Before we start, let me do a crash review. The Smith chart is a way of plotting impedances of the form R +/- X or their reciprocals, called admittances, and provides graphical ways of solving for impedance matching problems. Look at the top chart in part 1; the central axis (pure resistance) goes from zero to infinity, which means the right end is pretty packed and it’s impossible to distinguish, say 10,000 from 1 million or 1 billion ohms. Left to right, it’s low impedances to high impedances. The center, marked 1.0, on that chart is usually normalized to the impedance level you’re working with, and 50 ohms (pure resistance) is far more common than much higher or lower. The top semicircle is inductive impedances, while the bottom semicircle is capacitive impedances.
At the end of the first part of this series, I remarked that next we’d talk about electrical length on the Smith chart. The “secret” part of that is that the complete circumference of the chart is never more than 1/2 wavelength, 180 degrees. At 0 and every multiple of 1/2 wavelength, we’re at the left end of the central axis where the chart presents the same impedance and (zero). The same principle applies for 1/4 wave – the opposite side of the chart is an open circuit and the impedance repeats every half wavelength, or every odd multiple of 1/4 wavelength (1, 3, 5, 7... waves).
I put the word secret in scare quotes because it’s hard to be less secret. It’s printed right there on the left. Above the real axis, along the top of the chart it says, “wavelengths toward generator” while along the bottom radial scale it reads, “wavelengths toward load.” Some charts will say the same thing on the right side of the chart around the infinite impedance point.
This is where one of famous behaviors of a transmission line comes from. Let’s say you want to keep some interfering signal out of a receiver (TV, FM radio, whatever), you isolate the offending frequency and cut a piece of transmission line that’s 1/4 wavelength long, leaving the end open. It behaves like a dead short to the signal you cut the line for and any signal reflecting to the high impedance end sees an open circuit.
The line used for the shunt 1/4 wave stub should match the impedance of the circuit, say 75 ohms for cable TV or 300 ohms for an FM receiver, and that 1/4 wavelength means it’s shorter than a quarter wave in free space by the velocity factor of the transmission line. That means you calculate the length for your frequency, and look up the velocity factor for the transmission line you’re using. Typical coax would be around 0.6 to 0.7, so once you find that number, you cut the line to 60, 66 or 70% - or whatever the percentage of the computed length.
You can also impedance match on the Smith chart by modeling pieces of transmission line. A well known example is a quad antenna, which is one full wavelength loop. The impedance of a 1 wave loop is usually said to be 100 ohms, and it’s recommended to include 1/4 wavelength of 75 ohms for each loop in a multiband quad array. Unfortunately, that’s not exactly true. To really match 100 to 50 ohms requires 1/4 wave of SQRT(50*100) or 70.7 ohms (where SQRT is short for “square root of”), leaving a plot like this:
You can see that if you had 70.7 ohm line, it would work perfectly. In the case of using 75 ohm line, you get a slightly higher VSWR, 1.15:1. No big deal. This works for a 25 ohm antenna as well. It starts on the opposite end of the axis and SQRT(25*50) is 35.4 ohms. The problem is that 35.4 ohm coax is harder to find than 70.7 ohm line - except I can find no evidence either impedance line exists.
Also note this is for 10 MHz, and a quarter wave is much less coax than if you wanted to try this on 80 or 160m. Note how for the simplified model of the coax it says 19.18 feet long. For comparison, I took a real world plot of an electrically short vertical (my MA8040V, which you might read to be an 80/40 Vertical – as it is) on 160 m and put in 90 degrees of coax to make it a high impedance. That line was 113 feet long.
The antenna itself, measured at the radio end of my antenna coax and switches, is the pink arc at lower left – so very low impedance and entirely capacitive. The green arc is after the additional 113 feet of RG-8U coax. As you can see, it’s transformed to much higher impedances, but an uncomfortable thing happens. The curve goes from inductive to capacitive. The frequency where it goes from inductive to capacitive is just above the top of the 160m band at 2.000 MHz. I think a shorter transmission line would make the matching a bit less sensitive, but this is purely academic, not a ready to build project. I can’t see leaving (at least!) a hundred feet of coax in the radio room to get a not very efficient station on 160m.
Housekeeping: Something I didn’t say in part 1 is the Smith chart is named for Dr. Phillip Smith who published on it in 1939, and not Snuffy, just a cartoon I remember the name of from my little kid days. I mean like over 60 years ago.