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DISCONTINUITY IN FEED LINES 371

experimentally that is typically too costly, exhaustive and time-consuming, or to resort the help
of high-quality computer models. As such, we need
1. The analytical or numerical tool that allows describing uniformly a different kind of
microwave discontinuities in a wide variety of feed lines.
2. The algorithm or algorithms letting combine all discontinuities in the whole network.

In fact, we charted the path to the first problem solution by introducing and actively using the
term of reflection coefficient . Recall that this parameter was expressed as the phasor equals
to the ratio of the E-field complex magnitude of reflected mode to that of the incident wave of
the same mode. The propagation in line of multiple modes slightly complicates such description
due to possible interaction between modes making it a little bit bulky. As a result, the analysis
has to include multiple reflection and coupling coefficients for each mode. To simplify the
subsequent discussion, we limit our consideration by the case of single mode feed lines. It
means that all higher order modes are evanescent and exponentially decay as they move away
from discontinuities. If so, it is always possible to define the reflection coefficient as a complex
number not a function at arbitrary (but not very close to discontinuity) chosen cross sections of
line commonly called reference planes. The reader will find comprehensive material devoted
to this issue (as many other) in Chapter 14 of online book [3].
Our following and the slightly delicate goal will be to eliminate the unambiguous concepts of
voltage, current and impedance from the RF circuit description thereby keeping the
conventional and robust circuit technique untouchable. So let us start.

7.3.2 Generalized Scattering (S) Matrix

As we have demonstrated in Chapter 6 the traditional circuit theory concept of voltage, current
and impedance become immeasurable and ambiguous as soon as the dominant mode deviates
from TEM (see (6.9) in Section 6.1.2 of Chapter 6 and following discussion). It made us
normalize the impedances defining them straight through the measurable reflection coefficient
(check (3.89) in Chapter 3). To move forward in the same direction assume that N is the
number of single mode lines of semi-infinite length connected to each other through some
network within the volume V as it is
shown in Figure 7.3.1. Here the vector-
column of incident dominant modes of
complex magnitude =
propagates toward
( , , … , )
1
2

the reference plane =
{ , , … , } labeling as Port
1 1 2 2
= {Port1, Port2, … , PortN}. The vector-
column of reflected dominant modes
= ( , , … , ) in the same lines

2
1
travels away crossing the same
reference plane. Then the reflection
coefficients can be uniquely defined and
Figure 7.3.1 Network of discontinuities measured
as = { , , … , } =
connected to N single-mode lines ( / , / , … , / ) . 2 Remind
1

2
2
1
1
T
that the superscript indicates the