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196 Impedil1lce Matching
The functional multiplier A is defined as
b=Aa, (6.60)
and this is substituted in the Iplmax expression (6.57). Differentiating the
resulting expression with respect to parameter a and selling this to zero, as
necessary for a minimum, yields
A= tanhna
(6.61)
tanhnb'
Similar substitution of (6.60) into (6.59), followed by differentiation, yields
A= cosh a (6.62)
coshb'
Now A may be eliminated from (6.60) and (6.61) to produce the necessary
condition for minimum Iplmax:
tanh na tanh nb
(6.63)
cosh a cosh b '
which is still subject to the"- constraint in (6.59). Note that the integral
limitation in (6.45) was not used directly in this case; however, it does indicate
that the minimum must exist.
Simultaneous solution of (6.59) and (6.63) produces the values of parameters
a and b; thus the ripple parameter € and flat-loss parameter K are
obtained according to (6.55) and (6.56). The selectivity expression (6.52) is
then known, and all matching network elements may be found, as shown in
Section 6.4.1. The two equations to be solved are transcendental and thus
nonlinear. Newton's method from Section 5.1.5 will be applied.
Equations (6.59) and (6.63), respectively, define the functions
fl(a, b) = sinh a-sinhb-26sin 2:
(6.64)
and
f 2 (a, b) = h(a) - h(b),
where the defined function h with dummy variable x is
h(x)= tanhnx .
cosh x
(6.65)
(6.66)
Solutions are obtained by determining the values of a and b that make
f l
=0=f 2
• The Jacobian matrix requires expressions for the partial derivatives
of f l and f 2
with respect to a and b. It is helpful to employ the derivative
expression for (6.66):
n - (sinh nx)(cosh nx)(sinh x)1(cosh x)
h'(x)= .
(cosh 2 nx)(cosh x)
(6.67)
Estimated changes in variables a and b to approach a solution are obtained
according to (5.37) and (5.38).
Starting values of a and b for the Newton-Raphson method are especially