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To a first-order approximation, the length is 1/4 wavelength per leg. However, that assumes an ideal conductor in free space. We rarely have the luxury to suspend an antenna a few wavelengths from the ground, water, buildings, and anything else remotely conductive. The way the antenna interacts with those things in the near-field affects its impedance and makes it behave electrically longer.

The common formula you'll see is: Total Length (in feet) = 468 / f (in MHz). If we suppose exactly half wavelength and do the unit conversions, we would expect L (ft) = 492 / f (MHz). So why do the calculators use the shorter length? It's an empirical compromise. Taking 5-10% off total length is generally what's needed to account for things in the near-field. The 468 number has been repeated enough that it's stuck. In practice, I almost always cut dipoles for a full half-wavelength, hook them up to an antenna analyzer and then trim them down. With as cheap as hookup wire is, I'd rather not take the risk of being too short and having to field solder a splice (not fun on ARRL Field Day).

There's not really a general formula for finding length. The physics is nothing more than Maxwell's equations, but many of the deviations from an ideal dipole come from interactions with the environment. It's difficult to measure and/or predict how the environment will behave, so you're often better off building the antenna and then adjusting it in place. And so we end up with empirical rules of thumb like L = 468 / f.

Antennas are one of the harder topics for amateurs, for sure. The theory is well-understood, sitting somewhere at the intersection of EE and physics. I'm lucky enough to have a strong background in both fields, but there's a clear lack of curriculum for amateurs without that background. This website at least seems useful for building intuition, so hopefully it helps you some.

You're trying for a huge breadth of material, stuff that is covered in multiple specialities in electrical engineering. Multivariable calculus is really the entry point for engineering level antenna design.

Balanis is the book I used in my antennas class, https://www.amazon.com/Antenna-Theory-Analysis-Constantine-B... but you'll need multivariable calculus to get through it.

Balanis' book is a great reference but I wouldn't recommend it for learning. Honestly the ARRL antenna book and antenna handbook are the best practical books on putting an antenna together without getting bogged down in the details. The real hard part is getting some radio equipment so that you can experiment and learn since Spectrum and Network analyzers are waaaay out of most people's hobby budgets.

True. I already own a Baofeng, and an SDR, but for the rest I'm honestly thinking of using my Arduino in conjunction with some diy circuitry (if I can wrap my head around enough of it) to build an oscilloscope, and I found an article that used a noise generator in conjunction with an SDR as an SWR meter [0]

It's definitely not high end, but my wife would kill me if I she found out how much I'd have had spent a halfway decent SWR meter, if I went that route.

[0]: https://www.rtl-sdr.com/rtl-sdr-tutorial-measuring-filter-ch...

I bought a $20 oscilloscope for fun and learning, called a DSO 138. Probably not suitable for antenna stuff though.

You could also try a local hack space, if you have one. My local one, Vancouver Hack Space, has some very fancy oscilloscopes.

Not a bad idea but a few things to keep in mind. - The sample rate on an Arduino is pretty low, depending on which model you have it may only be a few MHz and the absolute best you can theoretically possibly achieve with your home o-scope bandwidth is half the Arduino clock frequency. You'll also have to cast off the Arduino language and start moving to C/assembly to get the full performance out of it. -The dynamic range on the Arduino ADCs is pretty low. It's going to be limited by the number of bits in the Arduino ADC. -The input on an Arduino ADC is going to be either very low or very high impedance. This is also common on lower frequency scopes (low frequency in the case being 1GHz or less bandwidth) but if you try driving your input directly the impedance mismatch from the RF is going to cause poor signal and lots of noise on your measurement.

I don't know what SDR you have but it might be suitable for the signal capture and you can probably gin up some filters in post processing to approximate a Spectrum analyzer. If you have a source and 2 directional bridge couplers you could use it to make a poor mans Scalar Network Analyzer. A noise generator or a swept tone (Chirp) can be used in both these cases. The article you linked basically did a 1-port Network analyzer using the bridge coupler and used to to measure VSWR.

A lot of modern oscilloscopes, even hobby ones, now include the ability to capture scope traces to a PC and an FFT math function. You're going to be limited in frequency without a downconverter but it's a good way to go on a budget when working on a bench. The Rigol's are a lot better than they were when they first came out and make a good hobby scope for a reasonable price. If you can be spendy, the Keysight hobby level scopes are a joy to use. Best bet on a budget is to troll the internet for an old Tektronix, Lecroy, or HP/Agilent/Keysight but you'll probably not get something with fast trace capture or a built in FFT.

All of that said, making test equipment out of your Arduino is a great hobby project that will teach you tons of useful engineering. I highly recommend it!

I had Stuzman and Theile in antenna class, but Balanis for grad level EM.

The main reason antennas are hard to understand is that their dimensions are comparable to the wavelength they are tuned for. Unless you are dealing with microwaves, the systems used to generate and detect RF are modeled using "lumped circuit elements". For example, resistors, capacitors, inductors, and transistors. And, at first, big sparks. These elements are small compared to the wavelength.

The design of those elements and the circuits using them was historically pretty independent from formal electromagnetic theory, as developed by Maxwell. The intersection was Oliver Heaviside, at the end of the 19th century.

Before Heaviside, RF electronic design was a largely a matter of groping though the practicalities of employing those circuit elements to create oscillations and couple them to resonant wires supported as high as possible.

As the peer posts explained, antennas are resonant structures in which electrons are caused to slosh back and forth. As they slosh, they accelerate, and as they accelerate, they radiate electromagnetic energy. But, it is best to stick to the rules of thumb at the level of the ARRL handbooks. To understand the EM theory related to radiation and antennas, you really need to work through to the final chapters of Griffiths, "Introduction to Electrodynamics".

But that is not necessary to get an intuitive understanding of antennas, to construct them, or to run the modeling software.

Resonant antennas are only one class of antennas, with the other being traveling wave antennas. A log periodic looks similar to a Yagi, but it’s not. Same with a biconical and dipole.

It’s an easy day when I have dimensions on the order of a wavelength. Usually it’s 1/10 or less, and shoved up against metal.

How do I develop my intuition regarding the low resistive impedance of such antennas? I understand the high capacitive reactance, but haven't got a grip on why the restive part is so low.

What is your general approach to matching these very short radiators?

When you bring an antenna close to a conductor, say a dipole next to a plate, the radiation resistance decreases. This is due to the currents induced in the plate, creating an image and reinforcing the currents in the dipole.

It’s really the ratio of radiation resistance to conductor resistance. You can shrink an antenna to infinitesimal size, made of perfect conductor, but as the radiation resistance decreases, it’s more difficult to impedance match. An infinitesimal antenna would have zero bandwidth. Sort of like Bode Fano criteria limiting bandwidth versus impedance.

There is a Chu theoretical limit which limits antenna efficiency and bandwidth given volume, hence 3D fractals and other stuff. Ain’t no free lunch. A lot of antenna research is who can get closest to the Chu limit. Sort of like coding and the Shannon capacity.

https://en.m.wikipedia.org/wiki/Chu–Harrington_limit

There is still a lot of snake-oil in antennas, some of it fractal related.

Thanks, I had not heard of that. Also, that link lead me to the WP article Electrically Small Antenna. It's remarkable how much they knew 70 years ago.

> I still struggle to understand basic concepts like: + Baluns + Why does the height of an antenna effect its effectiveness? + Gain + Circuit design + Transistors

That's a lot of topics. People spend a lot of time understanding each one. Keep working at it, you will traction eventually.

I'll try to make some helpful comments.

> Baluns

Well, a lot of the confusion comes from the fact that two fundamentally different widgets are called "balun". A form of transformer, and a form of choke. Bottom line: to feed a balanced antenna with an unbalanced feed line (coax) you want to keep common-mode currents off of the shield. The electric field should be entirely contained between the inside of the shield and the center conductor. The choke style balun (stack of ferrite beads) creates a high impedance on the outside of the shield, so the current flows inside. The transformer style accomplishes the same goal by different means. Usually a transmission line wound on a toroid.

> Gain

A finite amount of power is going into the antenna. Nothing you do in the antenna can increase the power, but you can direct it. It almost always works out to optimizing the phase difference among different radiating elements such that you get constructive interference in the desired direction, and destructive interference in undesired directions. To visualize, gin up some code that plots two sine waves of the same frequency and their sum. Alter the phase of the two source sine waves and observe the result. A 3 element Yagi-Uda works on this principal: The "reflector" is a bit longer than 1/2 wave, so the energy that it absorbs in the near field is reradiated with a phase lead w.r.t. the driven element. The "director" is a bit short, and reradiates with a phase lag. The radiated energy from the elements sums constructively going forward.

> height above ground

Some energy from the near field impinges on the ground, and is reflected with a phase reversal. It will sum constructively or destructively at various angles above the horizon as a simple trigonometric function of height. (N6BV's HFTA - HF Terrain Analysis program -- uses GTD - General Theory of Diffraction -- to model antennas above ground terrain. Good fun for optimizing antenna height.)

> Circuit design

Start with circuit analysis. Once you understand the building blocks, you will be able to synthesize something with them. This should feel a lot like programming eventually. Start by learning how to read simple schematics. The best advice in this regard that I ever got was from my first semester circuit analysis prof: "Keep redrawing the circuit until it makes sense." Which means: redraw simpler circuits representing each regime of operation. Start with DC: caps are open circuits, inductors are shorts. Draw that. Now you understand the DC bias, or can figure out the bias voltages pretty easily with Ohm's law. Then redraw at the operating frequency with freq-dependent impedances.

> Transistors

A bipolar transistor is a current-controlled current source, and FET is a voltage-controlled current source. So to analyze a transistor, you have a bunch of things to draw: the input side at DC to find the operating point (bias) the input side at AC to understand the controlling signal, the output side at DC to understand the operating point and output impedance, and the output at AC. Also identify the interstage coupling elements. Not sure if my comments on transistors are helpful yet until you are more confident at circuit analysis. Keep at it! It isn't so different from programming, just a different grammar.

This is very helpful! And thank you for the encouragement.

Why would you ever feed a balanced antenna with coax? I use open wire transmission line and have nearly zero loss in the transmission line.

A transmatch is a wonderful piece of gear.

> I'm hoping a little more exposure to electrical systems will help.

I'll give you a nugget that I think is correct.

The radiation efficiency is determined by the cross product of the electrostatic and magnetic fields the electrons are exposed to. No cross product no radiation. AKA why most circuits don't radiate very well.

A dipole is a resonator that generates a large electrical and magnetic cross product. Because it resonates the energy stored/flowing through it is many times the energy being pumped into it.

Lots of things effect the resonance peak of a physical resonator. That's the difference between theoretical antenna's and physical one.