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226 Impedance Matching
Table 6.9.
Typical Carlin Piecewise Resistance
Data for Fitting Program 82-5
WO R 4
W R 4
-1.5 0.0835 0.05 2.1728
- 1.0 0.6577 0.10 2.1456
-0.5 1.5800 0.15 2.0749
-0.3 1.8628 0.30 1.8628
-0.15 2.0749 0.50 1.5800
-0.10 2.1456 1.0 0.6577
-0.05 2.1728 1.5 0.0835
0.0 2.2000
"Xq=O for all w.
obtain the Levy (1959) coefficients for an appropriate lowpass rational polynomial
having a constant numerator and a sixth-degree denominator. The
pertinent linear system of equations may be solved by Program B2-1. The
rational polynomial coefficients of s for the data in Table 6.9 are: ao~2.1819,
b o = I, b,=O, b 2 = -2.0505, b 3 =0, b 4 = -2.7689, b,=O, and b,= -3.0330.
Note that the even input data produce the required even fitting function. This
polynomial is the basis for the Gewertz procedure in Section 3.5.1, which finds
the Z4(S) = ZRLC impedance function looking into the matching network from
the load interface (see Figure 6, I).
The last Carlin step is to synthesize the ZRLC input impedance function
obtained by the Gewertz method. This has been described in Sections 3,5.3
and 3.4, The result will be a network like that shown in Figure 3.8; it is similar
to an example given by Carlin (1977). Carlin and Komiak (1979) also give a
rule of thumb for estimating the required complexity of the rational polynomial
used to fit the optimal piecewise linear resistance function; this deter·
mines the matching network complexity as well.
6.7.5. Summary of Carlin's Broadband-Matching Method. Carlin identified
at least three important concepts applicable to the broadband-matching
problem. First, a piecewise linear representation of a resistance function can
be used in a closed-form application of the Hilbert transform to find the
corresponding reactance function, assuming minimum phase. The excursions
in the piecewise linear representation occur as coefficients in a linear combination
of easily computed resistance and reactance contributions. The technique
is equally valuable for computing the transfer angle given a piecewise linear fit
of transfer magnitude.
Second, the generalized gain function at the load interface is at most a
quadratic function of the resistance excursion variables. If a classical leastsquared-error
solution were employed, a standard quadratic program would
suffice. The gain function is well conditioned in any event.