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Network Objecti"" Functions 153
occurs when P-->oo, for suitable functions. ft is interesting to think of this
process in terms of I/P-->O, because the Richardson extrapolation to zero
considered in Section 2.3.2 for Romberg integration is also applicable here.
Thus, the minimax conditions can be predicted without actually making P all
that large. The proper extrapolation variable and other important parameters
will not be treated here; satisfactory minimax results often can be obtained by
simply setting P =2, 10, and 30 in a sequence of minimizations. This point will
be explored in the network optimization example in Section 5.5.4.
5,5.3. Objective Function Gradient. When finite differencing is used to obtain
partial derivatives, then the entire objective function-as in (5.88}­
should be employed in the difference functions. However, if partial derivatives
of the response function(s) R,.k are available analytically or, more likely, by
application of Tellegen's theorem, then (5.88) should be differentiated so that
the partial derivatives of the response function may be employed. Differentiation
of (5.88) with respect to x j
produces
aE ~ P-1 aR;
g.=-=p L." W(R-G) -.
1 aXj i= I I I I OX j
(5.89)
Again, note that response R, is a real quantity; e.g., if it is SWR and
derivatives of Z," are available, then identity (5) in Table 4.5 will be required
to express the derivative of R, needed in (5.89).
By the Tellegen method, partial derivatives of complex quantities are also
complex; thus 2N registers and additional computer coding will be required to
exploit this approach. Of course, the minimization time will be much less than
when using finite differences, because there will be no wasted calculations,
and the exact partial derivatives will speed convergence.
5.5.4. L-Section Optimization Example. The concepts in Chapter Five are
now brought together for a practical network optimization problem, which
will illustrate almost all fundamental techniques. The lowpass L section shown
in Figure 4.18b will be optimized to match a frequency-dependent load
impedance to a resistive source impedance over a band of frequencies. Design
methods for this impedance matching problem will be considered in Chapter
Six.
Appendix-B Program B5-2 is composed of Fletcher-Reeves optimizer Program
B5-1 lines 150-940; lines numbered less than ISO input data, and lines
numbered greater than 940 form an error function and its partial derivatives
(gradient vector). The general process is flowcharted in Figure 5.28a. Also, the
function and gradient computation are shown in Figure 5.28b, and the
sampled-error-function formation is shown in Figure 5.28c.
A brief discussion of Program B5-2 code should reveal the simple details.
The Land C values (in henrys and farads) are input into X(I) and X(2) by
lines 100 and 110, respectively. Line 120.inputs the value of P, which should
initially be 2. After minimization, the program is sent to this line (by line 999)