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and (6.34) can be written as
Lon/ess Unifonn Transmission Lines 187
( Zo)2=r_ L .
R J
. )-r
(6.37)
It is useful to plot loci of constant B and constant Zo/R J
from (6.36) and
(6.37) on the normalized Z2 plane, i.e., on an r+jx Smith chart grid. Jasik
(1961) has done so, as shown in Figure 6.13. This Smith chart is oriented
differently from that in Figure 6.12 (90 degrees clockwise), so that the negative
half-plane is on the left. Also, the chart perimeter scale for rotation from the
load toward the generator is in CiA-the fraction of a wavelength (360
degrees}--and the electrical length of the transmission line, (6.36), is similarly
labeled. Feasible solutions must be within either of the two circular areas.
Example 6.7. Suppose that an impedance of 15-j35 ohms must be matched
to 50+jO ohms. Normalizing Z2 and the Smith chart to R J
= 50 ohms gives
r+jx=0.3-jO.7; this is plotted on the left side of Figure 6.13 at the intersection
of the two circular coordinates. The corresponding wavelength scale reads
0.398. Using a compass, this point is rotated clockwise until it is within either
circular area. Suppose that the initial point is rotated at that radius to the
point corresponding to a wavelength scale reading of 0.474; this is the point
also corresponding to Zo/R J = 0.4 and ell. = 0.325 as shown. Thus, the 50­
ohm line rotation must be (0.474- 0.398) X360= 27.36 degrees. Then the chart
indicates that impedance (0.2 - jO.16) can be matched by a 0.4 X50 = 20-ohm
transmission line that is 0.325 X 360 = 117 degrees long. This network is shown
in Figure 6.14; there is an infinite number of other feasible solutions.
~
50+jO n
50 S1 27.36 0 15-j35 n
Figure 6.14. One transmission line network that solves Example 6.7.
6.2.4. Real-to-Real Transmission Line Matches. There are two transmission
line-matching cases that deserve special mention. The most important is a
lossless, 9O-degree line called an impedance inverter. As 1!~90 degrees, (6.28)
shows that y~oo. Also, (6.27) then shows that the input impedance of a
90-degree line is
Z6
ZJ = z . (6.38)
2
A 90-degree line is equivalent to 0.25 wavelengths, or half a Smith chart
rotation, as shown in Figure 6.13. Although (6.38) is true whether or not Z2 is