### Understanding Decibels

Posted:

**Sat Apr 30, 2011 10:02 am**It's been kinda quiet on the forum lately, so I thought I would add a small discussion that might be useful not only for the Technicians, but for hams of all stripes. Let's go over decibels and their use in ham radio. Have you ever picked up an issue of QST and looked at the advertisements and found items such as antennas and their gain being expressed in dBi, or a new transceiver with a dynamic range in dB and wondered what the heck it all means ? Every ham has !

Decibels allow us to talk about very large differences in power (or intensity) or voltage levels. This originally began back in the early telephone days as a way to quantify audio levels in telephone circuits. Before the word 'decibel' was used, it was called transmission unit (TU) and was defined as 10 times the common log (base 10) of the ratio of measured power to a reference power level. As we are talking about telephones, the TU was renamed decibel in honor of, you guessed it, Alexander Graham Bell ! OK, so what's that logarithm stuff mean ? Well, this is where I have to get my 'geek' on a bit. A logarithm quite simply is an exponent. For example, if we are given a base of 2 and an exponent of 3, then we can evaluate 2³= 8. Inversely, if we are given the base 2 and its power 8 (2ⁿ = 8), then what is the exponent that will produce 8 ? That exponent is called a logarithm. We say that the exponent 3 is the logarithm of 8 with base 2. 3 is the exponent to which 2 must be raised to produce 8. This is an example of binary logarithms - often used in computers. Now let's go to common logarithms which is, of course, base 10. Since 10^4 (10 to the 4th power) = 10,000, then log 10,000 = 4. 4 is the exponent to which 10 must be raised to produce 10,000. Logarithms began back in the 17th century and are now heavily relied upon by scientists and engineers to greatly simplify calculations by means such as slide rules and logarithm tables - up until the late 1970's. Remember the slide rule ? I sure do !! That's how we dealt with calculations to send men to the moon in the 1970's !

OK, let's take it back to ham radio. The maximum power output of a legal transmitter in the USA is 1500 watts and the noise floor of a modern receiver is 0.04 microvolts. The received power at this voltage level into 50 ohms is 0.00000000000000003 watts or 3 x 10 ^-17 watts. These numbers are not easy to deal with. In order to talk about decibels, you have to have a frame of reference. RF engineers refer to power levels as dB above or below 1 milliwatt in a 50 ohm system and call this result dBm. Thus, 1 milliwatt is 0 dBm. A rule in logarithms is that when subtracting logarithms, you can divide the factors - that's how we talk about differences in power. So log (A/B) = log A - log B. OK, so what is 1500 watts in dBm ? And how does this compare to the noise floor ? Behold these equations:

1500 watts in dBm = 10 log (1500 / 0.001) = +62 dBm

3 x 10^-17 watts in dBm = 10 log (3 x 10^-17 / 0.001) = -135 dBm

OK, so how many dBm is a 100 watt transmitter producing ? 100 watts in dBm = 10 log (100 / 0.001) = 10 log (100,000) = 10 x 5 = +50 dBm. Simple, huh ?

We can talk more about receivers and transmitters later. Let's go to one of my favorite subjects - antennas ! So how are decibels used for gain comparisons in antennas ? Well, a beam can have considerable gain. Typical HF Yagi beams can have 8 dBi gain or more and a large VHF beam can have 20 dBi gain or more. OK, but what's that 'i' mean ? Remember, decibels always has to have a frame of reference, and in this case that is the isotropic radiator. Huh, you ask ? An isotropic source is a theoretical point source of electromagnetic energy that radiates equally in all directions. Here are some easy numbers to remember:

☻1 dBi = 1.25 x power

☻2 dBi = 1.6 x power

☻3 dBi = 2 x power

☻10 dBi = 10 x power

So, a 20 dBi gain antenna would have 10 x 10, or 100 x power gain of an isotropic radiator. 1 watt fed into that antenna would be as loud as 100 watts fed into an isotropic source. But suppose we want to compare different antennas, that is, compare not to a theoretical point source ? This is done by changing the frame of reference to a dipole in free space, or dBd. The half-wave dipole in free space has a gain of 2.15 dBi, so gain expressed in in dBd is always 2.15 dB less than gain expressed in dBi. This does not change the actual gain of the antenna; the frame of reference is the only thing changed. Guess which number is used by manufacturers of antennas to sell their wares ? So why is the dipole in "free space" ? Well, in the real world, the ground affects the antenna performance by reflecting signals upward. This actually adds up to about 4 dB to the gain of the antenna. So a half-wave dipole antenna over ground can actually have about 4 dBd of gain ! The half-wave dipole over ground has 4 dB gain over a half-wave dipole in free space. You can conceptualize this effect by remembering that dish antennas are used to reflect and enhance signals from space to a receiving antenna (which may or may not be a dipole). Simply imagine the ground as the 'dish' that is reflecting this energy upward back at the dipole ! Neat, huh !?

There are lot and lots of examples of decibels being used in ham radio to make life easier to understand what's going on, whether under the hood of your transceiver or in the ether above your antenna. Can you think of some examples ?

73, Jamie

WB4YDL

Decibels allow us to talk about very large differences in power (or intensity) or voltage levels. This originally began back in the early telephone days as a way to quantify audio levels in telephone circuits. Before the word 'decibel' was used, it was called transmission unit (TU) and was defined as 10 times the common log (base 10) of the ratio of measured power to a reference power level. As we are talking about telephones, the TU was renamed decibel in honor of, you guessed it, Alexander Graham Bell ! OK, so what's that logarithm stuff mean ? Well, this is where I have to get my 'geek' on a bit. A logarithm quite simply is an exponent. For example, if we are given a base of 2 and an exponent of 3, then we can evaluate 2³= 8. Inversely, if we are given the base 2 and its power 8 (2ⁿ = 8), then what is the exponent that will produce 8 ? That exponent is called a logarithm. We say that the exponent 3 is the logarithm of 8 with base 2. 3 is the exponent to which 2 must be raised to produce 8. This is an example of binary logarithms - often used in computers. Now let's go to common logarithms which is, of course, base 10. Since 10^4 (10 to the 4th power) = 10,000, then log 10,000 = 4. 4 is the exponent to which 10 must be raised to produce 10,000. Logarithms began back in the 17th century and are now heavily relied upon by scientists and engineers to greatly simplify calculations by means such as slide rules and logarithm tables - up until the late 1970's. Remember the slide rule ? I sure do !! That's how we dealt with calculations to send men to the moon in the 1970's !

OK, let's take it back to ham radio. The maximum power output of a legal transmitter in the USA is 1500 watts and the noise floor of a modern receiver is 0.04 microvolts. The received power at this voltage level into 50 ohms is 0.00000000000000003 watts or 3 x 10 ^-17 watts. These numbers are not easy to deal with. In order to talk about decibels, you have to have a frame of reference. RF engineers refer to power levels as dB above or below 1 milliwatt in a 50 ohm system and call this result dBm. Thus, 1 milliwatt is 0 dBm. A rule in logarithms is that when subtracting logarithms, you can divide the factors - that's how we talk about differences in power. So log (A/B) = log A - log B. OK, so what is 1500 watts in dBm ? And how does this compare to the noise floor ? Behold these equations:

1500 watts in dBm = 10 log (1500 / 0.001) = +62 dBm

3 x 10^-17 watts in dBm = 10 log (3 x 10^-17 / 0.001) = -135 dBm

OK, so how many dBm is a 100 watt transmitter producing ? 100 watts in dBm = 10 log (100 / 0.001) = 10 log (100,000) = 10 x 5 = +50 dBm. Simple, huh ?

We can talk more about receivers and transmitters later. Let's go to one of my favorite subjects - antennas ! So how are decibels used for gain comparisons in antennas ? Well, a beam can have considerable gain. Typical HF Yagi beams can have 8 dBi gain or more and a large VHF beam can have 20 dBi gain or more. OK, but what's that 'i' mean ? Remember, decibels always has to have a frame of reference, and in this case that is the isotropic radiator. Huh, you ask ? An isotropic source is a theoretical point source of electromagnetic energy that radiates equally in all directions. Here are some easy numbers to remember:

☻1 dBi = 1.25 x power

☻2 dBi = 1.6 x power

☻3 dBi = 2 x power

☻10 dBi = 10 x power

So, a 20 dBi gain antenna would have 10 x 10, or 100 x power gain of an isotropic radiator. 1 watt fed into that antenna would be as loud as 100 watts fed into an isotropic source. But suppose we want to compare different antennas, that is, compare not to a theoretical point source ? This is done by changing the frame of reference to a dipole in free space, or dBd. The half-wave dipole in free space has a gain of 2.15 dBi, so gain expressed in in dBd is always 2.15 dB less than gain expressed in dBi. This does not change the actual gain of the antenna; the frame of reference is the only thing changed. Guess which number is used by manufacturers of antennas to sell their wares ? So why is the dipole in "free space" ? Well, in the real world, the ground affects the antenna performance by reflecting signals upward. This actually adds up to about 4 dB to the gain of the antenna. So a half-wave dipole antenna over ground can actually have about 4 dBd of gain ! The half-wave dipole over ground has 4 dB gain over a half-wave dipole in free space. You can conceptualize this effect by remembering that dish antennas are used to reflect and enhance signals from space to a receiving antenna (which may or may not be a dipole). Simply imagine the ground as the 'dish' that is reflecting this energy upward back at the dipole ! Neat, huh !?

There are lot and lots of examples of decibels being used in ham radio to make life easier to understand what's going on, whether under the hood of your transceiver or in the ether above your antenna. Can you think of some examples ?

73, Jamie

WB4YDL