A mapping of the right half of the impedance plane (all impedances with
a resistance that is not negative) onto the unit circle, using the
transformation (Z  Zo) / (Z + Zo). Smith Charts are a common tool for
doing impedance matching, a fundamental task in high frequency
circuit design.
This transformation plots all resistances onto the diameter of a circle
centered on Zo (the characteristic impedance), with a short circuit (0
ohms) on the left and an open circuit ("infinity") on the right. Pure
reactances are plotted onto the circumference of the circle, with
reactance increasing from zero towards infinite as you move away from
the short circuit impedance point towards the open circuit impedance
point. The top half of the circle represents increasing inductances and
the bottom half increasing capacitances. Impedances having the same
value of resistance will fall on circles contained within the unit circle
and tangent with the open circuit impedance point. Impedances having
the same value of reactance will fall along circular arcs passing through
the open circuit impedance point.
Reflection coefficients may be plotted on the Smith Chart in standard
polar format: the magnitude of the reflection coefficient (a number
between 0 and 1) is the distance out from the center of the chart; the
angle of the reflection coefficient indicates the amount of rotation
around the chart. 0 degrees corresponds to the open circuit
impedance; positive phase angles move across the top (inductive
portion) of the chart to 180 degrees at the short circuit impedance
point; negative phase angles move across the bottom (capacitive
portion) of the chart to 180 degrees at the short circuit impedance
point.
The mathematical relationship between reflection coefficient and
impedance is
Z/Zo = (1 + ) / (1  ) or = (Z  Zo) / (Z + Zo)
