How to calculate antenna beam width

[yanir]

Hi Antenna gurus.

I'm trying to figure out the beam width for a patch antenna at a specified gain. Let's say the gain of the antenna is 25dbi. If I take an ideal isotropic antenna and take all the energy from the lower hemisphere and direct it equally on the upper hemisphere I'd have a gain of 3dbi and a beam width of 180 degrees right? If I keep doubling my power and narrowing the beam by halving the beam width will I arrive at a decent approximation?

Is there a better way do this (without simulation)?

Is my line of thinking correct?

I question my results because the beam is gets extremely narrow at 25dbi using this method.

Thanks!

[ataradov]

Your thinking is correct.

and narrowing the beam by halving the beam width

Technically, you don't halve the angle, you halve the surface area and end up with some angle. But for high gain antennas you halve so many times, it does not really matter.

And yes, high gain antennas are very directional.

[ataradov]

Also, after doing some math, it look like angle of the ideal beam can be calculated as A=arcsin(2/G). This is valid for G >= 2.

So in this case A = arcsin(2/316) = 0.0063291 radians = 0.36 degrees.

[retrolefty]

25 dbi is a lot of gain and it counts both for transmit and receive, care to share the design and frequency?

[ataradov]

25 dbi is a lot of gain and it counts both for transmit and receive, care to share the design and frequency?

It is a typical claim for Chinese 2.4 GHz yagi antennas that are sold all over the place.

Just goolge "25 dBi yagi" and you will find a number of places selling them.

PS: I have not seen an actual performance measurement, so they very well may be "Chinese" 25 dBi, lust like 1000 W portable CD players

[retrolefty]

25 dbi is a lot of gain and it counts both for transmit and receive, care to share the design and frequency?

It is a typical claim for Chinese 2.4 GHz yagi antennas that are sold all over the place.

Just goolge "25 dBi yagi" and you will find a number of places selling them.

PS: I have not seen an actual performance measurement, so they very well may be "Chinese" 25 dBi, lust like 1000 W portable CD players

 Most likely a claim not to be belived in most cases. My attention was raised as the OP stated it's a 'patch antenna' and I wasn't aware that they could be made with that much gain. A quick but not through search brought up this nugget:

The directivity of patch antennas is approximately 5-7 dB. The fields are linearly polarized, and in the horizontal direction when viewing the microstrip antenna as in Figure 1a (we'll see why in the next section). Next we'll consider more aspects involved in Patch (Microstrip) antennas.
http://www.antenna-theory.com/antennas/patches/antenna.php

[yanir]

The directivity of patch antennas is approximately 5-7 dB. The fields are linearly polarized, and in the horizontal direction when viewing the microstrip antenna as in Figure 1a (we'll see why in the next section). Next we'll consider more aspects involved in Patch (Microstrip) antennas.
http://www.antenna-theory.com/antennas/patches/antenna.php

Ah interesting. I didn't know that a patch could not have that much directivity, perhaps it's more practical than theoretical?

I can't go into much detail on the source antenna, mostly because I don't have the detail. But I'm ultimately trying to estimate the beam width for an antenna with a gain of 25dbi. I usually use patches for this application but it would seem that something else is being used (i'd guess helical).

Either way it seems that it's quite narrow.

[yanir]

Also, after doing some math, it look like angle of the ideal beam can be calculated as A=arcsin(2/G). This is valid for G >= 2.

So in this case A = arcsin(2/316) = 0.0063291 radians = 0.36 degrees.

is G = Gain in dbi? If so where did 316 come from? 0.36 degrees seems too narrow.

Are you sure this equation is correct? If i use 25 for G I get 4.58 degrees. Seems possible.

[jh15]

Does it have side lobes?

[ataradov]

is G = Gain in dbi?

It is a linear gain. P(dBi) = 10*log10(P) => P = 10 ^ (P(dBi) / 10) = 10^2.5 = 316.23.

Are you sure this equation is correct?

This equation is derived from some basic geometry.

I worked with hemispheres (hence "/2"), since otherwise math get a bit complicated. Let me double check my calculations and I'll type them here after that.

[ataradov]

Area of a sphere A1 = 4*pi*R^2, Area of a spot where energy will be concentrated in a directional antenna is A2 = 2*pi*R^2 * sin(alpha), where alpha is a beam width angle. In this case math holds only for alpha = 0..90 degrees (hemisphere).

A2 = A1 / G. After some trivial transformations we get sin(alpha) = 2/G, so alpha = arcsin(2/G).

Simple check. For G = 2 (or 3 dBi) we get alpha = 90 degrees, or full hemisphere.

[Howardlong]

Also, after doing some math, it look like angle of the ideal beam can be calculated as A=arcsin(2/G). This is valid for G >= 2.

So in this case A = arcsin(2/316) = 0.0063291 radians = 0.36 degrees.

is G = Gain in dbi? If so where did 316 come from? 0.36 degrees seems too narrow.

Are you sure this equation is correct? If i use 25 for G I get 4.58 degrees. Seems possible.

I agree, 0.36 is waaay out of the ballpark. Even 4.58 degrees is hopeful, maybe 10 degrees off the top of my head.

Also, a single patch on its own is unlikely to have anywhere near 25dBi gain, more like about 7 or 8dBi.

[ataradov]

Well, I've shown my math, show yours if you have better version.

It is possible that 25 dBi is not actually achievable in real life, so your real life beams are way wider. But if you were to get 25 dBi, the beam would be very narrow.

[Howardlong]

This is the rule of thumb calculation for reasonably directional antennas:

Beamwidth (degrees) ~= sqrt ( 32400 / ( 10 ^ ( GdB / 10 ) ) )

[Howardlong]

Well, I've shown my math, show yours if you have better version.

It is possible that 25 dBi is not actually achievable in real life, so your real life beams are way wider. But if you were to get 25 dBi, the beam would be very narrow.

I also have the added benefit of being able to rely on something more than just equations ;-) somewhere in my career I designed antennas for S band satellite groundstations. Just sayin'

I think your maths is assuming only two dimensions, i.e. the beam is squashed like a doughnut in a press, spreading out uniformly in a plane, rather than being squeezed out of a toothpaste tube in one direction.

[ataradov]

Where does 32400 come from? And what is "reasonably directional"?

Actually in my case you get half of the full cone angle, so total beam width would be A = 2*arcsin(2/316) = 0.73 degrees.

[ataradov]

I also have the added benefit of being able to rely on something more than just equations ;-) somewhere in my career I designed antennas for S band satellite groundstations. Just sayin'

I always respect the real life experience, but then I want to see why math is wrong, especially if math predicts much worse results. Usually things are the other way around.

I think your maths is assuming only two dimensions, i.e. the beam is squashed like a doughnut in a press, spreading out uniformly in a plane, rather than being squeezed out of a toothpaste tube in one direction.

I don't understand .The math is for ideal conical section of the sphere.

[Howardlong]

Where does 32400 come from? And what is "reasonably directional"?

Actually in my case you get half of the full cone angle, so total beam width would be A = 2*arcsin(2/316) = 0.73 degrees.

Reaonably directional is so that

sin x ~= x

for small angles.

Fudge factor is quite a common one, but is here, https://en.m.wikipedia.org/wiki/Directivity

[ataradov]

Ok, it looks like I'm wrong somewhere. It is tool late today to think clearly.

One thing to note so far. That formula produces solid angle, which is not the same as an angle of a cone approximating the beam. Although they are reasonably close for 25 dBi gain.

The logic behind Wikipedia article is basically the same as mine, so it is very possible that I made some stupid mistake. I'll double check this tomorrow.

[Howardlong]

Ok, it looks like I'm wrong somewhere. It is tool late today to think clearly.

One thing to note so far. That formula produces solid angle, which is not the same as an angle of a cone approximating the beam. Although they are reasonably close for 25 dBi gain.

The logic behind Wikipedia article is basically the same as mine, so it is very possible that I made some stupid mistake. I'll double check this tomorrow.

If you take the square root of your result (in radians) you're not a million miles off, hence my analogy to squashed doughnut/squeezed toothpaste.

If it's any consolation, for me, it was a bit early.

[ataradov]

Ok, I found a mistake. My handwriting has failed me and I mistook R for A

After correcting this mistake final formula for the cone angle is theta = acos(1 - 2 / D). Or for D = 316, we get 0.1125 radian or 6.5 degrees.

This matches Wikipedia results, and they do essentially what I did - calculate ratio of surface areas of the full sphere and a directional spherical segment.

PS: Again, this is half of the angle, so full cone opening angle will be 13 degrees.

[Howardlong]

If the OP wants to get more gain from a patch antenna, there are two common methods.

(1) Create a patch array, feeding each element in phase with an appropriate power splitter/phasing harness. The downside of this, in a passive configuration, is that beyond a certain number of patches the losses on the power splitter can become significant. This is especially true of arrays fabricated on FR4 which is becomes quite lossy at higher frequencies. In short, it's the law of diminishing returns.

(2) Use a passive reflector, typically this would be a dish. The problem with using patches with dishes is that it's not always easy to efficiently illuminate the dish without some overspill, which leads to noise in your receiver from the warm earth if the antenna's space-pointing. Choosing the right dish f/D is also important for this. This is why horn antennas or their derivatives are more commonly used as feeds. Having said that, I've had a lot of practical success with patch feeds on dishes.