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On the November 12th 2017 OMARC "New Ham" Net, a topic came up about how long to make a wire antenna for a certain band.

I had prepared some math for the net in advance using the 2m band as an example, but decided to recalculate for the 40m band for a more practical application based on discussion.

If we use the 40m band as an example, there are "close enough" and "somewhat magic" lengths such as 33.7 feet for a quarter-wave wire antenna or 34.3 feet per leg of a 1/2 dipole that will have good resonance or low SWR in the voice portion of the 40m band.

The thing is, these are just starting values and there is no perfect wire length. A lot has to do with the height of the antenna, how it is fed and what type of conductor is used for the antenna elements. Also, the length of a 40m dipole in an attic compared to outside will be different. Every installation of the same wire dipole will be slightly different.

I was going to share some math on how we arrive at values like 33.7 feet on the net, but figured here is a good spot for that along with easy ways to figure out where to start when making a wire antenna.

**What the heck is 40m?**

The amateur 40m band is 7.000 to 7.300 MHz in most parts of the world. 40m is the metric length of how long one wavelength of a 7 MHz signal is. 40m is about 120 feet and that is not very small, but once upon a time it was. This so called "shortwave" band was much shorter compared to the longer wave 120 KHz, 500 KHz, etc frequencies in use in the early days of wireless circa 1910 when things were getting more mature. Today we know these as LW or "Long Wave". Amateurs for the most part are on "short wave", aside from some recent LW allocations and our VHF, UHF and SHF allocations.

**Why does the length matter?**

For all the below "math" I will use the frequency of 7.100 MHz or 7100 KHz, but you can easily substitute any other frequency too, such as 146.000 MHz

A 40m or 7.100 MHz signal gets that from taking the speed of light and dividing it by the frequency to figure out its wavelength. You can also do the opposite and take a wavelength and convert it to frequency. Here is how and why this matters why finding that perfect antenna wire length.

- speed of light (in a vacuum) 299,792,458 meters a second
- speed of light (in Earth average atmosphere) 299,702,547 meters a second
- speed of light (in water) 225,407,863 meters a second

We will use the in Earth atmosphere number here since its "slightly" more accurate for our purposes because air slows RF signals slightly and we do not live in a vacuum, or at least some of us.

299,702,547 divide by 7.1 equals 42211626. What the heck does that mean? You carry the digits over and technically a 7.1 MHz is 42.21 meters long.

This is too complicated and an easier way to do that is to use 300 divided by frequency. The "300" is sort of close to the speed of light (299), just without the rest of the digits, so now 300 divided by 7.1 gives us 42.25 meters.

If you did 299.7 (more accurate) divide by 7.1 MHz, you get 42.41m.

Most text will tell you 300/F. Now you know why. I would use 299.7. The additional .3 actually is a bit of a "fudge factor" when taking into account other factors, reviewed later.

**How long to make my antenna though?**

Now that we know how long a full wavelength is for 7.100 MHz is at 42.21m, a 1/2 wave antenna would need to be 21.105 meters long. A 1/4 wave antenna would be 10.55 meters long.

**Meters to Feet**

Since the United States is different than the "metric" world, here is the conversions for the "theoretical" math above for antenna lengths.

- 42.21m = 138 1/2 feet = full wave
- 21.105m = 69 1/4 feet = half wave
- 10.55m = 34 1/2 feet = quarter wave

**STOP! That is not correct.**

Technically, this is correct if you had magic wire that was able to conduct your signal at the speed of light. Guess what? There is nothing that exists to do that, so this is where the term "velocity" comes in.

If you have a conductor that allows electrons (your signal) to flow but has some resistance to it ( the wire alloy), that lowers the velocity of the wire and is part the reason why a thicker conductor usually is better, but not always practical for long wire lengths.

**The "Fudge Factor"**

Lets use 16 gauge wire comprised of 19 strands of copper-clad steel with a polyethylene jacket as an example. A manufacturer says this has a velocity factor of 92%.

We can use the 92% to give us a so called "speed of light" value of 275,809,061 meters/second. This comes from the speed of light in a vacuum at 92%.

For this, we are using against average earth atmosphere, but that does not matter because we are calculating in the speed just in your wire.

Now we can use 275809061 divided by 7.1 to equal 38846346. That is too complicated, so lets try 276 divided by 7.1 and we get 38.87 meters.

**Stay with me....**

Now that our 7.1 MHz frequency in the 16 gauge wire thinks it is 38.87 meters , we can recalculate our antenna lengths....

- 38.87m = 127 feet and 5 1/4 inches = full wave
- 19.43m = 63 3/4 feet = half wave
- 9.71m = 31 feet and nine inches = quarter wave

**This is still all theoretical though**

So lets say you want to make a 1/2 dipole, you make it 31 feet 9 inches on either end. You hoist it up at least 1/2 wave length in the air which would be 63 feet and you use the most magical 50 ohm coaxial cable as a feed line that is 80 feet long. How will your antenna work?

We wont know because there are too many variables like the velocity factor of your feed line, obstructions, the reflectivity of the ground below the antenna, the wire vendor 92% velocity factor wire not really being 92%, and so much more.

Chances are this antenna is likely going to resonate at a higher frequency than what you intend, so this is why you add some additional length to it and then trim it back because its easier to shorten an antenna that add to it.

You can only decrease antenna resonance by trimming the antenna shorter. If the antenna resonates at 7.4 MHz, that is not good because you will have to add wire to it. If the antenna resonates at 6.9 MHz, that is great! Easier to trim it to perfect resonance at 7.1 or 7.2 MHz.

**What about 248/F or 468/F or 492/F?**

248, 234, etc all fine and good. That number is derived from the function of the speed of light, velocity factors, etc. The idea with that simple equation you may see in the ARRL literature is make it easy to find a starting point for how long to make a 1/4 antenna.

**Long story short**

Start with a value 10% bigger than what I share above, even 20% if you want. You can slowly trim the antenna to be resonant at the frequency you want or just leave it "as is" and use an antenna tuner to "tweak" things.

Some say 33 feet, others say 34 1/2 feet, Really does not matter.

Just start somewhere and adjust. There is no magic length. Some will come close, but there is not 100% accurate length.

I put this together to figure out where the equations come from and if you followed one and your antenna did not work the way you thought, why. Its all in the math (and materials)

**Other Notes**

If wire has a lower velocity factor (80%), often times a shorter antenna is needed to be resonant. So, if you use a 12 gauge wire the antenna may actually be longer than if you used one with 18 gauge wire. This seems counter intuitive because then why not just use crappy 80% velocity wire all the time?

Reason is, low velocity wire wastes your energy as heat and then your signal wont get out as well unless its on the exact resonant frequency its designed for. Always go with the thickest wire you can afford and support.

This is why thicker wire is always better. Increased bandwidth is a function of less heat loss due to higher velocity factor.

**Best antenna tips for a 40m dipole**

Use nothing smaller than 16 guage wire

The more strands the better. Try to avoid solid core wire.

The outer covering will matter. That fancy "flex weave" antenna wire with UV coating is worth it. Do not buy cheap Home Depot wire unless you have no choice.

Use good quality 50 ohm coaxial cable. Thicker 9913, LMR-400, RG-8 is best, but RG-8X is ok too. You may just need to trim your antenna differently due to feedline velocity factors.

Use a 1:1 balun is optional. It helps, but is optional if the antenna is designed correctly. Any balun will lose some power since they are not 100% efficient.

Higher is better. There is less ground loss when an antenna is in the air.

**Share more tips and tricks below in the comments.**

Hope you enjoyed the "math" above.

This only scratches the surface, especially if designing for higher frequency antennas where dimensions and materials are even more critical.

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