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I have a router with a wifi antenna that can be turned in any angle. I wonder what difference does the direction of the antenna make to the electromagnetic signals propagation? Where is the signal strength the biggest?

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I had the same question once, and scoured the Internet for advice. All I got was conflicting information, much of it from "experts". I ended up getting a signal strength app for my smart phone and one for my laptop and experimenting. In my house, with my router, I found no detectable difference between horizontal and vertical.

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There is a nice video on youtube by Royal Canadian Air Force. That explains how directivity works with animations.

Antenna Fundamentals 2 Directivity

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Well a router antenna is simply a dipole. It will have maximum radiation in its broad side direction and it's radiation pattern looks like a donut.

Check the link below for the illustration of the dipoles far-field https://www.cst.com/Academia/Examples/Wire-Dipole-Antenna

Of course the router itself and any metallic objects nearby will influence the radiation pattern and also in small rooms you will have multiple reflections from walls, furniture, etc. so it is difficult to say what is the maximum.

But as long as you keep the routers antenna pointing upward you should get an optimal signal.

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