# Stacked Yagi Gain calculation

I had a quick look at a practice exam for the license in my country and had a question on stacked Yagi antenna. I have been looking around but the only thing I can find is the mention of an additional 3db gain, increased sidelobes and reduced take-off angle. But nothing authoritative on this.

So here is my question:

For horizontally polarized Yagi antennas stacked vertically, how to calculate the gain provided by each additional antenna in the stack?

I don't know your country's questionaire, but:

This is just asking for Array Factor for a specific array. There's nothing specific about Yagi, or any other antenna.

When arranging identical antennas in an array, and specifying with which phase they are fed, we can calculate the antenna array's pattern from the individual antennas' patterns, their position and phase.

Basically, what happens is that you calculate a pattern, the array factor, that says how much gain and phase you have in every direction $$\vec r$$,

$$AF(\vec{r}) = \sum_{n=1}^N a_n e^{jk\vec{r}\cdot\vec{p}_n}$$

where

• $$\vec r$$ is the direction that you're looking at (in 3D radians)
• $$N$$ is the number of antennas,
• $$a_n=e^{j2\pi\varphi_n}$$ is the phase vector of the $$n$$th antenna (can also scale this if you don't feed the same power into every antenna),
• $$p_n$$ is the position of the $$n$$th antenna in 3D space
• $$k=\frac{2\pi}\lambda$$ the wave number of the free-space wave of the frequency you're interested in

(loosely based on Balanis' "Antenna Theory" 2nd ed; that's pretty much the standard book for antenna studies.)

Long story short: if you squint, and imagine that $$\vec p_n$$ are just equidistant points on a straight line, this looks like a DFT; i.e. if you have a linear array and do a DFT on the phase vectors of that, you get (due to annoyingly necessary coordinate transforms) a pattern in $$\sin \theta$$ space. That's a radiation pattern "just like an antenna has one", and we call it array factor, because we multiply it with the antenna factor, which is the actual individual antenna's pattern.

Rough calculation for this case: assume your yagis are both looking in the same direction, and are mounted half a wavelength apart right next to each other, the first at $$(0,0,0)$$, the second at $$(\pi/2,0,0)$$. Then, $$k\cdot\vec p_n$$ reduces to to $$k\vec p_0=(0,0,0), k\vec p_1=(\pi,0,0)$$. If you feed them in-phase, then $$a_0=a_1=1$$ w.l.o.g., and we get a result that says "twice the amplitude at $$\sin\theta=0\implies \theta \in\{0,\pi\}$$ and no amplitude orthogonal to that".

Now, you multiply the "twice to the front, 0 to the side" thing with your yagi gain pattern and see that you actually had an array gain of 3 dB to the front, which was your Yagi's main lobe to begin with!

The "reduced take-off angle" belongs in the category of Dreckeffekte (German: dirt(y) effects) and doesn't spring from the analytic consideration with perfect antennas mounted on an infinitely high mast, but are more from the "either you know by experience, or you've run a larger simulation" category.

• sorry I have a hard time following. So in your quick example, you ended-up with a best case scenario of +3db adding the second antenna. Is a 3rd one going to add another +3db? There is no diminishing returns? – ITChap Oct 6 '20 at 11:26
• A third one certainly won't add 3 dB, but doubling the number of antennas would (in the end, a 4-antenna array is just an array of two 2-antenna arrays). – Marcus Müller Oct 6 '20 at 11:51
• @ITChap no, doubling again (to 4 antennas) will increase the gain another 3 dB. – tomnexus Oct 6 '20 at 11:53
• By the way, it's not really bad if you're not fully following the math here. The idea is simply that an array of antennas has a pattern that is the product of the individual antennas' patterns, and the array factor, which looks a lot like an antenna pattern itself, and is calculated in similar ways. That answers "how to calculate the gain provided by each additional antenna in the stack?", even if it's not an "easy" answer. – Marcus Müller Oct 6 '20 at 11:53
• Thing is, some things simply aren't that easy, and if you need to be able to calculate the gain of an arbitrarily shaped array of antennas, you'll need to understand the underlying math. For a very specific question "linear array, all antennas same phase, same power, same direction, same type, what to do to get 3 dB more gain" is "double the number of antennas"; that derives quite directly from the more complicated math for the general case, but it's a useful rule on its own, too. Sadly, your question *made no such simplifying statements". – Marcus Müller Oct 6 '20 at 11:55