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Loss/ess Unifonn Transmission Lines 183
r-
z, l
= z;
-----~
Figure 6.11.
The transmission line-matching problem.
convenient graphic design aid for trend analysis. Finally, two important
techniquesl'or matching real loads to real sources will he discussed. The
inverter (90-degree line) will be a main feature in Chapter Eight.
6.2.1. Input Impedance and Reflection Coefficient. The input impedance of
any two-port network was given in terms of its ABCD (chain) parameters in
bilinear Equation (4.18). The ABCD parameters for uniform, dissipative
transmission lines were given in (4.13)-(4.15). It is quite easy to show that
these lead to the following expression for 2\ in Figure 6.11 :
2 0 (2,+ 2 0 tanh yf)
2, = (20+2, tanh yf) , (6.23)
where ye=NP+jO; i.e., e is the transmission line length and NP is the loss in
nepers for this length.
It is useful to retain the dissipative factor in order to show the general
applicability of the Smith chart as a transmission line model. The reflection
coefficient in (6.22) can be normalized to 20 instead of R\ and then solved for
2,/Zo:
or (6.24)
A similar expression relates 2,/2 0 and p,. The expressions for 2\/2 0 , 2,12°'
and the identity
e-yr _e--yf
tanhye= yl ~yr
e +e
can be substituted into (6.23) to identify the reflection relationship
PI =P2e-2yf =P2e-2NPe-j20.
(6.25)
(6.26)
The Smith chart in Figure 6.10 was described as the reflection plane.
Certainly, load impedance 2, corresponds to a point p, on the Smith chart. In
polar form, the P2 angle traditionally is measured counterclockwise from the
real p axis in Figure 6.10. When 2, terminates a transmission line of length e
as in Figure 6.11, the input reflection coefficient corresponding to impedance
2\ is computed by (6.26). It shows that the angular length of the line is
measured from the p, radial in a clockwise direction with twice the angular
units on the chart plane. For dissipative (lossy) transmission lines, the Smith