Transmission Line Theory
Transmission lines are the circuits that deliver power from a transmitter to an antenna and from an antenna to a receiver. These are the impedance matching circuits. While transferring the RF waves on a finite transmission line, for maximum power transfer from source to load (i.e. transmitter to antenna or antenna to receiver), the impedance of source must be equal to the impedance of load. This is known as impedance matching. If there is impedance mismatching, part of the energy in the incident wave is reflected back giving rise to a standing wave. The ratio of the voltage of reflected wave and the voltage of incident wave is known as the reflection coefficient. It is denoted by Γ.
It is expressed as Γ= Vref/ Vinc.
The amount of reflected wave depends upon the mismatching of the source and load impedances. Therefore, from the fig., gamma can be expressed as,
Γ = (ZL - Zo) / ( ZL + Zo ) where, ZL is load impedance and Zo is the characteristic impedance of transmission line, a constant.
ZL= RL + jXL.
Normalized impedance is used for plotting on Smith chart. This is because the behavior of the transmission line depends on load impedance as well as characteristic impedance.
Normalized impedance, Z = ZL/ Zo = r+ jx, where r = R / Z0 and x = X / Z0.
Therefore, Reflection coefficient can expressed as
Γ = (Z-1) / (Z+1) or
Γ = (ZL-1) / (ZL+1) as Zo takes constant values such as 50Ω, 100Ω, etc.
Γ = [(r +jx) - 1] / [(r + jx) + 1]
Γ = [(r - 1) + jx] / [(r + 1) + jx] for r lies between 0 and ∞ , x lies between -∞ and +∞.
The above equation can also be written as
Γ = u + jv = Rejθ , where R is the radius and θ is the angle of incidence.
Z = r + jx
From the last two equation, we can see that there is one-to-one correspondence and for every Z we can obtain a unique Γ. Following fig. shows mapping of Z to Γ- plane.