual dimensions, radiating uniformly in all directions. A common dipole antenna has a slight increase in radiation energy at right angles off from the center of the length of the dipole, with less energy radiating off of the ends of the dipole away from the center. This comparison of increase in energy radiating at a right angle from the center (and less from the ends) compared with that which would be uniform from an Isotropic Antenna, constitutes what is considered to be a slight db Power Gain over that of an "Isotropic Antenna in free space". This "Power Gain of a Common Dipole" is usually considered to be approximately 2.16db over an Isotropic Antenna. This is referred to as a 2.16dbi Power Gain.

### Dbs for Power Gains in comparison with a Common Dipole:

Here again, this is the same concept as the first db Power Gain for antennas, and similar to the presentation above about "dbs for Power Gains over an Isotropic Antenna". The real difference here is that instead of comparing the antenna power gain against an Isotropic Antenna, the power gain of the antenna in question is compared with that of a common dipole. Where someone might want to impress you by saying that their antenna has a 12db Power Gain, are they relating that to an Isotropic Antenna or a Common Dipole? If it is actually being compared to an Isotropic Antenna, then the 2.16db Power Gain of the Common Dipole should be subtracted out, leaving only a 10db Power Gain over a Dipole. This then is referred to as a 10dbd Power Gain.