Electronics Theory

A couple of weeks ago, I received an e-mail from David, WD5BZN:

I was looking around for a book on radio math and came across some online discussion from about 10 years ago, in which you said you were contemplating writing such a book, and were soliciting ideas from people about what it should cover.

I was curious if you ever took a stab at that? If not, is there another book on the market that you recommend that covers the same topics?

Thanks, and 73, David

I replied:

I  never wrote one myself, but W5JCK has written one that covers the mathematics on the Extra Class exam – W5JCK Math Guide for Amateur Radio Extra Class Exam 2012-2016. Do you have a question about a particular topic?

David wrote back:

Thanks for the reply Dan.

No particular question at this point. Like many of the people who replied to your question about whether such a book would be helpful, I just have a general interest in beefing up my knowledge. I have a General license. Honestly, I passed that test by memorizing-not truly understanding-many of the questions and answers. So now that I have licensing out of the way, I want to actually learn some of the science behind radio. And I think math is required. Math seems to be the dividing line between “pop” science and real science.

I don’t come from a math/science background. My dad was a college history professor. My education was liberal arts all the way down the line. So, I don’t expect to be able to educate myself up to the level of someone  with, for example, a degree in electrical engineering. But I would like to do what I can. That was the motivation for my question.

To which, I replied:

I don’t know that I’d say that math is the difference between pop science and real science, but knowing the math gives you a better understanding of the technology. The mathematics describe how the technology really works. When you understand the math, you have a better understanding of the technology.

Having said that, the math we use in ham radio is no where near the math that engineers study. There isn’t a hint of calculus in ham radio, for example.

He then asked:

I am curious what your background is? I’ve always wanted to learn some calculus, whether or not its really needed for ham radio. I just think it would be cool to learn and I have always regretted stopping my math education with high school algebra.

I answered:

Well, I hate to say this, but I do have an electrical engineering degree. I got my ticket, though, when I was 16 years old, and had been interested in shortwave radio since the age of ten. And, I was always pretty good at math going through high school and college.

Having said that, I just Googled “calculus for non-mathematicians” and came up with the Web page, “A Gentle Introduction to Learning Calculus.. While I’ve only skimmed it, it does seem like a good introduction to the concepts of calculus. If you do get a chance to read it, I’d be interested in your reaction to it.

He e-mailed back:

No need to apologize! I sometimes wish I had gone in that direction. One thing about being a lawyer is that very often there just isn’t a single correct answer to a problem. I think that is what has always intrigued me about math/science—that there really is a correct answer. If you don’t do the math correctly, the bridge you are building, or the circuit you are designing, just won’t work. There is something very straightforward about that, that is lacking in my business.

I will definitely check out the resource you linked to below, and let you know what I think.  A couple of other books that I have had recommended to me are The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guides) and Engineering Mathematics​. The latter was recommended to me by a friend who was a math major. You may know of it, being an EE. It is apparently very popular in England. I’ve started reading it. It is interesting because it purports to contain all the math a first/second year engineering student would need. He has another book with more advanced math for those wanting to pursue graduate studies. Stroud’s book begins at the literal beginning, with basic arithmetic, goes through algebra, trig, calculus and ends up with an introduction to differential equations. It may be an ambitious project for me to get through, but what the heck-I’m approaching it like a hobby and going very slowly. If you are able to check out the table of contents I’d be interested in what you think.

Anyway, thanks again. Good stuff. What can I say-some guys manifest their midlife crises by buying motorcycles or speedboats. Apparently I am dealing with mine by trying to learn math.

I replied:

I understand what you’re saying about the certainty of mathematical solutions, but that’s really true only in the purely mathematical world or for cases that can be treated as such. Unfortunately, most of engineering, including amateur radio, cannot be treated this way.

Let me give you an example. The equation for calculating the resonant frequency of an LC circuit is f = 1 / (2π√(LC)). Simple enough, right?

It’s simple enough on paper, but in practice it can be quite another matter. For one thing, components have tolerances. You may think that you have a 100 pF capacitor, but depending on the type of capacitor and the individual capacitor that you pick out of the pile, that capacitor may actually be 93 pF or 110 pF. In addition, the component leads have both resistance and inductance, and there’s some amount of capacitance between coil windings. So, depending on how accurate you want that resonant frequency to be, you have to take all that into consideration. That’s why circuits have trimmer capacitors and trimmer resistors.

Even then, what’s to say that those component values are going to remain constant? In practice, what happens is that component values change as they heat up. When the capacitor heats up, the capacitance changes, meaning the resonant frequency of the circuit changes as well. That’s why when you hear guys on the air using older tube rigs, you hear them drift all over the place.

One of the characteristics of a good engineer is that he or she not only knows how to use the mathematics, but also knows the shortcomings of simply using the mathematics. Like most things in life, there’s more to it than is apparent.

David then wrote:

Thanks for keeping this discussion going. It’s very interesting and helpful for me.

Your example about the resonant frequency of an LC circuit illustrates your point nicely. And, it also indicates why I probably would have preferred to be an engineer as opposed to a pure math guy. For my money, it’s more rewarding to solve problems than to get bogged down in theory.

Of course, I didn’t pursue either of those goals—I went to law school instead, which is why we are having this exchange in the first place! And your example is also a perfect instance of the kind of stuff I want to learn. In addition to the math, I’d like to truly understand the electronics of radio.

As I mentioned, I think there is a good deal more memorizing than understanding when it comes to passing license exams. At least that was true for me-maybe I should not make that assumption about others. Anyway, now that I have licensing out of the way, I want to figure out what the heck is really happening from soup to nuts, from the insides of the radio all the way out to the radio waves propagating through space. Your LC circuit example fits perfectly with this discussion—it was probably covered somewhere in study materials for the test, but I honestly have no idea what such a circuit does, which piece of equipment in the shack it plays a role in, etc……..

And, I replied  just now:

Actually, while there might be a question or two on the test regarding LC circuits and how they work, my study guides do not cover material like that in any depth. There’s really no way that I can do that. Too get that kind of information, you need to purchase some books that cover RF circuit design. One book that gets good reviews is Experimental Methods in RF Design from the ARRL.

You should also get the ARRL Handbook, if you don’t have one already. In fact, I’d suggest looking for older editions of the Handbook as well as purchasing the new edition. The older editions cover many of the basics in more depth than the later editions.

As for Stroud’s Engineering Mathematics, I searched our local library’s catalog for it, but no luck. Here in Michigan, though, many libraries participate in what’s called the Michigan eLibrary. Because the Ann Arbor library participates, I can search for and then request books from just about anywhere in the state of Michigan. Searching through the Michigan eLibrary catalog, I found the sixth edition at Andrews University in Berrien Springs. It arrived in Ann Arbor yesterday, and I just picked it up about a half hour ago.

At over 1,200 pages, it’s quite formidable. When the librarian got it off the shelf behind the desk, I joked with her that this certainly isn’t light reading! You’re right, too, about this book starting at the very beginning. The first chapter covers arithmetic!

I skipped over to the chapter on differentiation and was quite impressed at the explanation. Stroud starts out by showing how the derivative of a straight line is its slope (it’s called “gradient” in this book, though, the author being English), then proceeds to show how the derivative of a curve is the slope of the tangent at a particular point, then generalizes it again further to explain how to calculate the derivatives of polynomials. It’s a very logical and a very good explanation.

If you haven’t purchased this book yet, I think that I can safely say that it’s worth the $60 Amazon is charging for it.

So, that’s where we’re at right now in this discussion. I probably should have mentioned that a lot of this radio stuff is stuff you only learn by experience. You actually have to build some circuits  and antennas or troubleshoot some gear to really learn this stuff. That’s part of the fun of amateur radio, though, isn’t it? You’re always learning.