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Fano's Broadband-Matching Limitations 197
important in obtaining Fano's optimal solution. Good estimates are
and
a =sinh-I [ 8(1.78 -06+ 1) sin 2: ] (6.68)
b = sinh-I [ 8(1.78 -0.6 - 1) sin 2"n ] (6.69)
for b greater than zero. These were obtained by the author by studying the
optimal-solution graphs of Green (1954, pp. 66-69). They will always satisfy
the constraints in (6.59) or (6.65). The estimate of the solution for f 2 in (6.65) is
usually close enough for engineering work without iterative refinement. This is
an important observation when using programmable hand-held calculators.
Program B6-2 in Appendix B implements the Newton-Raphson iterative
procedure just described. It is a small BASIC program, and usually converges
reliably. For very large values of Q and/or bandwidth, a damping factor of
0.5 in both variable steps (lines 480 and 490) may be necessary to obtain
convergence.
Example 6.10. Example 6.9 considered an infinite matching network. Program
B6-2 may be used to obtain optimal matching solutions for finite
lowpass matching networks. What range of SWR occurs over a 50% passband
for QL = 3 and degree n = 3. 5, 8, and 50? Running Program B6-2 produces the
performance data in Table 6.3, as illustrated in Figure 6.22.
Table 6.3.
SWR Ripple Over a 50% Passband for Networks of
Varying Degrees Terminated by QL =3
QL %BW· n MinSWR MaxSWR
3 50 3 1.3833 1.5486
3 50 5 1.3343 1.4006
3 50 8 1.3095 1.3385
3 50 50 1.2828 1.2840
(J BW = bandwidth.
1 '--- ---'---''-_L-_+
o
Figure 6.22. Lowpass response showing passband
SWR ripple.