Smith Chart - Mathematical Basis - The Normalised Impedance Smith Chart

The Normalised Impedance Smith Chart

Using transmission line theory, if a transmission line is terminated in an impedance which differs from its characteristic impedance, a standing wave will be formed on the line comprising the resultant of both the forward and the reflected waves. Using complex exponential notation:

and

where

is the temporal part of the wave

is the spatial part of the wave and

where

is the angular frequency in radians per second (rad/s)

is the frequency in hertz (Hz)

is the time in seconds (s)

and are constants

is the distance measured along the transmission line from the generator in metres (m)

Also

is the propagation constant which has units 1/m

where

is the attenuation constant in nepers per metre (Np/m)

is the phase constant in radians per metre (rad/m)

The Smith chart is used with one frequency at a time so the temporal part of the phase is fixed. All terms are actually multiplied by this to obtain the instantaneous phase, but it is conventional and understood to omit it. Therefore

and