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220 Impedance Matching
k'" 1, .. " n
'.
Slope ""
w~ W~_l
o'---__--""- -'- -.J
W~_l w~
w
Figure 6.38. Linear interpolation between sample points according to Equation (6.113).
useful choice for the necessary dependent excursion is
n-l
fn=-Lf k
o
k=O
Using (6.1\2) in (6.111), the reactance at some frequency w is
(6.114)
X(w)= 2: n rk- Ilooda -In k IY+W\ -- dy.
k~O 'II a dy y-w
(6.115)
The crux of Carlin's method is a broadly applicable linear combination of the
excursions r k that expresses the reactance function corresponding to (6.1\2):
The reactance contributions, b kl are
n
Xq(w)= 2: rkbk(w).
k=O
(6.116)
bk(w) = 1/'11 (w'lnIY+WldY. (6.117)
wk-wk_l )uJk_1 y-w
Note that the narrow limits of integration result from the single segment of ak
in Figure 6.38 having a nonzero slope. The integral in (6.117) has a simple,
closed-form evaluation, as given by Bode (1945, p. 319). Therefore, a final
2.5
2.0
R,
1.5
1.0
0.5
a a 0.5 1.0 1.5 2.0
r 3
(dependent)
Figure 6.39.
w
A piecewise linear representation of R q
with excursion variables fk'