download searchable PDF of Circuit Design book - IEEE Global ...

224 Impedance Matching
From (6.116),
ax,
-a- =bk-b,.
rk
(6.135)
6. 7.3. Optimization ofthe Piecewise Resistance Function. The preceding objective
function has been incorporated in the Fletcher-Reeves optimizer in
Program B5-1. The result is Program B6-6 in Appendix B. The input section
through program line 140 loads the breakpoint data required for the Hilbert
transform calculation of reactance from resistance, as in Program B6-5. All
but the last resistance excursion become the optimizer variables. The objective
function and its gradient are assembled by subroutine 1000 in lines 1000­
1260; this requires appeal to subroutine 3000 at every sample frequency to
compute Zq=Rq+jXq. Lines 3020-3040 set constraint (6.114), and lines
3050-3250 perform the Hilbert transform calculations as in Program B6-5.
Example 6.21. Input the data in Table 6.7·into Program B6-6 to obtain the
optimum resistance excursions for a gain of 1.0 at the four sample frequencies.
The program output is shown in Table 6.8.
The optimized excursions are ro = 2.2754, r, = - 1.0603, r 2 = - 1.1167, and
(constrained) excursion r 3 = -0.0984. Inspection of Figures 6.1 and 3.8 shows
that ro is the eventual generator resistance. If it is desirable to hold this at a
certain value, e.g., 2.5 ohms in this case, then all that is necessary is to add the
statement "1225 G(I)=O" to Program B6-6. A rerun of Example 6.21 shows
how the zero gradient holds the first optimization variable at its initial value.
The choice of starting excursion values is somewhat arbitrary. Carlin (1977)
suggests assuming reactance cancellation and setting the residuals to sustain
the dc gain at the in-band breakpoints.
6.7.4. Rational Approximation and Synthesis. At this point in Carlin's
broadband-matching method, an optimal piecewise linear representation of R,
is known. The remaining task is to realize a network that provides this
behavior. This is clearly the subject treated in Section 3.5. The Gewertz
method considered there began with a rational function of input resistance in
the form of (3.94), or (3.98) in particular. It is always in powers of w' or the
equivalent powers of S2, since resistance is an even function of frequency. The
next step in Carlin's method is to fit such a rational function to the piecewise
linear representation. This can be accomplished by the method in Section 2.5.
A table of impedance versus frequency and the form of the desired rational
polynomial were required in Section 2.5. In the Carlin method, the table of
data is created from the piecewise linear resistance function by (I) using
symmetric positive and negative frequencies for the even resistance function
and by (2) using zero reactance values at every sample. A typical data set is
given in Table 6.9. The data in Table 6.9 can be input into Program B2-5 to