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Fano's Broadband-Matching Limitations 195
where m is an integer. A useful identity is (4.16), which reduces the computations
to an evaluation of real sine and cosine functions. The process of
expressing the roots results in the defining equations for two important
positive parameters, a and b:
sinh'(na)= I +~2 , (6.55)
e
sinh'(nb) = (~ )'. (6.56)
The zeros of the reflection coefficient in (6.53) are given by (6.54), with b
substituted for a. With (6.55) and (6.56) substituted into (6.53), useful expressions
are obtained for IPlm" and Iplmin, corresponding to T~(w) values I and 0,
respectively:
I pi = coshnb
(6.57)
max
cosh na '
Ipl . = sinh nb
mm sinhna·
(6.58)
The poles and zeros of pes) are available from (6.54); these are significant
because p(s) = e(s)/f(s) and p(s) = I according to (3.52)-(3.55) in Section 3.2.4.
Choosing - a in (6.54) locates the required left-half-plane poles for e(s). Fano
showed that choosing only left-half-plane zeros for p by using - b in (6.54) in
place of ± a maximizes the broadband match for ladder networks.
It is now clearly possible to synthesize the network; this could be started
from either end. Usually, synthesis is not necessary.- However, there is a
crucial relationship involving loads with a single reactance (g, in Figure 6.18).
This relationship turns out to be
2~,sin:n =sinha-sinhb, (6.59)
where the connection to g, is through the decrement (6.49).
So far there is one degree of freedom remaining: given the bandwidth and
load Q, (6.59) relates parameters a and b; one of tbem can be chosen
arbitrarily. Then the flat loss and ripple in Figure 6.21 are determined by
(6.55) and (6.56). Other orders of parameter selection for using the available
degree of freedom are possible.
6.1.3. Optimally Matching a Single-Reactance Load The objective is to use
the one degree of freedom that is available for single-reactance lowpass loads
(RC or RL) to minimize the maximum reflection coefficient (6.57) over the
band. The constraint is the relationship in (6.59), and the variables are
parameters a and b. The number of elements (n) in the networks of Figure
6.18 includes the load reactance g,. Following Levy (1964), this minimization
is determined analytically by employing a Lagrange multiplier, as described in
many calculus textbooks.