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DISCONTINUITY IN FEED LINES 375

interconnected basic electrical circuit components or lumped elements such as resistors,
capacitors, inductors, which interacts with other circuits through ports”
3
1 11 12 ⋯ 1 1
⋯ 2
= or � 2 � = � 21 22 2 � � � (7.9)
⋯ ⋯ ⋯ ⋯ ⋯ ⋯
1 2 ⋯
Let us list without proving the most important features of S-matrix of the passive network:

1. 0 ≤ � � ≤ 1.

2. = or = (all combinations of i and j ). Such network is called reciprocal. This

fact follows from Lorentz’s reciprocity theorem considered in Chapter 3 and actually the
consequence of Poynting’s theorem.
3. = . That is formal definition passive and lossless networks. Here = ( ) and I is
†
∗
†
a unit or identity matrix with ones on the main diagonal and zeros elsewhere. Remind that
superscript * means here complex conjugate. This unitary is the consequence of the power
conservation law we formulated in Chapter 3 and follows from (7.5) as = 0.
Evidently, rewriting (7.5) in matric form we obtain
†
†
2
2
∑ (| | − | | ) = − = 0 (7.10)
=1

Then replacing vector b by its value from (7.9), we will get the declared expression after
some matrix manipulations.
4. ℎ = . Here R is the diagonal matrix describing the additional phased shift
∗
∗

of incident and reflected wave as the reference plane in the connected lines moves outward
1 1 0 … 0
= � 0 2 2 … 0 � (7.11)
… … … …
0 0 …
The success or failure in computer analysis, synthesis, and even simple optimization depends
severely on the number of independent parameters that describe the elementary blocks
organized into a more complicated network. The important feature of the majority of two-port
discontinuities in Tables 7.1 and 7.2 is that they are characterized by single quantity namely by
its reflection coefficient. To show it let look closely at the unitarity of S-matrix applying it to a
2-port reciprocal ( 12 = ) and lossless network
21
∗ ∗
11 21 11 21 1 0
†
∗
= ( ) = � � � � = � � (7.12)
∗ ∗ 0 1
21 22 21 22
The matrix multiplication in (7.12) yields
2
2
| | + | | = 1
11
21
| | + | | = 1 � (7.13)
2
2
22
21
∗
∗
+ = 0
11 21
21 22
3 https://en.wikipedia.org/wiki/Scattering_parameters