A customer asked if we could use passive UHF RFID to monitor if unauthorized employees cross a line in their warehouse to avoid steep fines. After hearing about the problem from my engineers, I had to write this article because it would let me do some real, honest to goodness, mathematics.

In my former life as a PhD student at UC San Diego, most every day I got to work with math in one way or another. But since I mostly run Telaeris these days, my opportunities to use higher math have become few and far between. But boy – do I ever love math! And because we solved the problem here, you get the solution for free, just by reading.

Looking at our customer’s problem initially, we decided that because of the high ceilings, we would likely have the reader antennas floor mounted.

The question that follows is this:

**How far away from the line do we install the RFID antenna into the floor?**

We chose wide RFID antennas, to minimize the number of antennas that would be used. Each antenna had beam width of 45 degrees. With employee badges worn around the neck, badges would hang about 4 feet above the ground. This is where the math comes in. We need to set up equations to calculate the distance X from the line that the reader has to be installed. The diagram is shown below.

Digging back thirty years to my trigonometry class at La Salle High School in Pasadena with Mr. Uejima, I recalled a couple of facts. Given one side and one angle of a triangle, it is possible to solve for any other side or angle.

But first, we need to get the angle α. Because α + θ is a right angle (90°) and we know the full beam width is 45° we solve for α with the following equations.

Then from the dark recesses of my mind an acronym came forth calling out “TOA….TOA…TOA” – tangent equals opposite over adjacent! With this, I was able to set up the equations to solve directly for the distance X.

Of course, when we use to do this at school, we had trig tables in the back of our math books. Today, I just asked my cell phone “what is the tangent of 67.5 degrees” and was rewarded with the value for my calculations.

The answer for the distance from the line is calculated to be** 1.66 feet or 20 inches** away from the line. This makes the **do not cross** zone rather tight and well contained.

I love the fact that with just a little bit of math and common sense, we are able to quickly characterize how a system should theoretically behave. Of course, this doesn’t account for the way passive RFID can reflect and bounce, but some issues can only be solved with a test in the field.

The next time we get into math, I hope to be able to discuss multi-variable optimization of real time location systems….but somehow I think I will have a much smaller audience for that article!

Dave,

I really enjoy your news letter & more importantly for the most part I understand what you are saying. So if you are trying to educate the under educated under achieving you are succeeding. Hope this finds you and your tribe are all well.

Steve