The Smith chart is really complex looking chart that is so common to radio frequency (RF) design that it's practically the symbol for RF design. There are so many books with a Smith chart on the cover, I have a few within arm's reach right now, that you'd think RF designers couldn't live without them. That may actually be true.
So what's up with these things? They are simply a very concise way of displaying a ton of information that has to do with the impedances in something you're working on and solving matching problems. What gets most people is there is only one straight line on a Smith chart, horizontal and straight left to right down the middle of the chart. The basic construction of a Smith chart showing the straight line and a couple of characteristics is shown here:
The only purely resistive impedances on the chart are located on that
line. The top half of the chart is impedances that are inductive (R+X)
while the bottom half is capacitive impedances (R-X). The impedance at
the left end of the chart is zero ohms. The impedance at the right side
is infinite impedance; an open circuit. That means trying to plot high
impedances gets pretty cramped in that area. If you look closely at the
middle you'll see that number (just to the left of my red line) is 1.0.
Ordinarily, the chart you use is normalized to put the impedance you're
working with at that center dot, so you divide every value by 50 ohms
(usually, in radio design), which makes the right end a bit less cramped.
It's important to say that impedance could just as well be plotted on a
regular Cartesian graph like you're used to for an XY plot. The plot
like this, with arcs and circles everywhere has advantages that are hard or
impossible to achieve in the Cartesian "Y is perpendicular to X" plot and the
Cartesian plot has the serious disadvantage of only giving you half the chart
to work with. On a Cartesian plot none of the resistances would be
negative, so you'd only get the positive values of R with +/- values for
X. You'd get half a plane. The first step in transitioning from a
Cartesian graph to the Smith chart is to transform everything into polar
coordinates: magnitude and angle.
Since every point on the above chart represents R+X, those are series
impedances. It's also possible to get a version of this chart that shows
the reciprocal of impedance, called admittance. These show parallel R
and X. The red circles represent constant reactance, the green circles
are constant susceptance, the reciprocal of reactance.
Perhaps an example of use would help without overloading you.
I've mentioned before that I had made a simple little circuit to ensure that my 80 and 40m trapped vertical would be usable on 30m. My previous radios had never blinked at matching to that antenna, but the new radio would measure the SWR, decide it was over the 3:1 it's rated to match to and just stop. My trick was to put an LC in series with the antenna coax inside the shack that dropped the SWR from just about 6:1 down to comfortably under 3:1. I made it fairly close to just under 3:1 so I wouldn't mess up the bands the antenna was designed for. The first thing I did was sweep the antenna with my old VNA, an AIM4170, and plot its impedance on the Smith chart. I put two markers on the plot and those impedances, the bottom and top of the 30m band, are at the bottom of this chart (I also circled them in blue for you). You can see they're high impedance because they're toward the right edge of the chart and they're capacitive because they're below the middle straight line.
The next step was to match those to be inside of 3:1. Smith charts allow you to plot a circle of constant SWR so you simply add a part or two to push the points you're interested in inside that circle. The SWR circle can be done on paper by drawing a circle with a compass that is centered on 1.0 (50 ohms), put the pencil tip down on the 3.0 marker and draw the circle with that radius. That's the 3:1 SWR circle, in black in this graph from the excellent freeware program SimSmith. The light blue trace shows the impedance of the antenna before and the dark blue trace is after adding a shunt capacitor series inductor combination (seen at the top left of this screen capture - the left most block is the antenna impedance).
The dotted pink trace uses the same color as C1 in the circuit diagram, and green trace shows the transformation from L1 in its color. I checked both ends of the 30m band and verified that SWR was transformed to under 3:1, and that adding my little circuit didn't affect the 80 and 40m bands. I've been using that for around 2-1/2 years now and it has been zero problems.
Now look a little closer at this plot. The dotted pink trace is very
close to an admittance circuit. That tells you adding a shunt capacitor
always moves clockwise on (or "parallel to") an admittance circle. A
shunt inductor will always move counterclockwise. Similarly, the green
dotted trace is along a circle on the impedance chart. A series inductor
always moves counter clockwise (toward the inductive reactance side of the
chart) while a series capacitor always moves counter clockwise or toward the
capacitive side of the chart.
Next time, I'll talk about how the chart can cover from zero on the left to infinite on the right and how those are only 1/4 wave apart.