A Smith chart is a circular plot with a lot of interlaced circles on it. When used correctly, matching impedances, with apparent complicated structures, can be made without any computation. The only effort required is the reading and following of values along the circles.
The Smith chart is a polar plot of the complex reflection coefficient (also called gamma and symbolized by Γ). Or, it is defined mathematically as the 1-port scattering parameter s or s11.
A Smith chart is developed by examining the load where the impedance must be matched. Instead of considering its impedance directly, you express its reflection coefficient ΓL, which is used to characterize a load (such as admittance, gain, and transconductance). The ΓL is more useful when dealing with RF frequencies.
We know the reflection coefficient is defined as the ratio between the reflected voltage wave and the incident voltage wave: