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Input ImpedafJce $ytrtlresis Fro", Its Real Part 57
different chain matrix and therefore a different network. They will all have the
same transducer magnitude function versus frequency, but their input impedance
functions will differ (see Ternes and Mitra, 1973).
Example 3.14 gave K=f/p so that p= 1. As noted in Section 3.2.4, any
roots of pes) occur on the j", axis for lossless ladder networks. A more general
case would be the K(s) given in (3.57), where the rool' of p are at W= ±2. A
lowpass function would then produce a network with "traps" to produce zero
transmission (loss peaks) at these root frequencies of pes). A very effective
method for designing networks of this sort without resorting to synthesis will
be described in Sections 9.2 and 9.3. The continued fraction expansion
described here will not suffice for the synthesis of these more general networks.
However, Ternes and Mitra (1973) provide a compact summary of
Orchard's elegant method for networks containing the four possible arrangements
of traps; the method is well suited for small computers.
Finally, as noted in Example 3.14, R, = R, is not the general case. However,
il is a fairly standard procedure to make this assumption, then derive one or
more elements by synthesis from opposing ends of the network, and then
decide (by any difference in answers) what the impedance scaling must be,
i.e., how R, is related to R,.
3.5. Input Impedance Synthesis From Its Real Part
Sections 3.2 through 3.4 developed a method of doubly terminated network
synthesis, along with the introduction of various computer aids for a variety of
engineering applications. The specification related to power transferred from a
source to a resistively terminated lossless network, and the power was relative
to the maximum available from the source. There are many situations where
the source impedance has no real part, so that the maximum power available
is infinite in theory. An equivalent case is the situation where the complex
source is connected to an unterminated lossless network, so that no power can
be transferred to the network. In either case, there is often an interest in the
output voltage function versus frequency. These cases arise from singly
terminated networks.
It is important to understand that the discussion of singly terminated
networks and the synthesis of input impedance from its real part are the same
thing. The need to realize an input impedance function might occur, for
example, in building a lumped-element dummy antenna to approximate the
real antenna behavior over a band of frequencies. Suppose that a constant
current source is connected to the singly terminated lossless network, as shown
in Figure 3.7. The input power must be P, = If,I'R in , and the power in the
output resistor must be P, = /V,/'/R,. Since the network is lossless, P, = P"
and the impedance transfer function is thus
(3.93)
Networks with only one possible signal path are called minimum-phase