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Fono's Broadband·Matching Limitations 199
identifying which lowpass LC network terminated by a resistor corresponds to
the physical load being matched. It Was suggested that the methods described
in Section 2.5 and Chapter Five were applicable; here, it is assumed that the
Fano load network is known. For practical applications, his lowpass network
structures are terminated by no more than a series L, followed by a parallel
RC or a parallel C, followed by a series RL, i.e., one- or two-reactance
lowpass loads.
Fano's bandwidth limitations apply to lossless, doubly terminated networks;
i.e., they have resistors on both ends. Then the magnitude of the
generalized reflection coefficient in Section 3.2.3 must be constant at a
particular frequency at all network interfaces, especially at their input and
output ports. Certainly, a small input reflection coefficient magnitude corresponds
to a good input impedance match. Fano showed that the integral over
all real frequencies of the return loss is equal to simple functions of the load
components. The ideal reflection coefficient behavior would be some small
constant value over the frequency band of interest, and unity (complete
reflection) at all other frequencies. Then the integration of this constant
provides a simple estimate of the best-possible matching using an infinitely
complicated matching network (given the one- or two-reactance-Ioad network).
The classical load parameter was defined as the load decrement; it is
the ratio QBW/QL' where QBW is the geometric-mean bandpass frequency
divided by the bandwidth, and QL is the series X/R or parallel R/X at the
band mean frequency. For the lowpass case, the decrement is equal to l/QL'
computed at the band-edge frequency.
An equal-ripple approximation to the ideal "box" shape for the reflection
frequency function is obtainable as a Chebyshev function; it was defined as a
transducer function and converted into a reflection function according to
Section 3.2.3. The expression for the s-plane poles and zeros of the rational
reflection function was given in terms of the two defined parameters a and b.
The maximum and minimum values of the reflection magnitude were derived
from the equal-ripple Chebyshev function. Because the poles and zeros of the
reflection coefficient were available, it was noted that matching network
synthesis was possible. However, for the present application, this was mentioned
only to justify the first stage of such a synthesis, which could produce
an algebraic expression for the first load reactance. This expression is a
constraint on the reflection relationship, which still leaves one degree of
freedom in the matching analysis.
One application for the single degree of freedom of single-reactance loads is
to minimize the maximum passband reflection coefficient magnitude while
satisfying the load reactance constraint. One function was obtained by using a
Lagrange multiplier to minimize analytically the maximum reflection magnitude;
the constraint was a second function. Then Newton's method was
applied to solve the two nonlinear functions for the values of parameters a
and b. Expressions for their starting values were given; these are sufficiently
close to a solution so that the Newton iteration may be dispensed with when