# Formulae for monopole antenna inductance/capacitance

I am looking to make a wire antenna for a transceiver module I have. My target frequency is ~915MHz, and so with a quarter wavelength model I should be using an antenna of ~81mm.

If I wanted to use a longer antenna (between 1/4 and 1/2 wavelength), I've found out that I need to use impedance matching to bring the resonance of the circuit back in line with my target frequency, however I cannot find any formulas to be able to calculate what capacitance/inductance the antenna has.

ChatGPT has provided a formula for antenna capacitance, although I'd rather base this off a better source.

$$C_{ant} = \frac{2l}{\eta c}$$

where $$l$$ is length, $$\eta$$ is impedance of free space (with an approximate value of 377$$\Omega$$ ) and $$c$$ is the speed of light

NOTE: I noticed that the 81mm antenna did not work very well, and through trial and error i found a length of ~105mm to work much better. I think this is because of imperfect connections (since this is still just a prototype) introducing some reactance to the circuit. If this is a poor assumption I'm open to an explanation for this as well.

EDIT: I am using the transceivers to talk between a remote and a module for indoors (~10-15m) and outdoors (~100m). The module is completely metal, and all my electronics have to be inside of it, which is why I need a longer antenna since the 1/4 or 1/2 wavelength would not be enough to reach outside of it. I don't have an easy way of using a different type of antenna, and I don't have a ground plane.

I did have a thought that maybe only the length of the antenna sticking out of the box matters, as the electromagnetic flux inside the closed conductive surface is 0 by gauss' law, however I'm not sure if it works like that

• I would be happy to help, but I need more info. What is this antenna being used on? HT, Mobile, indoor, outdoor, ground plane, or no ground plane to use. Commented Jun 11 at 2:36
• Hi, Electrical Engineer here, if you think you'll learn anything factually correct from ChatGPT, or let it do your calculations, you will at best waste your time and at worst get both wrong results and learn incorrect things, which you'll have to "unlearn". Simply. Don't. ChatGPT is designed and trained to produce plausible-sounding text, not correct text. It has zero understanding of anything, especially not the physics of waves. Commented Jun 11 at 15:10

As @Dereck points out, it would help to know about the device and your intended usage since it can determine the format of an antenna. @user98055, you say, "If I wanted to use a longer antenna (between 1/4 and 1/2 wavelength), I've found out that I need to use impedance matching to bring the resonance of the circuit back in line with my target frequency..." Just to be clear, actual antenna resonance only occurs at intervals of a 1/4 wavelength (minus the velocity factor caused by the antenna material and surroundings, like air, mentioned by @tomc; I've found shortening by 5% is usually extremely close to resonance) So designing an antenna in-between intervals of 1/4 and a 1/2 wave, is trickier, and requires you to figure out the excess capacitance or inductance you'd have to add to make the antenna "look like" it is resonant to the transmitter; as a general rule, ground plane antennas where the radiator is shorter than a 1/4 wavelength, or odd multiples thereof, have excess capacitance and need inductance to be brought into resonance, and radiators longer 1/4 wavelength, or even multiples thereof, have excess inductance and need capacitance to be brought into resonance. Also don't confuse resonance with impedance matching, as @tomc indicates, center fed 1/2 wave antennas, like a 1/4 wave ground-plane, is fed at a current loop (high current, low voltage), and end-fed 1/2 wave antennas are the opposite, high voltage, low current.

Here's a good impedance matching calculator: https://leleivre.com/rf_lcmatch.html

I'd like to know where ChapGPT got 377Ω, as the impedance of an antenna, unless it was a non-resonant antenna, since end-fed 1/2 wave antennas are ~2500+Ω, and center-fed 1/2 wave dipoles / 1/4 wave ground-plane antennas are generally 37Ω-73Ω, depending on the angle between the elements, and the distance to reflective objects around the antenna.

Just keep in mind that in the microwave region, tiny changes make a difference, and except for very local communications, antennas in that region are usually colinear, or co-phased, to increase pattern gain in desired directions, since even a double 1/2 wave colinear antenna may only be ~30cm long.

• I do have a good guess where ChatGPT has that number from: Free Space impedance is ca 377 Ω. The fact that it even mentions that is an excellent illustration of "ChatGPT is a random generator for words that usually appear next to each other, and 377Ω appears often in the neighborhood of all the words mentioned here, so it must be what the user asked for!". Commented Jun 11 at 15:13
• I searched that, and see that is exactly what comes up! Commented Jun 11 at 15:19
• I'd wish people understood that better: ChatGPT is essentially what happens if you refine "autocomplete" until it can build good-sounding, intelligible sentences. It's kind of the opposite of "artificial intelligence" – it's a random number generator, where the numbers are letters, and they're not uniformly likely, but the more likely you'd see them in an answer to a text whose letters are given to you. Commented Jun 11 at 15:42
1. get the ARRL Antenna Handbook

tldr.

If you are not going for quarter wave, the next point to try to hit is just shy of 5/8 wave. as 5/8 is not resonant you would need to add some inductance.

You most likely do not want a 1/2 wave because that puts a high current node at the input which causes high impedance.

note: monopoles switch between capacitive/inductive for odd/even multiples of 1/4 wave.

You can also just try odd multiples of 1/4 wave if there is room use something with low resistance like copper.

last edit: you mention 81mm fo quarter wave. this is too long for inside the antenna. Since the speed of light slower when going through something like a metal wire you have to account for what they call "velocity factor" or "end effect" at minimum call it 95% the speed of light "in free space" and multiply by .95 I get 77.868mm for your quarter wave.

there are tables for different materials, some coax goes down to .66

Your latest edit makes it sound like you may need to think about a transmission from your device to the outside of your metal enclodure before you worry about your antenna. Your metal enclosure may function as a ground plane.

In the application you describe, I strongly recommend a 1/4 wave monopole antenna. There's no benefit to a longer antenna, and there will be all sorts of pain trying to match it. It sounds like you don't have the necessary equipment, and anyway there's no reason to go there.

Nothing wrong with trial and error! If you have a way of measuring the performance, like testing the range of your device, then make a longer antenna and clip it shorter 5 mm at a time, until the performance peaks and gets worse again. What you are doing by experimenting is finding the antenna length that presents the best match to the transmitter, in your exact application. All real-world antennas are fine-tuned in-situ, after some theoretical design, and the monopole only has one parameter to adjust, so go for it.

Note that for a gadget in a metal case,

• the Antenna is the piece of wire sticking out of the metal case. This part should be a quarter wave long, about 80 mm. The metal case forms the groundplane, the length is measured from the groundplane to the tip of the antenna.
• you need a way of carrying the signal from the radio to the wall of the box, this must be done with coaxial cable. The coax outer conductor is connected to the PCB ground and the enclosure. The inner goes from the transmitter pin, and the monopole. You can't just use a random piece of wire from the transmitter to the antenna*.

'* of course you can but then the matching becomes extremely tricky, unpredictable and sensitive to everything in the box. In antenna design we try very hard to exclude problems like this, because we can't re-tune the antenna every time the user changes the batteries, etc.