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Fano's Broodband-Matching Limitations 189
segment should have 20=50 ohms and the output segment should have
20=75 ohms. Evaluation of (6.41) using Program A6-2, key D shows that
each line should be 29.33 degrees long at the specified frequency.
6.2.5. Summary of Transmission Line Matching. The dissipative transmission
line ABCD parameters presented in Chapter Four were employed to
show how a clockwise spiral locus on a Smith chart models the input
impedance or reflection coefficient of a terminated transmission line as a
function of line length. Also, the same function was obtained for lossless lines.
Transformations of load to input impedance when at least one is complex
are not always possible with a single line segment. Examples for both complex
and real source impedances show that it is possible to rotate a given complex
impedance until it can be transformed to the specified resistance by a second
transmission line segment. The process was graphically illustrated for realsource
situations.
Finally, the 90-degree line transformer (inverter) was mentioned with
respect to its Smith chart behavior and importance in direct-coupled filters
(Chapter Eight). Then a simple two-segment transmission line-matching network
was described that is less than 60 degrees long and matches resistances
over narrow frequency bands. Its derivation emphasized the fact that
matched, doubly terminated lossless networks of any kind exhibit conjugate
impedance matches at any interface. Computer Program A6-1 was provided
to evaluate these important lossless transmission line relationships.
6.3. Fano's Broadband-Matching Limitations
Fano (1950) described a complete theory for the design of optimal lowpass
matching networks when the load impedance could be specified as that of
some LC subnetwork terminated by a resistance (see Zz in Figure 6.1).
Previously, Bode had given the gain-bandwidth-matching restriction for load
impedances consisting of a series LR or parallel CR. The next three topics
presented here will involve .the most practical of these load networks. They are
shown in Figure 6.16 in lowpass form.
-­
z,
: =c,
L,
< '
: =c,
lal
Ihl
Figure 6.16. Lowpass load impedance forms having two reactances. (a) Series resistance; (b)
parallel resistance.