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218 Impedance Matching
passbands extending to infinity on the frequency scale. This does not preclude
Fano integration to a finite limit when the load consists of lumped elements
that truncate (i.e., band limit) the response.
6.7. Carlin's Broadband-Matching Method
Figure 6.1 pictured the environment for broadband impedance matching: load
impedance Z, must be transformed by the network to some desired Zi"
function of frequency. The quality of the match was indicated by a low
magnitude of the reflection coefficient Pi. versus frequency. The difference
between designing filters and broadband-matching networks is the frequency
dependence of load impedance Z,; it is a resistor in filter design. By Fano's
classical method, Z, was assumed to represent the input impedance of an LC
subnetwork terminated by a resistance. For practical results, the lowpass
model of the load impedance must not consist of more than one or two
reactances and an end resistor, as shown in Figure 6.16.
Given some arbitrary physical load impedance modeled by impedance data
measured at several frequencies, the first-and often difficult-task in applying
Fano's method is to classify the actual load, i.e., fit it to the most
appropriate lumped-element lowpass model. For loads with bandpass behavior,
this usually requires identification of a synchronous bandpass subnetwork
and then its corresponding lowpass prototype. Furthermore, the power transfer
of the classical method is constant over the band; however, a sloped or
other-shaped response often is required.
Fano's method depends on the fact that the magnitudes of the generalized
reflection coefficients in (3.46) at any interface in a lossless, doubly terminated
network are all equal at a frequency. In fact, his reflection coefficients are
conventional, since they are located adjacent to the resistive terminations.
Carlin (1977) noted that IPql is equal to Jp,"1 in Figure 6.1. His greater
contribution was in noting that a piecewise linear approximation to R q
, the
real part of Zq= R q +jX q , enables a simple computation of X q using the
Hilbert transform. Furthermore, he showed that power transfer in terms of
generalized P. is at most a quadratic function of the R, piecewise linear function
variables. Thus a nonlinear optimization program will usually succeed in
obtaining power transfer and/or several other impedance-dependent objectives
by a piecewise fit of R q , the real part of Zq. The Gewertz method for
finding a resistively terminated lowpass network, given the real part of its
input impedance, was described in Section 3.5. By Carlin's method, such a
network would be the required matching network in Figure 6.1, where the
source impedance would be the terminating resistance.
This topic will begin by describing the basis for BASIC language
program for finding··the imaginary part of a minimum-reactance impedance
function from a piecewise linear representation of its real"part. The