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.