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ImpedanCi! Matching
only hand-held computers are available. BASIC language Program B6-2 was
provided to determine optimal matching performance. A comprehensive
graph of SWR/reflection versus load decrement for varying network degrees
was obtained. It is clear that the number of lowpass network components,
including the single-reactance load, should not exceed about eight, because of
rapidly diminishing returns. There are other ways to use the single degree of
freedom available for single-reactance loads. These will be exploited in Sections
6.4.2, 6.4.3, and 6.6.
6.4. Network Elements for lbree Source Conditions
The network LC-element values will be determined in this section by recursive
formulas. Three types of sources will be considered (see Figure 6.18). First, the
resistive source consisting of ~ + I will be considered, as originally assumed;
the first matching network element is then gn' Second, a single-reactance
source, consisting of both gn.1 and gn' will be specified so that the first
matching network element will he gn-I' Finally, an ideal current source will
be considered. In all of these cases, the load will have a single reactance,
namely g, in Figure 6.18.
The poles and zeros of the reflection coefficient have been given as
functions of design parameters a and b (see (6.54)). It was noted that the
synthesis of the network element values by the methods described in Chapter
Three is straightforward but tedious. However, Green (1954) carried out
detailed calculations for cases of low degree and guessed an expression for
element values of networks of any degree. It has since been discovered that
Takahasi published a complete derivation and proof of general results in
Japanese in 1951; the interested reader is referred to Weinberg and Slepian
(1960). The closed formulas for element values are easily evaluated once the
single degree of freedom is assigned; i.e., parameters a and b are chosen.
6.4.1. Resistive Source Optimally Matched to a Single-Reactance Load. A
resistive source optimally matched to a single-reactance load is illustrated in
Figure 6.18, where the source real part is gn + I; the matching network includes
g. through g" and the load consists of g, and 150= I. For the lowpass network
in Figure 6.18 with a passband edge at I radian, (6.49) and (6.59) yield
where angle 0 is
=2 sinO
gl sinha-sinhb'
(6.70)
0= 2"n' (6.71)
Parameters a and b are found approximately, from (6.68) and (6.69), or
exactly, by the Newton-Raphson iterative procedure in Section 6.3.3.