A single stub tuner uses a piece of transmission line (the stub)
connected in series or parallel with the main transmission line to
match a load. By matching the load impedance to the transmission
line's characteristic impedance, we can avoid reflections and other
signal degrading nastiness.

A similar way to match impedances is by using
a double stub tuner. The advantage of the double stub tuner is
fabrication; it's easier to change the length of the stub (having one
part nested inside the other) than it is to change the distance of the
stub from the load. This is problem comes up in the lab, where you are
constantly changing the operating frequency and load, but not as much
in equipment designed to operate at a fixed frequency and with a well
defined load.

Single stub tuners have two degrees of freedom. One is the length
of the stub (l), and the other is the distance the stub is placed from the
load (d).

Single stub problems are often solved using a Smith chart. Using
the Smith Chart is quick and easy once you get the hang of it, and
pretty accurate too.

The general steps for solving a single stub tuning problem using a
Smith chart are:

- Normalize the load impedance by dividing it by the characteristic
impedance of the line.
- Plot the normalized load impedance on the Smith chart.
- If your stub is in parallel with the transmission line (this is
common), convert the impedance to an admittance by rotating it a
quarter of a wavelength around the Smith chart (half a turn, 180
degrees). This is the load admittance. If the stub is in series with
the transmission line, stick with impedance.
- Using a compass, rotate the admittance (or impedance) around the
circle of constant VSWR toward the
generator (the directions ``toward generator'' and ``toward load'' are
marked on the Smith chart). The circle will intersect the circle of
unit resistance (the circle on the left hand side of the chart) in two
places. These intersections correspond to the distance the stub has to
be from the end of the load.
- Select one of these points. Measure the electric distance
traveled around the circle (refer to the wavelength measurements on
the outer ring of the Smith chart). Multiplying this number by the
wavelength will give you the distance the stub has to be from the
load (d).
- At the distance d from the load, we have an admittance (or
impedance) value in the form of 1 + jA, where A is the value of the
reactive component. To get the impedances to match, we need this
number to be 1 + j0. So, we have to add 0 - jA to 1 + jA to get a
matched impedance of 1.
- Note the value of the reactive component (inductance or
capacitance). Find where the negative of this value is on the outside
of the Smith chart.
- If the stub is an open circuit, locate the point of zero
admittance (or infinite impedance) on the Smith chart. It will be
either on the left or right most side. If the stub is a short circuit,
locate the point of infinite admittance (or zero impedance).
- Starting from the point of infinite or zero admittance
(corresponding to the end of the stub), move along the outside of the
Smith chart toward the generator until you reach -A.
- The distance traveled in the previous step, when multiplied by
the wavelength, gives you l, then length of the stub.

The important facts to consider are whether the stub is in series
or parallel with the transmission line, and whether it is an open
circuit or a short circuit. Knowing these two things, and possessing an
understanding of waves on a transmission line, it should be
possible to reason out the solution to any single stub problem.