RF Basic Principles

Let's start at the beginning of a few key concepts and see where it all leads us, but I'll tell you now that this is gonna be a long, confusing post.

In 1880, the world of RF was officially discovered by a man called 'Heinrich Hertz' who proved Maxwell's theories on electromagnetic radiation were true. He used a spark gap attached to an induction coil and a separate spark gap on a receiving antenna. When waves created by the sparks of the coil transmitter were picked up by the receiving antenna, sparks would jump its gap as well. 

This was huge, probably the biggest thing to happen since Michael Faraday or James Maxwell at the time. This was confirmation that there was a whole other field of electricity out there based around electromagnetic radiation. And the premise was really stupid too, it was basically: "If we wiggle electrons here, then they also wiggle over there."

So yeah, welcome to RF, the only time a conductor says "Screw your physics" and can stop being ~0Ω. 

Antenna Principles 

What are RF waves?

RF (Radio-Frequency) signals are high frequency AC signals, and they are used for radio/wireless communication. Officially, RF frequencies start once you get into the ~3kHz range, and you can start looking into using an antenna to transmit that signal wirelessly. Slight hiccup though - an antenna for 3kHz will have to be around 25km so it can capture enough of the RF wavelength and transmit it wirelessly. That is a stupidly long antenna and not at all practical in any way. So, why is it that long?

Antenna Size vs RF Wavelength

Well, let's start with the simple formula (c = fλ), it's very obvious that if you decrease the frequency of a light wave, its wavelength will increase. So for 3kHz, that's a wavelength of 100km. Ok, so I can do basic maths, what does that 25km figure have to do with anything?

So, on a very basic level, antennas operate on the principle of resonance and standing waves - I'm gonna assume you're familiar with light waves and their periodic waveforms. 

If you find the quarter period of that wavelength, you'll end up with the electrical signal being a maximum at that point. So, if the antenna terminates at that length precisely, you find that the wave reflects back in a way that creates a standing wave. Boom, the standing wave is a resonant frequency/wavelength for the antenna, so it causes huge EMI radiation off the antenna - which can then be picked up by some other receiving antenna. So, an antenna is just a piece of metal that is insanely sensitive to specific frequencies of radiation. 

There are other resonant lengths like 0.5λ or λ that cause antennas to achieve a level of resonance in different ways, but the specifics really depend on  the antenna type in question, as they operate on different principles and will exhibit specific properties.

When you design an antenna, it's only really suitable for its main resonant frequency and the close range of frequencies surrounding it; this is called the bandwidth of that antenna.

Antenna Types

Have a look at these two photos, one of them is an antenna, and the other is not an antenna. Can you guess which is which?

If you guessed that the spiraly one on the left is an antenna, then you're right. If you guessed that the single wire on the right is an antenna, then you're also right; sorry I lied.

If a simple wire can be an antenna, then is there really a limit on how many antenna types there are? Well, probably, but I don't think we're going to hit it anytime soon. Have a look at this legendary website (antenna-theory), it's definitely not an exhaustive list of every single discovered antenna topology, but it covers the important details on the main antenna types. You will rarely find yourself using some other unique antenna in practical use cases, so I wouldn't worry too much about any of the other antenna types.

In reality, there's like 4 or 5 antennas that are used the most, and that's because they're good enough for almost all use cases, or they have a lot of commercial support and datasheets about them. So, pick a few popular topologies and learn about them and get very familiar with the theory behind them and you can then cycle between them for your projects depending on the specific context of use  - do you want something for long range? or do you want a wide capture of frequencies? etc.

Anecdotally speaking, here's my list of antennas that I can recommend learning about and using:

• Patch Antenna (Microstrips)

I've mostly had to get comfortable with this because it's the one that your degree teaches you about the most (at least in the early years). In reality, I've found that it's not the ideal antenna to use for many cases. It needs a lot of space for that big ground plane, and is unflexible, and rigid in shape. However, it's very easily and cheaply manufacturable (it's just a glorified PCB), and it can be mounted quite flush to objects due to its flat shape. Also, a major plus about patch antennas isn't actually about the antenna, but about the background reading you do in order to understand it. 

To learn about patch antennas, you have to learn about the microstrip concept, which becomes immensely useful for high speed PCB design and PCB antenna design. For example, there are transferrable skills with microstrips when it comes to making integrated F antennas on your PCBs, or for impedance matching purposes. 

• (Random) Wire Antenna

If I have no proper antenna lying around and I need a quick prototype to test a transmitter or reciever, then literally nothing has beaten the random wire antenna for me so far. You don't even need an RF connector connect it to the Tx/Rx device, and can resort to just soldering it straight to the antenna pin. It won't have the best performance and definitely is not suitable for long-range communication (although many HAM operators might disagree...) unless done properly, it's still more than good enough for short-range testing and everyone has a piece of wire lying around.

The general rule of thumb I've found is to stay away from multiples of 0.5λ wire lengths. Aiming for around 0.4λ for the frequency I'm using seems to work good enough.

• Monopole Whips

This falls under one of the antennas I'd never make on my own and I'd always buy off the shelf. Their shape is super compact so ideal for a lot of scenarios, and they're very mechanically robust for their cost. They come with rubber shields for mechanical safety, and have good performance for their designed frequency. A thing I've found useful is how their orientation is very easily adjustable on the fly since a lot of them come with swivelling a RF connector instead of a rigid one.

• QFH Antenna

This is the ultimate GNSS antenna and have fun trying to change my mind. I discovered this during my time in Sunride back when we were still researching and designing our custom radio/GPS system. This is much better as a 'custom' DIY antenna than a commercial one it has really good omnidirectionality and good VSWR (one of the early sunride prototypes had a 1.37 VSWR, which is insanely good). The pain of designing it can be eased by using 4nec2 (super simple antenna simulator) and some online QFH calculators. I will admit it is also as annoying to build and I recommend 3D printing a custom jig that allows you to wrap wires in designated areas to create the loops easily.

• Yagi Antennas

This is the RF equivalent of one of those super powerful focused flashlights. This is a very high directional antenna and is quite easy in concept to understand and practically use. Unfortunately,  a Yagi will be one of the biggest antennas you will ever own and will require some forearm strength to hold for extended periods of time. It's important to have active orienting and pointing of this antenna - in less fancy terms, you need to hold and point it towards the Tx/Rx device or else it won't work as intended.

Antenna Interfacing & Feeding

I won't talk too much about Antenna feeding or connectors in this post as I've covered them in other areas on this site. Check out the HAM radio section on the website as I go into detail on feeding antennas there. For RF connectors, I've made another post about them that you can find on this website.

Antenna Properties 

The three main properties to care about when evaluating antennas are Gain, Radiation Pattern, and S-parameters. After I explain them, I'd like to give an honorable mention to two more properties that are a bit more subtle and rely on intuition yet will still fuck your shit up if you aren't careful.

Antenna Gain & Radiation Pattern

Gain is quite an intuitive concept, but the way it's characterised and measured can seem dumb as hell at first glance, so bear with me while I start at the beginning with an explanation of an ideal antenna.

The physicist who first discovered the land of antennas first did so theoretically, so he said that an ideal antenna has an isotropic radiation pattern; meaning that the radiation spreads out equally in all directions from the antenna; essentially a perfect sphere of radiation. Another way to say this is that an ideal antenna is perfectly omnidirectional.

He then decided that from now on, every antenna he made in real-life would have a gain that is measured relative to this perfect sphere of radiation. And thus, an isotropic antenna was defined to have a gain of 1dBi (dB relative to isotropic) in all directions.

If that's confusing then don't worry because it first confused me as well, but all it's trying to say is something along the lines of:
"How many dB does this antenna's range extend past my isotropic antenna?"

For example, if you somehow defy the laws of physics and make a perfectly omnidirectional antenna, then that would have a gain of 1dBi; otherwise 0dB.

Eventually, you'll start to understand why he chose to characterise it this way, and that's because gain doesn't actually mean much in terms of real-life radiation power or signal strength. The thing antenna gain is describing is the direction in which the antenna will vomit out RF, and how far that vomit will splatter; so, antenna gain is kinda meaningless beyond those two descriptions. This is a good enough explanation for now I think, you'll find that it's a bit more complex than just this, but I think this is a good starting point for now.

The last thing I'm going to touch on is the close link between Gain & Radiation Pattern. The gain of an antenna will define its radiation pattern, take a look at this diagram:

There is always a tradeoff between gain and directivity of an antenna, and you also can't 'create' gain out of nowhere.

An antenna with low gain like the omnidirectional one doesn't mean that its a bad antenna; an antenna isn't defined by how far it can reach #justiceforlilantennas. It just means that it inherently can't reach very long distances, however it can very easily accept signals from all different directions which is a valuable property in some scenarios. 

If you increase your antenna gain, then you increase your directivity in only a certain direction; that means the radiation pattern changes from a omnidirectional one that can reach short distances in many directions, into one main lobe that can reach far. Your antenna gain value in dBi will then only be true inside that main lobe, but will be very low outside the lobe.

So, how come when you increase the gain, it suddenly changes the radiation pattern of my antenna? What about my omnidirectional RF laser?

Unfortunately, it's just the way it is, and as an engineer you accept it. If you want to understand it, then go join the physicists in not having showers and sex. When you increase your gain, you're actually just kinda refocusing the already existing radiation pattern of an isotropic antenna radiation into one beam. So you're not really generating gain, more like repurposing the already existing gain for your needs.

S-Parameters

Now this is a tough one to explain without explaining impedance matching, so for now I'm just going to explain what S-parameters would mean if you stumbled upon it on a datasheet. The only S-parametre you care about is S11 usually, sometimes if you have multiple feed antennas then you start to worry about S21/S12 isolation between the ports but that's a problem for future you to tackle.

You may have seen this interesting graph a few times when looking at an antenna characteristics:

All you need to know at this stage is that the more negative of an S11 you get, the more efficient your antenna is. The reason for the negative is to do with the maths behind the derivation, which there are a ton on youtube you can watch if you wanna feel dumb.

S11 is a measure of how much signal reflection there is from the antenna. This won't make sense until you reach the section below on Controlled Impedances & Return Loss, so patience, child, just accept the fact. 

That graph says that at a frequency of ~2.46GHz, the antenna shows an S11 of -17dB, so the antenna would have the best performance if operated at that frequency. In fact, the widely accepted standard is an S11 of <-10dB (90% of the RF signal makes it to the antenna), so technically you can also use it at 2.43GHz - 2.49GHz with acceptable performance. 

By far the best way to improve your S11 is through impedance matching, or changing a physical property of the antenna itself (bear in mind that can also change the operating frequency, or the bandwidth of the antenna, so careful).

Antenna Placement

Alright, onto the boogeyman of any RF system: the environment.

Sometimes you can do everything correct in your RF system; from your beautifully transmitting radio, to your smooth, efficient, transmission line, to a marvelous antenna. 

Then you go mount the antenna in its home, which happens to be a tiny, hollow carbon fibre tube, enclosed by steel plates, and held in place by threaded rods. (RIP Karman Alpha Telemetry). Then you wonder how the hell your signal is so bad, that at this point it would be more effective to strap yourself into the carbon fibre tube and yell out the information really loudly.

This is really hard to explain because this is one of those moments where I literally cannot think of a set of rules to simplify this concept that would still hold true in the majority of situations. You honestly just need to buy a VNA, and go wild with the testing and learn through personal observation.

The effect of conductors & insulators on an antenna

If you start moving a piece of metal close to the antenna from far away then there are various different scenarios that can happen and all of them could potentially be true. And there is absolutely no way to know unless you simulate (which is not accurate enough most of the time unfortunately), or measure the RF signals in the air at all points.

As the piece of metal gets closer, you might find that it suddenly pulls the radiation pattern of the antenna towards, thereby acting as a director for the antenna. As it gets even closer, you could suddenly see a bizarre effect where the metal blocks all the signals and instead reflects them backwards acting as a reflector, causing an increase of gain in a backwards direction. Or hell, once you get it close enough to the antenna that near-field effects come into play you could find that the antenna is so significantly de-tuned that it is no longer functional at the desired frequency; and this trick I learned from HAM, which is that sometimes if your antenna makes contact with a metal whilst active, it could cause a huge spike into your antenna which reaches your radio and blows it up.

Ok fine, it's like that? Fuck metals - Let's make a glass fibre housing around the antenna instead, that's an insulator so it shouldn't affect electricity and RF. Oh, you naive child, did you think it would be that easy? Thanks to near-field effects - which I don't understand well enough yet - if the dielectric is close enough to the antenna, it will still mess with the environmental capacitance or antenna properties and can cause de-tuning to your antenna. So, no conductors and no insulators? Do I just cry?

Well yes, but after you listen to this: the best rule of thumb is to just keep any metals as faraway as possible from your antenna, and don't put your antenna too close to ANYTHING else in general. If you have no choice due to constraints, then whip out your VNA and test every single part of the housing to find the best spot.

The effect of antenna polarisation on antenna placement

A very useful piece of advice that engineers might not be familiar with compared to HAM people is the relation between antenna height and its polarisation. If you're antenna is on Earth, vertical polarisation is standard as the electric field of the RF wave aligns with the surface of the Earth and gets a signal strength 'boost' along the surface. If you find that your antenna is high above the ground, then horizontally polarized makes the most sense to get the greatest possible signal strength to the Earth's surface. 

Polarisation

Hello, signal polarisation my old friend, I've come to cry alone again.

Now this is a weird one, and it doesn't even sound believable at all; I spent like 5 months trying to convince the Sunride avionics team that polarisation is important because it's just hard to imagine the effects of it, that is until Sam got an SDR and immediately found out first-hand that a -10dB drop in received signal is indeed a bad thing to have.

All RF (also known as EM) waves are made up of two orthogonal Electric (E) & Magnetic (H) field components. In this context, we basically only care about the E-fields and we decide what polarisation our signal is based off the E-field's angle with respect to the Earth's own E and H fields. So, if the E-field of the wave is perpendicular to the Earth's surface, then it is vertical, and if it's parallel to the Earth's surface, then it is... you get it. Look up the physics online if you want some more detail, it's super cool. 

Linear Polarisation (LP) - Vertical and Horizontal Signals

Linear Polarisation is by far the most simplest, and most common way an EM wave can be polarised. Any simple antenna, in which the electrons move back and forth along the antenna (e.g. a dipole) will generate this linearly polarised wave, which can be oriented horizontal or vertical; notice that there is only one plane of movement which the electric field oscillates on. The idea is that when you come to receive that signal, your antenna has to be polarised in the same exact way as the signal to ensure a seamless reception. Think of it like a revolving door, where you're walking towards it and happen to time it perfectly so that you don't stop moving for a second as you go through. That's kind of the same thing if you have a signal and antenna that mesh in terms of polarity.

For practical purposes, all you need to know is that for an incoming RF wave to be fully received by an antenna, the fields of the antenna and wave must align in a way that generates maximum induction in the antenna; in the case of RF, it happens to be orienting both E-fields parallel to each other or both H-fields perpendicular to each other.

So, what happens if you don't get a polarisation match? Well, then you get a lovely penalty called smashing your face into the glass of the revolving door and falling on your butt in a shameful, painful manner - in RF terms, you get an immediate loss in signal power by a whole -10dB.

Matching polarisation is all about perspective really (at least of linear polarisation), meaning that if your Tx antenna is horizontally polarised (==) and your Rx antenna is vertically polarised (||), then you can easily still make them match by rotating one of the antennas by 90 degrees with respect to the other one, so both E-fields are now oriented the same way 

Phew, it's a simple fix, no problem, then? Well, in this case yes, it's all dandy, but the problem occurs when you don't have control over the exact orientation, or one of the antennas is changing orientations very quickly and in an uncontrolled manner, then how the hell do you match up both antennas? This is where it might be time to look at different polarisations.

Circular Polarisation

The other main option is called circular polarisation, and you'll rarely find it in the wild, since it's usually much harder to create. You can create a circularly polarised signal through physically changing the antenna geometry, or messing with additional out of phase RF feeds into the antenna. Circular polarisation can help eliminate this matchup problem by sending a signal that is actually polarised both horizontally and vertically. Again, I'd recommend you do some reading, as the resulting shape is not the (+) signal you'd expect from the horizontal and vertical components and instead it mimics more a corkscrew, where the RF signal propagates through the air spinning the E-field into all planes.

Transmission Lines

Let's talk transmission lines; they are the basically the medium in which RF signals travel in; technically, air is a valid transmission line, albeit a shit one. The very first thing you should do is read until point 2 in the "RF Principles for PCB Design" post. I explain my understanding of AC signals and the importance of capacitance in RF applications. However, I only do it in the context of a Microstrip transmission line, which is mostly applicable for PCBs. So, let's expand this to a more general explanation.

What are Transmission Lines? Well, you can think of them as wires specially designed to 'carry' any AC signals. You may be confused as to why AC needs specially designed wires, I mean why does it matter, it's all just solid metal making some sort of electrical contact... r-right?

To explain the distinction, let's talk about what's likely to happen if you use a normal wire to carry an RF signal. So, you're shoving an oscillating current in a wire, so basic electronics states that a rapidly changing electric field is formed will produce a rapidly-oscillating magnetic field in the vicinity of the wire, which will excite electromagnetic waves in air (radio waves) that radiate away especially nearer to the resonance frequency of the wire- Waaait a second, that's just an antenna!! A normal wire just turns into an antenna!!! We don't want our wire to pick up or transmit any unwanted signals, it's just extra power losses and furthermore your transmitted signals present an EMC nightmare. So, we use transmission lines which are specially designed wires that isolate and shields the signal it carries.

There are a few common transmission line topologies e.g. a relatively famous one is parallel/twisted pairs (which is how USB cables worked). You're unlikely to ever need to use more than a few in your life. The ones I've come across the most are the following three:

• Microstrip Line

I've already kind of gone over this in the other RF post, so I won't talk about it too much. This transmission line is what you will always use to carry RF signals in PCB-like structures. Remember to always control its impedance; use online calculators and strive towards your manufacturer's own list of controlled stackups. There are various expansions and techniques on microstrips to enhance shielding or minimise reflections at branches, but I won't (can't...) go into them here. 

• Waveguides

Once you reach the high GHz region, this is probably the most common transmission line there is; waveguide is basically synonymous with high frequency. The reason it's not used at lower frequency is that the dimensions can get stupidly big, even at low GHz so it's not really used much for low frequencies. I highly recommend reading up on the antenna-theory website above if you're interested, or check up my SURE project post to see my messing around with slotted waveguides.

• Coaxial Cable

This is the first RF transmission line you'll probably hear about even if you didn't understand what it was at the time. It's called a 'coaxial cable', and is probably the cheapest and most common way to properly carry an RF signal 90% of the time. Take a look at this diagram below which shows the biology of a coaxial cable:

What's the idea here? Why is the wire wearing a condom? Well, that's how it achieves low losses and interference. The RF signal goes through the main centre wire, if you completely surround that wire with any conductor then any emissions will get absorbed and attenuated by that surrounding metal, so essentially nothing can pass through that shield. The dielectric defines an insulating separator between the main conductor and the shield, so you don't accidentally short your whole system.

It's important in practical use that you don't surpass the bending radius of your coaxial cable; basically, don't bend it too far, it can snap the dielectric inside or add kinks to the shielding braid, and that ruins the controlled impedance nature of the channel. The dielectric maintains a constant capacitance by controlling the gap between the braid and the wire, and the wire thickness maintains a constant inductance, so overall, the impedance of the channel is said to be controlled.

Controlled impedance and shielding seems to be the shared secret that all transmission lines use to allow efficient transfer of RF signals with little losses/interference. They might sound like random buzzwords, so let me try to explain the concepts and why they're important. 

Characteristic Impedance

Alright, it's time to talk about impedance again in even more detail. Yay.

Continuing on the transmission lines above, they can all be represented in EEE world as a circuit approximation of a series inductance (wire/trace) and parallel (shunt) capacitance. That ends up being the famous following diagram:

So, the key property you see here is called 'Characteristic Impedance' (Zo), which is the ratio of voltage and current of a wave propagating through a lossless transmission line; hence it's measured in ohms. The first idea is that voltages and currents are not equal across one length of transmission line at high frequencies, so if you try measuring the voltage at different spots, you'll get multiple different values and start having a panic attack when the voltage on one line aren't equal.

The second idea is the important one, and unfortunately the one that is hard to grasp - characteristic impedance has nothing to do with the concept of lumped impedance or resistance as you are familiar with it (which is a passive component with a fixed resistance value that dissipates electrical energy as heat) - e.g. the characteristic impedance of an RF trace on a PCB would not be the same thing as the resistance/impedance of the trace.

A really good explanation I read once said that just because things have the same units does not mean that they mean they represent the same thing. For example, trans-impedance amplifiers are designed in way such that you input a current and get an output voltage. So, the gain of that amp would be Output / Input, in this case Voltage / Current, which would give a gain with units of ohm. Does that mean that the amplifier is the same as a resistor?

That same logic applies here, characteristic impedance is a ratio of a wave's voltage to current which is given in Ω, that doesn't mean it has any other behavior in common with a resistor, and you shouldn't expect it to behave like one. When we say that the characteristic impedance of our transmission line is 50Ω, then that means that if we input a wave with a voltage / current ratio of 50Ω then it will pass through with absolutely no distortion and reflection backwards. With any other ratio and you'll see that the wave no longer perfectly travels through the transmission line, and some of it will starts travelling backwards where it came from.

Now that we've made the distinction, let's try to actually understand the concept of characteristic impedance, so let's do it how we do it in the bedroom; roleplay. Imagine you are a wave with legs that propagates by taking steps - each step you take is identical in size and that size happens to be your wavelength.

The characteristic impedance is the resistance per step. Taking a step in low resistance might feel like walking normally, but high resistance would feel like walking through honey.

That per step is key, and that is why it's said that waves will always see the same impedance across the whole length of a transmission line, because although the total energy lost is of course going to depend on the length of the transmission line, we don't really care about total loss when talking about characteristic impedance. In other words, it doesn't matter how far you walk, it's about how each step you take is always going to be of the same difficulty depending on the resistance you're walking through. See what's happening here? The characteristic impedance is the impedance felt taking one step. The number of steps there are to take does not change that difficulty per step value.

To bring things back into actual electronics and not kinky roleplay, characteristic impedance is the impedance that one wavelength of a propagating EM wave will 'feel' through a given transmission line. At any given step, the wave will see the same impedance between it and the next step.

Hopefully that helps solidify the concept of characteristic impedance a bit more, because it leads on to the next concepts: Return and Insertion Loss.

Return & Insertion Loss

This is super important, and you'll be happy to know that it's already been introduced. Yep, remember all that talk about Scattering (S-) parameters? To put your mind at ease, S-parameters is the negative value of return loss. Return loss is concerned with power, while S-parameters looks at the actual signal. I would recommend this for proper reading.

In the Characteristic Impedance section, we briefly talked about what happens if a wave with a voltage/current ratio gets input to a transmission line that doesn't support that ratio. We said that "that the wave no longer perfectly travels through the transmission line, and some of it will starts travelling backwards where it came from."

These reflections are terrible for wireless RF performance; like imagine losing 30% of your signal before you even get to radiate it from your antenna. Have a look at the following diagram and tables which should show the basic idea:

Ok, so ideally a transmission line and source/load will be designed so that they have the same characteristic impedance. How about in real-life, where it's virtually impossible to do this perfectly?

So, engineers figured out that they can actually observe the signal reflections and quantify various aspects like power, phase etc. These observations can be used to deduce conclusions about the performance of the system (this is basically the invention of the Vector Network Analyser - aka the VNA). So, the main use case of a VNA is to see how well-matched the impedance between the transmission line and source/load are by observing signal reflections.

The idea is you take a transmission line and badly matched antenna. You then send a signal through the transmission line and see how much is reflected back. The results come back in the form of a S-parametre graph and some weird looking venn diagram (Smith Chart). You look at your S-11 graph and probably think something like:
"Wow, that's horrible, about 50% of my signal reflected right back. I need to match this antenna right now."

So, you take a look at the Smith Chart and after some maths, and drawing circles with a finnicky compass, you find out that you can add 23Ω of inductance to change the impedance of the antenna to one that perfectly matches the transmission line. So, you pick out a suitable inductor and connect it between the antenna and transmission line.
This time when you run it through a VNA, and you get much better performance, with only 5% of the signal reflected back and a much smaller, centered Smith Chart.

So, this Smith Chart thing seems pretty important, I'll get to it in its own section after this. For now though, let's talk more about that "23Ω of inductance" and the practical ways of impedance matching.

Impedance Matching

In real-life, it's usually very easy to design a transmission line for a certain impedance since the common methods are standardised (microstrip lines, coaxials etc.)

The tricky bit is making sure that your antenna or radio output can also be seen as that impedance since the design is a lot more complex. So, there are a few known ways in the industry that are used to impedance match your load/source. The most common two options are an "Impedance Transformer" and a "Passive Matching Network".

Impedance Transformer

This will require a very good understanding of transmission line theory which I can't go too far into for the sake of keeping this post finite. So, I highly recommend you go step-by-step through Transmission Lines (antenna-theory.com) and you'll gain a much better understanding.

I've never actually had to design an impedance transformer, thank god, but you'll find that the most common one you'll learn about early is the Quarter-Wave Transformer. The  reason I've never used this is because this will only work for the designed frequency and for narrow-band applications only. That means that you have to know all the parametres of your system beforehand, and you also can never change it to accomodate different frequencies or bandwidths which I'm not too comfortable constraining myself to. 

But on the other hand, it's a super simple and cheap way to match two peripherals if you already know all the specs about them. 

So, let's say you just sat down to design a comms system from scratch. You already have a radio which is 50Ω, and you've just designed an antenna which you've measured to be an impedance of 100Ω. Using the equation above, you can actually find out what impedance the transmission line needs to be in order to facilitate a successful impedance match between the antenna and the radio; which turns out to be a 70Ω transmission line.

So, boom you design a 70Ω microstrip transmission line and now you have an RF system where the source and load are matched through the transmission line.

So, yes it's super simple in theory but not practical if you don't know the exact specs of your system. That brings us on to the second solution, which happens to be the one I've used the most due to its flexibility.

Passive Matching Networks

This is the other way to match the impedance of your load/source. In this method, you basically design your RF system normally assuming a transmission line of 50Ω (or whatever), but now you have to add in empty 'spots' both ends of the transmission line where inductors and capacitors can be placed at a later date, and these will alter the impedance to match the network. 

You won't know exactly what values the inductors and capacitors need to be, so what happens is that a VNA is used on the bare transmission line to find S-parameteres and using a Smith Chart, you can work out the inductance and capacitance values needed to match the system, then you can plug in the required components. So, it's essentially glorified LC networks and filters for unwanted frequencies.

It can be as simple as one shunt capacitor, or all the way to complex filter topologies; the most famous being the Π and T networks. I'd recommend you check out the ".(digikey) Pi, T Filters Match RF Impedances DigiKey" article in my Personal Collections folder. It goes through the two common topologies and has real-life component examples too.

The diagram above is a simplified Smith Chart, meaning that only the key components of the graph are there. Let's take a closer look at them:

1. The centre/origin of the chart where z = 1 represents your ideal impedance point; so your complex impedance at that point would be a perfect 50Ω.

• You might be confused as to why it says z = 1 instead of z = 50Ω, and the reason is that the Smith Chart is always normalised with respect to your ideal impedance.
  So, in this case the ideal impedance we care about is 50Ω, so 50Ω/50 = 1,

2. Anywhere on the blue circle shows a constant resistance, and anywhere on the green circle shows a constant reactance.

• Sliding to the left and right will change your resistance, while sliding up and down the curved lines will change your reactance.

• Note that following the drawn curved line doesn't change your resistance even though you're technically moving to the right/left - it's just the nature of a circular graph. 

So, usually your VNA will spit out a very similar plot to this. The impedance seen will be listed as a point anywhere on that whole Smith Chart.
To match impedances, your goal is to perform a series of translations/transformations that will move that impedance point to the centre where z = 1.

That's why usually the algorithm revolves around a two step game plan:

1. Move your point so it intersects with the Re(z) = 1 blue circle

2. Slide it down/up the circle to reach the z = 1 point.

See? Not too complicated as a concept. The tricky bit comes with figuring out the best translations/transformations and doing the maths to find out what that means in terms of real component values. 

Instead of trying to explain the concept poorly, I'd recommend you check out the following resources which have already explained it much better than I could hope to:
Smith Charts (antenna-theory.com) ||| !(Infineon) Antenna & RF PCB Design - page 27 ||| The Smith Chart: A Vital Graphical Tool | DigiKey 

RF Signal Propagation

I said in the other post to start thinking in terms of fields, and this is why, because current isn't just the movement of charge - it's arguably about the communication of the charges via the fields. This is how a circuit with a capacitor still allows current to flow, even though there is a physical gap in the capacitor.

Once you get into high AC or RF, the period of signal oscillation is shorter than the time taken for the signal to propagate along the wire length.

This can also be rephrased to the wavelength of the RF signal becomes shorter or of the same order of magnitude as the wire length.

To elaborate, 

Essentially

RF Wave Propogation

(Explain how lots of maths, Maxwell, Telegrapher etc. is needed to truly understand and I'm not going into those cuz I don't understand em lol.)

Talk about path of impedance vs path of resistance. Impedance function of resistance + Reactance. Reactance depends on capacitance and inductance.