Cherish (mareserinitatis) wrote,

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Friis Transmission Equation

The title gives it away, but I've been itching to write about the Friis Transmission equation, the favorite equation of antenna engineers everywhere*.

But let's start with a joke, told to me by my friend Mike (who actually works on power supplies): Did you hear about the antennas that got married? The wedding was okay, but the reception was great!

We'll revisit this later.

When someone is trying to acquire a signal be it listening to the radio (that thing we had before iPods), getting a signal from a satellite so they can watch the Vikings lose football game, or read an RFID tag on a pallet, the people designing those systems have to know how much power they can expect in order to read the signal. Admittedly, systems to read these signals are quite sophisticated and there is a lot of fancy stuff that can be done using advanced signal processing techniques. However, from an electromagnetics point of view, the Friis Transmission equation is of utmost importance.

So what is it? The Friis Transmission Equation tells us how much power an antenna can receive from another antenna that is transmitting. We'll call the receiving antenna Pr, and the transmitting antenna P t.

If you look at the ratio Pr/Pt, you'd really like this value to be 1. That is, you'd like for the receiving antenna to capture everything that the transmitting antenna is sending out. If the receiving antenna is not receiving everything the transmitting antenna is sending out (which is how things really work) then this value will be something less than 1 but, hopefully, greater than 0. The less it receives, the closer to zero we get. The closer to zero, the more we'll have to rely on those fancy signal processing algorithms to get the information we want.

Let's look at the equation and try to figure out what things will decrease the reception.

The left-hand side of the equation is easy: this is the same thing we've already looked at, which is the ratio of our received power to our transmitted power, or how efficient we are at capturing the transmitted power. As a short-hand way of stating this, I'll just call this our reception efficiency.

The right hand side will give us a value for that power, but we should examine each of the terms. Let's start on the far right. The symbol lambda (λ), which looks almost like the Chinese character for people or some dude doing straight-armed jumping jacks, stands for the wavelength of our signal. Wavelength is defined the traditional way; that is, it is the speed of light divided by the frequency, or, in symbols, λ = c/ν. Because wavelength is in the numerator, we can see that increasing the wavelength of our signal will do a lot to increase our efficiency. The fact that it's squared means it will have a bigger impact than increasing those G terms (which I'll explain later). This is one reason why it's easier to pick up AM signals (which have a longer wavelength and smaller frequency) than FM signals at a large distance from the transmitting tower.

There is one problem with this, however: increasing the wavelength of your signal often means you have to increase the physical size of your transmitting antenna to radiate more efficiently. You often cannot do that because of restrictions on your space. You may also be forced to keep the same wavelength (or corresponding frequency) because you are required by the Federal Communications Commission (FCC) to stay in a certain transmission band. So, very often, increasing your wavelength is not something that can be changed.

If we look in the denominator, we see a factor 4πR, which is also squared. R, in this case, is the distance between the receiving and transmitting antennas. (I won't go into details on the 4π factor, as I'd be here all day. If you want, you can read this Wiki article for the particulars.) We can see that if we increase the distance between the antennas, the value for our efficiency will decrease pretty quickly because of that squared factor. This is the reason why, as you drive out of town, you loose the signal from your local radio station. Increasing the distance means you're going to capture less power. Putting the antennas closer together (or driving back into town) means you get better reception.

Again, distance may not always be something you can change. When I was working on my masters, we were developing RFID tags to go on cattle. There are already tags that many ranchers use, but most of these are the kind that require you to stand next to a cow a wave a wand by it's body. We were trying to develop a system that could track cattle as it moved through gates. Because it had to read several cows in a short span of time, we had to mount the antennas above the cows. There was also the additional problem that cows will chew on anything: if we put them on the fence, they would come up and start chewing the cables. So our system had to be able to read them at a distance of 4-6 feet away.

So what about those G terms? What are those?

The G stands for gain, and going with the previously mentioned notation, Gt means gain of transmitting antenna while Gr means the gain of the receiving antenna. But what is gain?

The gain is a combination of two things, radiation efficiency and directivity. The radiation efficiency is similar to our reception efficiency above. It is basically the ratio of how much power an antenna radiates versus how much power you put into the antenna. As I mentioned above, increasing the wavelength of an antenna often means you need to increase the physical size. This is because of the radiation efficiency. As a general rule, the longer an antenna is, the better it will take its input power and convert it to a radiating wave. A long antenna (relative to the wavelength) will work better than a short antenna.

The directivity is a little harder to explain, but I'll give it a go. (If you want a more technical discussion, check here.) Basically, the directivity looks at the "shape" of the electric field created by an antenna, which is determined only by the shape of the antenna itself. If the shape of an antenna is a perfect sphere, we call that isotropic radiation, and say the directivity of the antenna is 1. That doesn't happen in reality, however, because nothing can generate a time-varying field evenly over space. (An electron, however, can generate an isotropic static that doesn't change in time.) The closest we can get, usually, is a field that is shaped like a doughnut (or toroid), and that is created by a straight antenna, or dipole. If we assume this antenna is really tiny (what we call a Hertzian dipole), the directivity of this antenna compared to our perfect isotropic radiator (that doesn't exist in reality) is about 1.5.

If you try different shapes, they sometimes can become more directive. One type of antenna that is highly directive is called a Yagi-Uda and is commonly used to pick up television signals. The electric field coming from an antenna like this will often have a shape that looks something like a balloon so that most of the field is radiated in one direction. These can have very high directivities.

The gain factor is where being an antenna designer comes into play. Can you increase the efficiency and/or directivity of your receiving antenna? It can take some ingenuity to come up with an antenna with a shape that will increase the directivity and increase the length to up the efficiency given a set of physical constraints. But if you can increase one or both of these factors on your receiving antenna, you may be able to increase your reception efficiency.

Doing so is not an easy task, which is why so much has been invested in signal processing algorithms. It's very hard to make a bigger antenna on your cell phone, so companies instead opt to add extra hardware (which in comparison, is pretty small) or outfit their towers with several antennas and more sophisticated equipment.

Also, the Friis Transmission Equation is often a simplified case. There are other factors that can be added to the right-hand side of the equation such as polarization mismatch, impedance mismatch, multipath (reflections from buildings, cars, etc), and many others. The case where you're likely to get the closest scenario to the equation is when looking at satellite communications as there are few to no obstructions between a dish on Earth and one in space.

Going back to the joke, we have to conclude that the antennas must be very well matched in many ways. I would hate to think of what would happen if they attempted a long-distance relationship, however.

*Or maybe just a few, like the author of this post.
Tags: antenna, electromagnetics, friis

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