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204 Impedance Matching
which may be processed using decreasing subscripts: r= n - I, ... , I. The g,
starting value comes from (6.75), with gH' = I, and (6.76):
(6.82)
The design procedure requires that parameter a be determined from either
power variation (6.80) or load reactance (6.82). Then (6.81) determines all
element values, including the dependent g,. Often, g, is also specified; the
problem has no solution if the calculated g, is not at least as large. By the
duality principle, this method may be extended to the zero-impedance (ideal
voltage) source or load problem.
Example 6.14. Consider the singly terminated network in Figure 6.25 for
n = 5. Suppose that g, = 0.8 and g, = I/o,= 0.59. By (6.82), sinh a = 0.5238.
Then (6.81) yields g,,= 1.2668, g3= 1.5743, g,= 1.6014, and g, = 1.3868. The
computed g, is greater than the given g, by 0.5868 farads. This shunt padding
element is placed at the matching network's input in a manner similar to the
arrangement in Figure 6.24. The power variation will be 1.03: I, or 0.11 dB,
according to (6.80).
6.4.4. Summary of Broadband Matching Under Three Source Conditions.
The topic of load impedances consisting of one resistor and one reactance has
been considered. The sources considered had just one resistor, or an addi·
tional reactance, or a reactance and no resistance. The source condition
determined the relationship of parameters a and b. They were found by
Fano's transcendental optimal equation, from specified termination decrements,
or by equating them so that one decrement was zero.. Lowpass
prototype element values were obtained for each case by a well-known
recursive relationship that avoids network synthesis. This is sometimes called
"direct design," since closed forms determine element values.
Program B6-2, which iterated Fano's transcendental solution, was extended
by adding the prototype element recursive equation. The dependent source
resistance was also calculated. Programs for sources incorporating a single
reactance would be quite similar; anyone of these would fit in a conventional,
hand-held, programmable calculator. The only complexity arises from the
order in which prototype elements must be calculated. The resistive source
case works from load to source; the load reactance is g,o The single-reactancesource
case works from the end with the lesser decrement associated with gJ.
The singly terminated (ideal or lossless source) case works from load to
source, but the prototype element gn is always the load reactance, so that the
elements are computed in the order of descending subscripts; !he source
reactance is dependent. The last two cases involve the possibility that the
source reactance may need to be increased to obtain the best solution. This is
accomplished by increasing the g, value (by making part of g, the input
element in the matching network).