Calculating a Wavelength
So you're building your own wifi antenna and the first thing you find out is that all the measurements are in wavelengths. What is a wavelength? How long is that? Why does it matter? Well, it does matter and this page has all the information you'll need on that particular subject. I'm not going to go into antenna design here, that's a huge subject all on it's own. If you're interested in that, and obviously you are since you're reading this, the place to start is the ARRL Antenna Book - it contains just about all the information you'll need on the subject. I was lucky enough to find a library copy; I got all I needed from it in the couple of weeks that I had it.
Relevant Links
What is a Wavelength?
A wavelength is the length, in some unit you can understand, of one wave. Is that clear enough? No, I thought not. Here's a dictionary definition: The distance between successive points of equal amplitude and phase on a wave (for example, crest to crest or trough to trough). Basically, radio travels on electromagnetic waves; and wifi is, of course, radio. These waves go up and down, like waves on water. The length of one wave is the length between the peaks, or between the shallows - either way, it's the same length. If you're building or designing your own wifi antenna then a wavelength is the length of one wave in the frequency spectrum used by wireless access devices (802.11a/b/g/i/n/x). So what is the frequency spectrum? That's the various frequencies used by your particular wireless devices. Here is a short table showing you what they are for 802.11a,b and g:
| WiFi Type | Min. Freq. (Ghz) | Max. Freq. (Ghz) |
|---|---|---|
| 802.11a | 5.15 | 5.35 |
| 802.11a | 5.725 | 5.825 |
| 802.11b | 2.4 | 2.4835 |
| 802.11g | 2.4 | 2.4835 |
These frequency waves are measured in Hertz which is the cycles per second, or waves per second that you send or receive. Can you groove to that? I knew you could. Hertz are abbreviated as Hz (Hertz=Cycles/Second), Khz (Kilohertz=Thousands of Cycles/Second), Mhz (Megahertz=Millions of Cycles/Second) or Ghz (Gigahertz=Billions of Cycles/Second). So where does length come into the picture? That's because of the well-known formula wavelength * frequency = speed of light, which allows you to calculate the length of one wave at a particular frequency, thereby giving you, yep, you guessed it, a wave length. Now let's break that formula down a little so you can really get your head around it.
Basic Formula
The speed of light is 299,792,458 meters per second, or 186,000 miles/second. That's smoking fast, wouldn't you agree? It breaks down to 982,080,000 feet per second, which is definitely breaking the speed limit in your town. So, knowing the speed of light and knowing the frequency, you can derive the formula for a wavelength like this: wavelength=speed of light / frequency. Of course, now we have to get down to units. Using meters, we have 299,792,458 meters/second divided by the frequency in Hertz to get a wavelength in meters. Now that you know what units your wavelength is in (meters) you can convert to inches, feet, millimeters, or whatever units your little heart desires.
Now, what comes along to complicate things? Channels. Yes, there are specified channels that are used for wireless access, or wifi. What are these channels? Each channel is an individual frequency, or rather the center point in a small range of frequencies. If you've ever set up an access point, you probably saw a setting for channels. So we'll need to know what frequencies are assigned to which channels to calculate our wavelength. Fortunately for you, that information is publicly available and presented right here, in the next table.
| Channel | Frequency (802.11b+g) | Frequency (802.11a) |
|---|---|---|
| 1 | 2.412 Ghz | 5.18 Ghz |
| 2 | 2.417 Ghz | 5.2 Ghz |
| 3 | 2.422 Ghz | 5.22 Ghz |
| 4 | 2.427 Ghz | 5.24 Ghz |
| 5 | 2.432 Ghz | 5.26 Ghz |
| 6 | 2.437 Ghz | 5.28 Ghz |
| 7 | 2.442 Ghz | 5.3 Ghz |
| 8 | 2.447 Ghz | 5.32 Ghz |
| 9 | 2.452 Ghz | NA |
| 10 | 2.457 Ghz | NA |
| 11 | 2.462 Ghz | NA |
| 12 | 2.467 Ghz | NA |
| 13 | 2.472 Ghz | NA |
| 14 | 2.484 Ghz | NA |
You no doubt noticed that 802.11a has only 8 channels. It is also modulated differently and has greater available bandwidth, but all that is outside the scope of this article and doesn't really matter when trying to calculate a wavelength anyway. What you care about is that the FCC allows the use of channels one to eleven (802.11b+g 1-11) in the U.S. while most of Europe uses channels 1-13 and Japan is stuck with one channel: 14. For our purposes, though, the channel number is only good for one thing: the channel tells us the frequency we're going to use to calculate a wave length. So let's get busy and calculate! We're going to do one simple calculation, the wavelength for channel one on the 802.11b+g frequency spectrum.
Frequency (Channel 1) = 2.412 Ghz = 2412 Mhz = 2,412,000,000 Hertz
Speed of Light = 299,792,458 meters per second
Therefore.......
Wavelength=Speed of Light divided by Frequency
And so.......
Wavelength (meters) = 299,792,458 / 2,412,000,000 = .124292063847 Meters
Which converts to.......
12.43781cm = 124.37811mm = 4.89678 inches
So one wavelength for Channel 1 on wifi 802.11(b or g) is 4.89688 inches!
Now you know how to calculate a wavelength for any specified frequency. That will be a major help in any antenna design, let me tell you. Knowing all that, you'll be able to figure out the dimensions of your antenna in wavelengths; so you can optimize for different frequencies. Doesn't that make you feel all warm and fuzzy inside? But wait! There's more! Just to save you all of that tedious calculation, and because I got an absolutely stellar calculator for Christmas, I'm going to give you a table of all the wavelengths for all the 802.11b+g channels.
| Channel | Frequency (802.11b+g) | Wavelength (Meters) | Wavelength (Inches) |
|---|---|---|---|
| 1 | 2.412 Ghz | 0.12438 | 4.89678 |
| 2 | 2.417 Ghz | 0.12412 | 4.88665 |
| 3 | 2.422 Ghz | 0.12386 | 4.87656 |
| 4 | 2.427 Ghz | 0.12361 | 4.86651 |
| 5 | 2.432 Ghz | 0.12336 | 4.85651 |
| 6 | 2.437 Ghz | 0.12310 | 4.84654 |
| 7 | 2.442 Ghz | 0.12285 | 4.83662 |
| 8 | 2.447 Ghz | 0.12260 | 4.82674 |
| 9 | 2.452 Ghz | 0.12235 | 4.81689 |
| 10 | 2.457 Ghz | 0.12210 | 4.80709 |
| 11 | 2.462 Ghz | 0.12185 | 4.79733 |
| 12 | 2.467 Ghz | 0.12161 | 4.78761 |
| 13 | 2.472 Ghz | 0.12136 | 4.77792 |
| 14 | 2.484 Ghz | 0.12077 | 4.75484 |
The first thing you probably noticed about the above table is that the differences in wave length are just not that large! So if you're building an antenna for wireless access, which number(s) do you use? Since the differences aren't that large, and you will probably want the ability to switch channels, what you would do is take an average value and use that for the standard value of a wavelength in your design. The mean value of a wavelength from the above table, then, would be 0.123 meters or 4.83 inches. The median value would be the same when rounded off for three decimal points. You could also use the maximum value (4.897 inches) or the minimum (4.755 inches), whatever floats your boat.
Now you have the first formula you need to begin designing your antenna. As you get into it, you'll find a lot of other formulas, some based on this one and others a bit more involved.