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182 Impedance Matching
Example 6.1, which involves negative reactance and susceptance travel on the
Smith chart. Negative travel is toward the bottom half of the chart instead of
toward the top half (positive half-plane).
It is strongly recommended that the reader write a small program to
accomplish the calculation of (6.22) and its inverse, so that Smith chart
plotting is simply a matter of locating rectangular and/or polar computed_
numbers. Cases involving complex sources may be treated as in the analysis
above and in Example 6.4. However, a more general treatment will be offered
in the next chapter, where the chart center can represent a complex number.
Many engineers associate the Smith chart with transmission line solutions; this
will be shown in Section 6.2.
6.1.6. Summary of L, T, and Pi Matching. The topic of L, T, and pi
matching began with a comment on the fact that lossless matching networks
exhibit conjugate impedance matches at every interface because of conservation
of power. Then, functionally similar equations were given and verified for
solving T and pi resistive matching network problems in terms of the currenttransfer-angle
parameter. The two possible L-section configurations were
treated as special cases of the T and pi configurations when the output branch
was omitted. A small BASIC language program was provided to calculate
element reactance at an assumed frequency.
Series-to-parallel impedance conversions and parallel combination of reactances
were described in order to always work with impedances as opposed to
mixed impedance/admittance units. The former strategy has been found
superior because engineers more readily recognize practical ranges of elements
in a single unit of measure. A hand-held computer program was provided for
these simple relationships, and examples were worked. These tools are vital
parts of the complete set of solutions obtained for an example that involves
both complex load and complex source, utilizing L-section matching networks.
Finally, a brief comment was provided on the value of graphic visualizations
in general and the Smith chart in particular. A much more general
treatment of the Smith chart will be furnished in the next chapter.
6.2. Llssless Uniform Transmission Lines
The matching network in these sections will consist of a lossless, uniform
transmission line, as shown in Figure 6.11. The load impedance Z, and the
desired input impedance Zl are given; the unknowns are the real transmission
line characteristic impedance Zo and electrical length O.
An expression for the input impedance Z, will be derived from Chapter
Four equations. A related reflection equation will be derived for relationship
to the Smith chart. The lossless case will then be examined to produce
solutions for a complex source and a complex load. A more simple result will
be obtained for the case of a real source and complex load; this will result in a