Ryan LeFebre
Terminated Lossless Line
 The ratio of voltage to current at z=0 must be ZL, to satisfy
this waves may be reflected
Terminated Lossless Line
 Total voltage on the line:
 Total current on the line:
Terminated Lossless Line
 Total voltage and current at the load:
 Solving for the reflected voltage:
Terminated Lossless Line
 Reflection coefficient:
 Total voltage and current:
Terminated Lossless Line
 We know the characteristic impedance of the transmission
line (Zo) and the load (ZL) but what is their impedance
together?
Terminated Lossless Line
 Transmission line impedance equation:
Impedance Matching
 Impedance matching is important for a number of
reasons. One example, maximum power is delivered to
the load when it is matched to the line
 As long as the load impedance has a positive real part,
a matching network can always be found
Quarter Wave Transformer
 Characteristic impedance of the feedline and load do
not match
Smith Chart
 Invented by Phillip Hagar Smith (1905-1987) in late
1930s at Bell Labs
 Graphical aid to assist in
solving transmission line problems.
Smith Chart
 Based on a polar plot of the voltage relfection
coefficient, Γ
 Let
 |Γ| is plotted as a radius (|Γ|≤1) from the center of the
chart
 θ (-180 ≤ θ ≤ 180) is measured counterclockwise from the
right hand side of chart
 Normalized quantities are generally used
Smith Chart
 If a lossless line Zo is terminated with a load
impedance ZL
 z is the normalized impedance
Smith Chart
 Using real and imaginary parts
 Reducing two real equations and rearranging
Smith Chart
 Family of orthogonal circles in the Гr, Гi plane
 Resistant circles have centers on horizontal Гi=0 axis
and pass through Г=1 on right hand side of chart
 Reactance circles lie on the vertical Гr=1 line (typically
off chart) and also pass through Г=1
 The Smith chart simultaneously represents both
a value of z and Г
Impedance Matching
 Real lossless feedline with a complex load?
Impedance Matching
 Turn the complex load real by adding a length of line
to it
 How do you find the length?
Smith Chart
 Recall generalized reflection coefficient
 This is the same form as
 If you plot the reflection coefficient at the load, the
normalized input impedance seen looking into a length l of
transmission line terminated with z can be found by
rotating the point clockwise by an amount 2βl around the
center of the chart
Impedance Matching
 Once the load is real, a quarter wave transformer can
be used