I've been studying for my amateur radio license recently, and this article is a great introduction to the basics.
But really, if you want to get your hands dirty with some practical electronics, and also want to be able to communicate without relying much on nearby infrastructure, amateur radio is a great hobby.
Do yourself a favor and study for both your technician and general at the same time (I’m assuming you live in the US). HF is exponentially more fun than just VHF/UHF.
The US ham test question pools are fully public. Your test will be a mixture of questions from the pool. HamStudy basically lets you churn the question pool, and then will offer explainer text / references to back up each question and correct answer.
I went on a vacation and used their phone app any time I was standing in a line. You can set it to just keep spinning through the questions, with a bias towards ones you're getting wrong.
You need to get 37+ correct to pass. Another way to think of that is you can get up to 13 wrong and still pass.
Within each category there are subcategories. "Antennas and Transmission Lines" for example has 8 subcategories. The 8 questions in "Antennas and Transmission Lines" are one from each of those subcategories. The question pools for these subcategories each have 10-14 questions.
If you compare to the closest corresponding categories/subcategories from the General and Technician exam you'll probably find that there are a few cases.
1. The Extra is just more of the same. It's not harder per se. "Commission Rules" for example.
2. The Extra goes goes deeper and also adds new material that is more advanced.
3. The Extra is in new territory.
If you get to the point where the Technician and General are going to no problem, then you will probably have no trouble getting to the point where case #1 is also no problem, and case #2 is also well in hand. It is #3 where you might have trouble.
But remember that you can get 13 wrong and still pass!
Pick say 10 subcategories that are in case #3 that look like they would be the hardest to get good at and just write them off.
For example in "Antennas and Transmission Lines" you might decide that the "Smith chart" subcategory, which has a pool of 14 questions, would take a lot of time to get good at. So skip it. That's 14 less potential questions you have to be prepared to answer, leaving more time to study for things in class #2 and the class #3 things that look most doable.
It doesn't cost extra to take the Extra test at the same session that you take the Technician and General tests, and there is no penalty for failing, so might as well go for it.
Huh, I never heard that one. Extra gets you more frequency privileges (nice not having to worry so much about band edges) but IMO the real benefit is being able to enjoy reciprocal operating under CEPT when traveling abroad.
Only the entry level license (Technician in the US, covering UHF/VHF) is substantially different from the German one, and it's also much more restricted. Germany in general is a much better country for radio, especially if you wanted to ever do high power broadcasting.
I think the one perk to the US is that the FCC has basically stopped caring about all but the most important frequencies. This makes HF particularly fun, since HF pirate radios are often the best listening stations in the entire RF spectrum. I have no idea what that's like in Germany, but I would imagine given the general ordnung culture and veneration of rules, German hams are much less tolerant of flagrantly unauthorized broadcast stations and your regulatory bodies are more proactive in shutting them down.
To get an idea about radios, I made a crystal radio when I was in 7th grade, I only had few components. The only component I had difficulty in getting was the crystal oscillator (I was living in a rural town).
It was mind blowing when I first heard the audio through IEMs ! It felt magical that this contraption was working without any battery source.
As someone who's always dabbled in electronics, skimmed and read some books, my primary complaint abot most electronics texts is that they just talk about individual topics: oscillators, amplifiers, etc.
What they never talk about, is putting them all together.
But as witnessed by this list, that's what a radio is. A collection of these "meta" components into a whole to get a better radio experience.
A radio built like this, with individual subsystems connected together, is much more understandable. Many (not all) radio schematics are presented as a whole, rather than the parts, or why you might (or might not) want to change one part or another (not components, but one, say, filter circuit to a different one).
It just seems to me that once you get past some basic theory, starting with a radio, and then systematically taking it apart is a better way of approaching electronics education.
"A radio built like this, with individual subsystems connected together, is much more understandable."
Yes! this has been my experience too, building something from first principles and given some tools and direction to experiment you get the chance, and experience, to really learn.
I've been looking for resources like this for building amps but they're either small signal or the whole design. You understand how they work but not where and what to change if you wanted to tinker or build your own.
Haha I know! When I was even younger, we had a radio that could recieve SW,AM,MW,FM(in TV range as well).
I used to hook up the antenna to various things like wire mesh or tv antennas etc and used to listen to short wave and AM for hours. I even got signals from far away countries, it was really fascinating!
Also I had seen some recon antennas in a certain campus (can't say much about that) when I was a kid. Those were like long wires hanging from towers. I believe they used to receive/decode SW/AM signals from far away. I realised this much much later in my life. But fascinating nonetheless.
And adding to all these is SDR! That's a whole different thing.
Oh, thanks, good to know. Now I feel more motivated. Because actually it’s not as easy as it looks from the text books. It’s like with drawing an owl. Yeah, pass the signal through a mixer and feed recovered carrier to its LO port and you’re done. Sure. Simple. Now just recover the carrier. So far I have built a PLL that locks to a clean signal but stops locking when the signal is modulated too much. Aargh.
I wish it was easier to buy a portable radio with one. Though admittedly I tend to use mostly vintage radios - as such I do most of my shortwave listening on a Zenith T-O which is pretty wonderful both in audio quality and capacity to pull in stations.
This is a video about the diode, in this case a Schottky diode made with a coil of copper wire filed to a fine point touching a piece of galena sitting loose in a bottlecap found by the side of the road.
The video is notable for demonstrating the original "breadboard" technique, where you connect your wires and components by clamping them to a wooden board with the head of a wood screw. The book I learned the technique from recommended using a dished washer under the head of the screw so that the screw head doesn't push the wires sideways as you tighten it.
I've been learning about radios for a while, and this article explained one of the key questions I had: why can't you turn on and off some single frequency waveform faster to transmit data faster? (answer: changing amplitude messes with the spectrum and makes it no longer a single frequency...)
I tend to prefer these visual and intuitive explanations to the mathematically based ones usually given in lectures. The "open capacitor" example was something I hadn't thought of before.
My grandfather, when he was a boy in the 1910s or 1920s built a radio out of a round oatmeal box, copper wire, and a crystal. I never saw the original, but he built a replica of it (featured in a book in the 1970s) that I saw as a child.
If you dig deeper, you end up with I/Q signals. If you take the input from the antenna and delay it by 90 degrees, you then have something which can have a positive or even negative frequency. Using multiplication of complex terms, you can add or subtract frequencies without generating unwanted images.
You could also use signals from 2 antennas 90 degrees apart to get I and Q. This gives you the added ability in that signals from one side has a negative frequency. It's some really useful stuff.
This is a great review, but don't let theory interfere with practice. There's an art and feel to radio and antenna building. I've built antennas with scrap wire and baluns without double checking . Just a rough length of wire tweaked until you hear someone.
The theory seems overly complex (because it is). but for practical radio you can tune up nearly anything and chat -- and be way more productive with it.
Imagine obsessing over the mechanics of baseball or fishing without ever tossing a ball or casting a reel.
> In today’s article, I’m hoping to provide an introduction to radio that’s free of ham jargon and advanced math
…(scroll)…
> The identity for cos(α + β) can be trivially extended to cos(α - β), because subtraction is the same as adding a negative number:
…(scroll)…
> From the formula we derived earlier on, the result of this multiplication necessarily indistinguishable from the superposition of two symmetrical sinusoidal transmissions offset from a by ± b, so AM signals take up bandwidth just the same as any other modulation scheme.
I got my amateur radio license last year, and this is precisely why I haven't been able to do much with it: seemingly all the guides, even the license study materials, use vocabulary I'm not familiar with. I have two CS degrees and a solid foundation in math, but I can't understand how radios work because of vocabulary more than the math.
My uncle who's been building his own radios for over 60 years, tried to explain to me how antennas work, and even to him it comes down to "black magic".
I'm told the way they work is not really intuitive, so you just have to math it out.
I work in software now, but I have an electrical engineering degree and started my career on a project developing a radio. Our project probably had ten or more electrical engineers on it, and only one or two of them really understood the RF side of it. It's a very specialized skill -- even EEs with >20 years of experience would describe things as black magic.
So I don't think you're alone feeling this way. Even with a good foundation in the theory and math, I think most people hit a wall with radios at some point. All the people I worked with who intuitively "got" RF stuff had been doing nothing else professionally for over a decade.
A long time ago, worked on a comm satellite program. It used a whack of tuned cans to combine high powered transmit signals with harmonics in each others' frequency bands to feed into the antenna. I once asked how they worked. The answer was 'magic'. I mean, they were physical RF filters, but no one could explain or reproduce how they worked. There was this one guy who could tighten the screws that adjusted the inside baffles so they 'worked'. No one else could.
Antennas are really black magic: optimizing an antenna requires stocastich method like genetic algorithms, simulated annealing, etc.
Moreover if you want to model the radiation patterns and the electrical characteristics you need to use finite element calculation methods.
So, you need a lot of computation power as antenna are not a problem that can be solved in a closed form.
Source: I almost burnt my PC on simulating a dipole array while studying for the antennas course at the university
I assure you, that doesn't go away in electrical engineering or even theoretical physics. Electromagnetism exhibits a large degree of behavioral emergence. It's one of the most well studied aspects of physics, but remains a rather convoluted and seemingly arbitrary puzzle box of nonsense especially at a macroscopic level.
Even just the theory is kind of mind expanding. I've done a little signal processing and ideas like "negative frequency" sound absurd up front and then seem reasonable once you've worked with them.
I've had my ham license for ten years, but I've only ever used a basic car-based mobile setup and my handhelds. My morse code speed is abysmal. QRP and all that are really cool, but I just use ham radio to supplement my fire/ems handheld in a natural disaster.
I'd love to see more people on the air. My advice is to get a radio with a good tuner, build a simple dipole with the online calculators, and try to make contacts that way.
This is basic math, not advanced math. If you don't know it, don't worry, you can learn it pretty quickly. Maybe watch some 3Blue1Brown videos and do some textbook exercises and/or game programming. Git gud. There is no royal road to radio modulation.
I learned that subtraction was the same as adding a negative number sometime around second grade, and I learned (then forgot) the trigonometric angle-sum identities in tenth grade. And that was even with the handicap of having to attend school in the US.
And, just above the text you're complaining about, he even provides a straightforward geometric proof of the angle-sum identity! So you don't even have to know it to read the article! You just have to know what a cosine is! I learned what a cosine was in eighth grade because I wanted to program a game where objects would fly across the screen at a constant velocity but a varying angle. You can learn it too!
He's not, like, invoking the convolution theorem or anything in those quotes. Although he does get into it a bit.
I think that, if you know the convolution theorem and Euler's formula, things like the production of sum and difference frequencies from the multiplication of sinusoids start to seem obvious rather than sort of random. When I was in high school they seemed sort of random. My uncle had tried to explain Euler's formula to me, along with the Taylor expansions for sine, cosine, and the exponential, but I hadn't really understood, because I didn't have the background knowledge to appreciate them then.
So much EE-related math becomes trivial (or at least not-hard) once you've internalized this formula.
What I am trying to decide is 1) Did I zone out in class when Euler's formula was introduced or 2) Did my secondary school mathematics classes just kind of gloss over it?
I lean towards 2 but unfortunately none of my college classes reintroduced the formula and I ended up making a lot of problems harder than they should have been (I have an EE undergrad).
I'm not sure any of my secondary-school classes taught it, but I dropped out of secondary school to go to the university, so maybe they would have in the following year. In secondary school I was in the "smart kids" math track, though, and they also had an "average kids" track and a "dumb kids" track, and I really doubt those tracks covered it.
Are you objecting to the trivial extension to cos(α - β)? The cos(α + β) identity itself is nicely explained in the text.
The third thing you quote is the result of fairly simply symbol manipulation that requires no new knowledge apart from the original cosine identity and the obvious corollary about subtracting an angle being the same as adding a negated one.
There is zero advanced math there. No complex numbers, no calculus, no limits, no Fourier, no "functions are vectors, too".
EDIT: I am offended that you guys think my awesome explanation is from GenAI.
Imagine a circuit. Like a flashlight. The electrons flow from the battery to the lightbulb and back. It’s like a race track. They proceed in an orderly fashion.
There are some other electrical components. If you hook them up in just the right way, you get something called an LRC circuit. The electrons don’t flow in an orderly way now. They go back-and-forth. In spurts and fits. You’re making them wiggle. There are some very nice equations that allow you to specify exactly how much and how fast the wiggling is.
One cool thing about a circuit with wiggling electrons is that if you put some wires close by those electrons will also start wiggling.
I've been studying for my amateur radio license recently, and this article is a great introduction to the basics.
But really, if you want to get your hands dirty with some practical electronics, and also want to be able to communicate without relying much on nearby infrastructure, amateur radio is a great hobby.
Do yourself a favor and study for both your technician and general at the same time (I’m assuming you live in the US). HF is exponentially more fun than just VHF/UHF.
A random plug for https://hamstudy.org/
The US ham test question pools are fully public. Your test will be a mixture of questions from the pool. HamStudy basically lets you churn the question pool, and then will offer explainer text / references to back up each question and correct answer.
I went on a vacation and used their phone app any time I was standing in a line. You can set it to just keep spinning through the questions, with a bias towards ones you're getting wrong.
Take a look at Extra too. You may find that it actually isn't much harder than General.
The Extra exam is 50 multiple choice questions broken down by category as follows:
You need to get 37+ correct to pass. Another way to think of that is you can get up to 13 wrong and still pass.Within each category there are subcategories. "Antennas and Transmission Lines" for example has 8 subcategories. The 8 questions in "Antennas and Transmission Lines" are one from each of those subcategories. The question pools for these subcategories each have 10-14 questions.
If you compare to the closest corresponding categories/subcategories from the General and Technician exam you'll probably find that there are a few cases.
1. The Extra is just more of the same. It's not harder per se. "Commission Rules" for example.
2. The Extra goes goes deeper and also adds new material that is more advanced.
3. The Extra is in new territory.
If you get to the point where the Technician and General are going to no problem, then you will probably have no trouble getting to the point where case #1 is also no problem, and case #2 is also well in hand. It is #3 where you might have trouble.
But remember that you can get 13 wrong and still pass!
Pick say 10 subcategories that are in case #3 that look like they would be the hardest to get good at and just write them off.
For example in "Antennas and Transmission Lines" you might decide that the "Smith chart" subcategory, which has a pool of 14 questions, would take a lot of time to get good at. So skip it. That's 14 less potential questions you have to be prepared to answer, leaving more time to study for things in class #2 and the class #3 things that look most doable.
It doesn't cost extra to take the Extra test at the same session that you take the Technician and General tests, and there is no penalty for failing, so might as well go for it.
As I understand it General gets you most everything and Extra is mostly useful for people who want to teach amateur radio classes.
Huh, I never heard that one. Extra gets you more frequency privileges (nice not having to worry so much about band edges) but IMO the real benefit is being able to enjoy reciprocal operating under CEPT when traveling abroad.
You can operate abroad with General in most countries.
it’s stuff like this that make me wish i lived in the US. i have to memorise a long questionnaire in german if i want to get my license
You can take your licence exam in any CEPT country and swap it (i.e. that countries licence) for a German one under reciprocal licencing agreements.
https://www.cept.org/ecc/topics/radio-amateurs
Only the entry level license (Technician in the US, covering UHF/VHF) is substantially different from the German one, and it's also much more restricted. Germany in general is a much better country for radio, especially if you wanted to ever do high power broadcasting.
I think the one perk to the US is that the FCC has basically stopped caring about all but the most important frequencies. This makes HF particularly fun, since HF pirate radios are often the best listening stations in the entire RF spectrum. I have no idea what that's like in Germany, but I would imagine given the general ordnung culture and veneration of rules, German hams are much less tolerant of flagrantly unauthorized broadcast stations and your regulatory bodies are more proactive in shutting them down.
You have to memorize a long questionnaire in the US (though in English) to get it there, too.
Great advice!
[dead]
To get an idea about radios, I made a crystal radio when I was in 7th grade, I only had few components. The only component I had difficulty in getting was the crystal oscillator (I was living in a rural town).
It was mind blowing when I first heard the audio through IEMs ! It felt magical that this contraption was working without any battery source.
You can use a germanium diode or even Shottky now instead of a “crystal”.
Such a simple radio can be a gateway drug to a very complex and deep hobby. In my case it went like that:
1. Built a simple radio
2. Could hardly hear anything, need to add an amplifier to it 3. Now it’s better but captures a lot of noise
4. Design a filter to select just that one station
5. Now I want to listen to more stations.
6. Ugh, you can’t design a good filter with variable frequency. Enter the superheterodyne world.
7. Now finally got something that resembles a tunable AM radio, but it kinda whistles / hums a lot. Ah, so the mirror image is a real thing?!
8. Need a higher IF to be able to better reject the image before the mixer. Ok, let’s make a double conversion superhet then.
9. Buy a set of ceramic filters and play with them to get the best selectivity.
10. Try to add more amplification only to learn if you go too far you get an oscillator instead of an amplifier.
11. The sound level is not stable. Add AGC.
12. Pick up some stations from 5000+ km away. Nice. But there is some weird distortion. Oh, I’ve been a culprit of frequency selective fading…
Fast forward and now I’m building a PLL synchronized AM product demodulator with a squaring loop for carrier recovery.
Fun. Lot of fun! Wholeheartedly recommend!
There's something to note here.
As someone who's always dabbled in electronics, skimmed and read some books, my primary complaint abot most electronics texts is that they just talk about individual topics: oscillators, amplifiers, etc.
What they never talk about, is putting them all together.
But as witnessed by this list, that's what a radio is. A collection of these "meta" components into a whole to get a better radio experience.
A radio built like this, with individual subsystems connected together, is much more understandable. Many (not all) radio schematics are presented as a whole, rather than the parts, or why you might (or might not) want to change one part or another (not components, but one, say, filter circuit to a different one).
It just seems to me that once you get past some basic theory, starting with a radio, and then systematically taking it apart is a better way of approaching electronics education.
"A radio built like this, with individual subsystems connected together, is much more understandable."
Yes! this has been my experience too, building something from first principles and given some tools and direction to experiment you get the chance, and experience, to really learn. I've been looking for resources like this for building amps but they're either small signal or the whole design. You understand how they work but not where and what to change if you wanted to tinker or build your own.
Haha I know! When I was even younger, we had a radio that could recieve SW,AM,MW,FM(in TV range as well). I used to hook up the antenna to various things like wire mesh or tv antennas etc and used to listen to short wave and AM for hours. I even got signals from far away countries, it was really fascinating!
Also I had seen some recon antennas in a certain campus (can't say much about that) when I was a kid. Those were like long wires hanging from towers. I believe they used to receive/decode SW/AM signals from far away. I realised this much much later in my life. But fascinating nonetheless. And adding to all these is SDR! That's a whole different thing.
> You can use a germanium diode or even Shottky now instead of a “crystal”.
Or a razor blade or a piece of roofing lead and a needle.
Now I’m interested…
Diodes occur naturally, all you need to do is find them!
Good places to look for them are conductors and crystals.
https://en.wikipedia.org/wiki/Crystal_detector
https://rimstar.org/science_electronics_projects/razor_blade...
Good luck! And don't sneeze once you've found a good spot.
> Diodes occur naturally, all you need to do is find them!
Even true for LEDs!
https://en.wikipedia.org/wiki/Silicon_carbide#Natural_occurr...
https://en.wikipedia.org/wiki/Light-emitting_diode#History
I have used a synchronous AM detector once in a radio, its a game changer for listening to shortwave.
Oh, thanks, good to know. Now I feel more motivated. Because actually it’s not as easy as it looks from the text books. It’s like with drawing an owl. Yeah, pass the signal through a mixer and feed recovered carrier to its LO port and you’re done. Sure. Simple. Now just recover the carrier. So far I have built a PLL that locks to a clean signal but stops locking when the signal is modulated too much. Aargh.
I wish it was easier to buy a portable radio with one. Though admittedly I tend to use mostly vintage radios - as such I do most of my shortwave listening on a Zenith T-O which is pretty wonderful both in audio quality and capacity to pull in stations.
The crystal radios I know don't have crystal oscillators - the word "crystal" refers to the diode.
Perhaps I'm getting confused cuz this was long back. But I think it was a oscillator for sure...
https://www.youtube.com/watch?v=-GxL13rid1w
This is a video about the diode, in this case a Schottky diode made with a coil of copper wire filed to a fine point touching a piece of galena sitting loose in a bottlecap found by the side of the road.
The video is notable for demonstrating the original "breadboard" technique, where you connect your wires and components by clamping them to a wooden board with the head of a wood screw. The book I learned the technique from recommended using a dished washer under the head of the screw so that the screw head doesn't push the wires sideways as you tighten it.
Crystal radios don't normally contain crystal oscillators, which do require a battery source. Perhaps you mean a crystal rectifier.
Yeah perhaps I'm confused cuz this was long long back.
I've been learning about radios for a while, and this article explained one of the key questions I had: why can't you turn on and off some single frequency waveform faster to transmit data faster? (answer: changing amplitude messes with the spectrum and makes it no longer a single frequency...)
I tend to prefer these visual and intuitive explanations to the mathematically based ones usually given in lectures. The "open capacitor" example was something I hadn't thought of before.
Previously:
Radios, how do they work? - https://news.ycombinator.com/item?id=39813679 - March 2024 (109 comments)
My grandfather, when he was a boy in the 1910s or 1920s built a radio out of a round oatmeal box, copper wire, and a crystal. I never saw the original, but he built a replica of it (featured in a book in the 1970s) that I saw as a child.
https://sg30p0.familysearch.org/service/records/storage/dasc...
If you dig deeper, you end up with I/Q signals. If you take the input from the antenna and delay it by 90 degrees, you then have something which can have a positive or even negative frequency. Using multiplication of complex terms, you can add or subtract frequencies without generating unwanted images.
You could also use signals from 2 antennas 90 degrees apart to get I and Q. This gives you the added ability in that signals from one side has a negative frequency. It's some really useful stuff.
Never thought about the AM bandwidth thing before that is interesting and seems obvious now (from Fourier Transform point of view)
Here's a video of AM modulation. SDR transmitter (running GNU Radio) connected to an SDR receiver.
https://www.youtube.com/watch?v=9LsJn0CyyZI
This reminds me of a beautiful book written by Paul Nahin, 'The Science of Radio'.
Bought it but never read it - is it worth pushing through?
I did many years ago. Absolutely worth it.
This is a great review, but don't let theory interfere with practice. There's an art and feel to radio and antenna building. I've built antennas with scrap wire and baluns without double checking . Just a rough length of wire tweaked until you hear someone.
The theory seems overly complex (because it is). but for practical radio you can tune up nearly anything and chat -- and be way more productive with it.
Imagine obsessing over the mechanics of baseball or fishing without ever tossing a ball or casting a reel.
Cool article; I’m still baffled though.
I was listening to story of QualComm in Acquired podcast. This is highly related.
> In today’s article, I’m hoping to provide an introduction to radio that’s free of ham jargon and advanced math
…(scroll)…
> The identity for cos(α + β) can be trivially extended to cos(α - β), because subtraction is the same as adding a negative number:
…(scroll)…
> From the formula we derived earlier on, the result of this multiplication necessarily indistinguishable from the superposition of two symmetrical sinusoidal transmissions offset from a by ± b, so AM signals take up bandwidth just the same as any other modulation scheme.
I got my amateur radio license last year, and this is precisely why I haven't been able to do much with it: seemingly all the guides, even the license study materials, use vocabulary I'm not familiar with. I have two CS degrees and a solid foundation in math, but I can't understand how radios work because of vocabulary more than the math.
My uncle who's been building his own radios for over 60 years, tried to explain to me how antennas work, and even to him it comes down to "black magic".
I'm told the way they work is not really intuitive, so you just have to math it out.
Maybe I should have gotten an EE degree.
I work in software now, but I have an electrical engineering degree and started my career on a project developing a radio. Our project probably had ten or more electrical engineers on it, and only one or two of them really understood the RF side of it. It's a very specialized skill -- even EEs with >20 years of experience would describe things as black magic.
So I don't think you're alone feeling this way. Even with a good foundation in the theory and math, I think most people hit a wall with radios at some point. All the people I worked with who intuitively "got" RF stuff had been doing nothing else professionally for over a decade.
A long time ago, worked on a comm satellite program. It used a whack of tuned cans to combine high powered transmit signals with harmonics in each others' frequency bands to feed into the antenna. I once asked how they worked. The answer was 'magic'. I mean, they were physical RF filters, but no one could explain or reproduce how they worked. There was this one guy who could tighten the screws that adjusted the inside baffles so they 'worked'. No one else could.
Antennas are really black magic: optimizing an antenna requires stocastich method like genetic algorithms, simulated annealing, etc. Moreover if you want to model the radiation patterns and the electrical characteristics you need to use finite element calculation methods. So, you need a lot of computation power as antenna are not a problem that can be solved in a closed form.
Source: I almost burnt my PC on simulating a dipole array while studying for the antennas course at the university
Related: https://en.wikipedia.org/wiki/Evolved_antenna
I assure you, that doesn't go away in electrical engineering or even theoretical physics. Electromagnetism exhibits a large degree of behavioral emergence. It's one of the most well studied aspects of physics, but remains a rather convoluted and seemingly arbitrary puzzle box of nonsense especially at a macroscopic level.
Even just the theory is kind of mind expanding. I've done a little signal processing and ideas like "negative frequency" sound absurd up front and then seem reasonable once you've worked with them.
I've had my ham license for ten years, but I've only ever used a basic car-based mobile setup and my handhelds. My morse code speed is abysmal. QRP and all that are really cool, but I just use ham radio to supplement my fire/ems handheld in a natural disaster.
I'd love to see more people on the air. My advice is to get a radio with a good tuner, build a simple dipole with the online calculators, and try to make contacts that way.
It gets crazier when you start talking about things like a Tarana BN. The amount of processing in them is pretty insane.
But yeah, black magic is right!
This is basic math, not advanced math. If you don't know it, don't worry, you can learn it pretty quickly. Maybe watch some 3Blue1Brown videos and do some textbook exercises and/or game programming. Git gud. There is no royal road to radio modulation.
I learned that subtraction was the same as adding a negative number sometime around second grade, and I learned (then forgot) the trigonometric angle-sum identities in tenth grade. And that was even with the handicap of having to attend school in the US.
And, just above the text you're complaining about, he even provides a straightforward geometric proof of the angle-sum identity! So you don't even have to know it to read the article! You just have to know what a cosine is! I learned what a cosine was in eighth grade because I wanted to program a game where objects would fly across the screen at a constant velocity but a varying angle. You can learn it too!
He's not, like, invoking the convolution theorem or anything in those quotes. Although he does get into it a bit.
I think that, if you know the convolution theorem and Euler's formula, things like the production of sum and difference frequencies from the multiplication of sinusoids start to seem obvious rather than sort of random. When I was in high school they seemed sort of random. My uncle had tried to explain Euler's formula to me, along with the Taylor expansions for sine, cosine, and the exponential, but I hadn't really understood, because I didn't have the background knowledge to appreciate them then.
> Euler's formula
So much EE-related math becomes trivial (or at least not-hard) once you've internalized this formula.
What I am trying to decide is 1) Did I zone out in class when Euler's formula was introduced or 2) Did my secondary school mathematics classes just kind of gloss over it?
I lean towards 2 but unfortunately none of my college classes reintroduced the formula and I ended up making a lot of problems harder than they should have been (I have an EE undergrad).
I'm not sure any of my secondary-school classes taught it, but I dropped out of secondary school to go to the university, so maybe they would have in the following year. In secondary school I was in the "smart kids" math track, though, and they also had an "average kids" track and a "dumb kids" track, and I really doubt those tracks covered it.
Are you objecting to the trivial extension to cos(α - β)? The cos(α + β) identity itself is nicely explained in the text.
The third thing you quote is the result of fairly simply symbol manipulation that requires no new knowledge apart from the original cosine identity and the obvious corollary about subtracting an angle being the same as adding a negated one.
There is zero advanced math there. No complex numbers, no calculus, no limits, no Fourier, no "functions are vectors, too".
To be fair, I think most people regard “not advanced math” to exclude trig, and maybe algebra too. Fear and loathing of math is a thing.
skill issue
To be fair, they did qualify it as "advanced" math :-)
What do you consider advanced math? This isn't solving differential equations...
How I wish I could quit my job and go back to school.
needs more pictures!
The article is interesting but some images have Error 400 so I can’t see them.
Probably just a network hiccup.
I ran into the same thing earlier, but it worked fine after a couple refreshes.
EDIT: I am offended that you guys think my awesome explanation is from GenAI.
Imagine a circuit. Like a flashlight. The electrons flow from the battery to the lightbulb and back. It’s like a race track. They proceed in an orderly fashion.
There are some other electrical components. If you hook them up in just the right way, you get something called an LRC circuit. The electrons don’t flow in an orderly way now. They go back-and-forth. In spurts and fits. You’re making them wiggle. There are some very nice equations that allow you to specify exactly how much and how fast the wiggling is.
One cool thing about a circuit with wiggling electrons is that if you put some wires close by those electrons will also start wiggling.
This is called radio.