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Load Effects on Passive Networks 369
follows:
(9.63)
(9.64)
(9.65)
9.4.3. Summary ofSome Lumped-Element Transfonnations. There is a large
collection of lumped-element equivalent branch topologies; a set by Zverev
(1967) is contained in Appendix H. In addition, Norton transformations may
be used to produce two- and three-pole equivalent bandpass networks that can
radically alter topologies and element values without changing frequency
response.
A first-order equivalence based on Taylor series provided several approximate
transformations involving traps that produced attenuation poles
(notches). It was shown that two elements cannot replace one on the hasis of
equal susceptance and equal slope at a frequency. This amounts to a statement
of Foster's reactance theorem (see Van Valkenburg, 1960, p. 123).
However, element values were given for (I) replacing two elements by three
and adding a notch, and (2) replacing four elements by three and removing a
notch.
9.5. Load Effects on Passive Networks
The image of the right-half load plane as seen at the input terminals of a
linear, active two-port network was the circle described in Section 7.3.2. The
Y L plane appeared as a Smith chart with complex normalization in the Y;n
plane. Filters usually have a high efficiency in the pass band and thus do not
shrink the load-plane image in the input plane at these frequencies. However,
the input image of a more limited load-plane neighborhood, typically a
constant SWR circle, is still of interest. This case occurs in such questions as
the effect of a spacecraft antenna SWR on transmitter output in the passband.
Another important question arises at filter stopband frequencies. The load
impedance often is specified at passband frequencies. However, both filter and
system designers make the assumption that the load impedance in the stopband
is either 50-ohms or is so reactive that the filter selectivity will be
increased if changed at all. The fact is that an antenna SWR in the stopband
is usually very large, and it is equally likely that filter selectivity will be
seriously degraded at certain stopband frequencies. Complaints about transmitter
excessive harmonic output in the field are common.