A few words about complex numbers seem to be appropriate.
If you reread (or read) my intro to Smith charts, you’ll notice that I didn’t mention at all that impedance is a complex number; that is, instead of the (R, X) that I used, most text books will talk of impedance as R + j*X, where j is an imaginary number, the square root of -1.
Decades ago, I ran into a math article somewhere that argued the concept of imaginary numbers kept a number of people from being comfortable with complex numbers and all of the useful applied math they bring to the world. Admittedly, there just isn’t a good word for a number that multiplied by itself comes out negative, since every other negative number times itself (or times any other negative number) gives a positive result. The problem is that we need two component numbers with resistance and reactance to be useful. The author of the article I read said we could just as well refer to these instead of as the “real and imaginary” parts of the number as the “humpty and dumpty” parts. The terms real and imaginary are just as useful as calling them humpty and dumpty.
(In the last year or so, when the conversation started up that math is independent of who’s doing it, so it couldn't possibly be racist, some teacher made a big deal about arguing 2+2 doesn’t always equal 4. His argument was that 2 apples and 2 bananas didn’t add to 4 anything, which simply proved he can do elementary school math at a 2nd grade level. I honestly don’t recall if I learned that in 2nd grade, but the point is that when you deal with units, any numbers must be in the same units before you can add them. If you say you’re 6’ tall and 180 pounds, those two units can’t be added in any way whatsoever to come up with a single number.)
Impedances require attention because resistance and reactance both use ohms as their units. Practically, impedances are an ordered pair; that is, an impedance where R=40 ohms and X=10 ohms can be written (40,10) and the plus sign is implicit for both parts without specifically writing it. If Z was capacitive instead of inductive, that would be written (40,-10) and those are not the same coordinates on a Cartesian XY plane or the Smith chart. The only reason I can think of to keep and use the “j” in (R + j*X) is as a reminder that inductive reactances are positive and capacitive reactances are negative. If you have two components with (R, X) values you can add parts in series by adding (R1 + R2) and (X1 + X2) to find the total impedance. To add them in parallel, you need to turn them into the reciprocals of their impedances, called their admittances. The concept is the same as combining resistors in parallel.
This is where I need to point out that doing Smith chart work with drafting tools and a paper blank is pretty much dead. The company I was working for 25 years ago threw out their paper Smith Chart blanks (the red/green kind in the original post - I know because I fished them out of the dumpster) and everything was done in software. Software is the way to go. I’ve used a Smith chart display output of all the professional level CAD programs, and a few different dedicated Smith chart programs. Lately, I’ve been using the very good freeware package I mentioned called SimSmith, by Ward Harriman, AE6TY. His latest release of the program is now called SimNEC, which merges his Smith chart program with a NEC2 engine. NEC is the Numerical Electromagnetics Code that’s used for antenna analysis. It’s a natural combination.
You don’t need to solve for series or parallel complex impedances; you just choose the circuit model you want to use, enter the component values and let SimSmith (or SimNec) tell you what it’s going to do. Like most things in life, the more you use it, the better you’ll get with it.
Oh, yeah. You can do things like sweep a filter design in SimSmith to see how it should work.
This is 4 pole 2 meter, narrowband filter I designed for a 2m transverter back in the days when I was fending off roaming velociraptors while not working in the lab. Red is insertion loss, S21, and blue is input return loss, S11.
2 apples plus 2 bananas would be 4 pieces of fruit. I suspect that teacher wouldn't have liked that response. :-)
ReplyDeleteSo create a new unit out of nothing so you can call it 4 something elses? Why stop there? Why not 1/4 fruit salad or 4 micro DelMontes? They make canned fruit salad, right? Four millionths of the amount of fruit salad they make in a year? Maybe 4 pico DelMontes or pico Doles?
DeleteA picoDelMonte is a mouthful!
Delete;P
Because apples and bananas ARE fruit. Your choice of feet and pounds is a good example of things that can't (or at least, shouldn't) be added together. The teacher's choice wasn't such a good example.
DeleteAlthough I did quite well with algebra and geometry, when I took calculus my eyes would just glaze over when they would talk about imaginary numbers. They still do. My brain just doesn't wrap around them for some reason.
ReplyDeleteI would have failed calculus if I hadn't quit tech school to take a full time job in television. I was tired of going to school in the morning then working full time running a service station.
Much like the story about calling them the humpty and dumpty parts instead of real and imaginary parts, another quote that stuck with me was that everyone finds some level of math at which it just doesn't seem to come as easily. For some folks it's fractions, for others it's partial differential equations or tensors. At that point, the only thing to do is sit and work at it harder, which sounds like what you did. It wasn't until I was taking differential equations that I met a math professor who said they threw out a lot of paper working through some problems.
DeleteI got my BS going to school taking two classes per term at night while working in the day. I wouldn't recommend it to anyone, but I know sometimes it's the only way.
Random comments: the realization that three eggs and three rocks had something in common (three) even though they were different things was the key to developing mathematics.
ReplyDeleteWhile SimSmith is an incredibly capable tool it is not easy to use and obfuscates the principles of the Smith chart. To teach Smith charts I prefer JJ Smith (Tonne Software, free) which makes playing with them easy.
And paper Smith charts are useful - in the same way that paper maps are still useful even if you can use a smart phone or pc to do mapping. Both let you visualize "how to get from here to there" whereas a computer doesn't.
Interesting, and thanks. I haven't run across JJSmith. I'll have to go look and play.
DeleteThe thing about the units thing (eggs are not rocks) is that Units Matter. There are entire weeks in science classes on "dimensional analysis" - converting units back and forth. For example, you can't divide lbs per square meter. Well, you can, but nobody will find your results meaningful. Convert your PSI, to Pascals and that kind of thing. I had those in Chemistry and Physics 101/102- type classes, but there's no reason they can't do them in high school (maybe they did and I forget).
Likewise, RF Safety limits are specified by the FCC as dBm/(cm^2). You'll probably start with watts at the transmitter and area in square meters - or square yards. Converting them is essential.
As for paper Smith charts, I find that the faint impedance and admittance curves on SimSmith work fine for me. I'm sure, though, that I don't do everything that can be done on a Smith chart with them, though.
Of course units matter - I learned to use the "factor label method" to do chemistry and physics, too. But seeing the number beyond the object is transcendent - the aha moment that created mathematics as a subject. It is also the step into analysis that enables engineering (and warps the engineering mind so it will never be normal again).
ReplyDeleteI was incorrect to say computers don't let you see how to get from here to there. I should have said that the computer selects a route and doesn't promote your own consideration of the different ways to get from here to there.
I see immediately the big difference. I've never used software that chose a route and designed things. I've just used the computerized chart as if it was paper.
DeleteThe last two JPEGs in the first article (of this - #37) are an example. I looked that the starting (high) impedance, and from that I saw the L network had to have the shunt part on the antenna side. Shunt L or C? Since the antenna impedance point is almost on top of a shunt C circle, went with that, then picked a series inductor that got me into the 3:1 circle.
Faster and easier to do than explain.