A logarithm is the exponent to which the number 10 must be raised to reach some given value. If we are given the number 1000 and asked to find the logarithm (log), we find that log 1000 = 3 because 10^3 = 1000. Notice that our logarithm, 3, is the exponent. An important thing to note about logarithms is that the logarithm of a negative number or of zero does not exist.
Suppose we have a GPS satellite transmitting 50 Watts of power, with a gain of 2 in the direction of a receive antenna with a gain of 2. Suppose the antennas are separated by 20,000 kilometers (as is the typical orbit of a gps satellite). The GPS satellites operate at a frequency of 1.575 GHz. How much power is received?
Using the Friis Transmission Equation and the fact that wavelength equals c/f, we can calculate the received power to be:
Of the 50 Watts transmitted, about 2.3-16 % of that power gets through. Incidentally, this might seem frighteningly small, but yes, antenna systems can operatte with this less power.
Engineers don't like to use linear units when the quantities can vary by such large amounts. Its real tough to picture in your head the difference between 1017 and 1018. To work around this, we use the decibel system.
By definition a decibel is a simple logarithmic transformation - units in linear are easily converted via:
Recall the Friis Transmission Formula:
To convert this equation from linear units in Watts to decibels, we take the logarithm of both sides and multiply by 10:
A nice property of logarithms is that for two numbers A and B (both positive), the following result is always true:
Equation (*) then becomes:
Using the definition of decibels, the above equation becomes a simple addition equation in dB:
The above representation is easier to work with, which is kind of cool.
Another common unit is dBm. This means "decibels relative to a milliWatt". In this case, we are talking about power, and the power is just specified in milliWatts instead of Watts.
1 mW = 0 dBm
1 W = 1000 mW = 0 dB = 30 dBm
0 dBm = -30 dB = 0.001 W = 1 mW
The relationship between the decibels scale and the mW scale can be estimated using the following rules of thumb:
• +3 dB will double the watt value:
(10 mW + 3dB ≈ 20 mW)
• Likewise, -3 dB will halve the watt value:
(100 mW - 3dB ≈ 50 mW)
• +10 dB will increase the watt value by ten-fold:
(10 mW + 10dB ≈ 100 mW)
• Conversely, -10 dB will decrease the watt value to one tenth of that value:
(300 mW - 10dB ≈ 30 mW)
Another common variation on dB in antenna theory is dBi which means "decibels relative to an isotropic antenna". This just specifies the gain of an antenna relative to the isotropic gain, which is 1. So, the unit of measurement dBi refers only to the gain of an antenna. The “i” stands for “isotropic”, which means that the change in power is referenced against an isotropic radiator.
So really nothing changes...:
Gain of 10 dB = Gain of 10 dBi
This means "decibels of gain relative to a standard half-wave dipole antenna" or "decibels relative to a dipole antenna".
The gain of a half-wave dipole is 2.15 dBi.
Antenna with a gain of 10 dBi = 10 dB = 7.85 dBd
Gain of half-wave diple antenna = 2.15 dBi = 0 dBd
Hence, 7.85 dBd means the peak gain is 7.85 dB higher than a dipole antenna;
this is 10 dB higher than an isotropic antenna.
If P2 > P1 : loss (negative number)
if P2 < P2 : gain (positive number)