40 dB = 10 Log (10/0.001).

Seemingly this is a much easier expression to use in calculations and eliminates the possible confusion of zeros (place holders). So, gain expressed in dB is simply 10 times the log of the ratio of two power levelsGain can also be expressed in minus (-) dB, generally we call this attenuation or loss, in some cases. Take for instance a resistive network that attenuates a 10 watt power level to a 1 mw power level. The ratio is simply reversed from when expressing gain as 1 mw/10 w or 0.001/10:

-40 dB = 10 log (0.001/10)

The nice thing about the expression of power levels in dB comes when a series of gains or losses are required to be calculated. Arithmetically the gain ‘figures’ would have to be multiplied where as dB can simply be a series of algebraic additions:

40 dB + (-40 dB) + 10 dB = 10 dB

Now, gain as referred to an antenna has somewhat of a different twist. If an antenna measured on an antenna range exhibits a gain of 15 dB over the gain of a dipole (or 15 dBd), just what is being said? If the 15 dB is reduced to a gain ‘figure’ by dividing the 15 by 10 and taking the antilog:

15/10 = 1.5 antilog = 31.62.

This simply means that the measured antenna has 31.62 times the ‘directivity’ of the dipole. In other words if 1 watt of power were to be applied to the dipole and used as a reference, then the same 1 watt applied to the measured antenna would appear as 31.62 watts at the measurement equipment.

Now, how is this possibly. Well, the term ‘directivity’ was used in the latest paragraph and at this point needs to be examined. What is ‘directivity’? Directivity is the property of having the induced power to an antenna placed more in one direction than any other. Often for the purposes of calculation the reference is the ‘isotropic’ source (an assumed entity) or a source that radiates equally well in all directions. The dipole actually has a gain ‘figure’ of 1.64 or (2.14 dB) over an isotropic source so the dipole is not the best reference that can be used; however, in practice it is usually adequate. So let me further explain the nature of the isotropic source.

If an ‘isotropic’ source (a point source) were to be placed inside a ‘unit’ sphere that has a radius of one radian 57.3° and have a circumference of 2π radians (2 X 3.1416 X 57.3°) or 360°, and a surface area of 4πR^2 radians or 41,259 steraidian square degrees (sr), then every degree on the inside of that sphere would be illuminated. Kraus suggests rounding off to 41,000 sr degrees in his second edition of “Antennas” McGraw-Hill. If all 41,000 sr degrees are illuminated on the surface of the 41,000 sr degrees of the unit sphere the directivity of the point source becomes “1” (41,000/41,000) and therefore the gain being; A = 10 Log of directivity is then “Ø”!

So how do we know that an antenna has a specified gain (most manufacturers usually fudge on this)? Let’s take an actual antenna with the specified gain figure. The M^2 2M9 Yagi antenna has a specified gain of 12 dBd or 14.14 dBi and a specified beamwith(s) of 35° E-plane and 40° H-plane;

A(dBi) = 10 Log ((41,000/(ΘE X ΘH))

14.666 dBi = 10 log ((41,000/(35 X 40))

The calculated figure of 14.666 dBi as compared to the specified figure of 14.14 dBi (12 dBd) is a lot closer than most published antenna gain figures. As a personal note I might add that having known Mike Staal (M^2) for over 35 years, he has always been quite honest with his antenna gain figures. The slight error may be attributed to the rounding off of E and H plane angles as these are difficult to measure in the first place.

I would like to think that this exercise has been useful to the reader in the understanding of how gains are arrived at and made useful.

**Dave W6OAL – **

**Olde Antenna Laboratory 41541 Dublin Drive Parker, CO 80138**