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DISCONTINUITY IN FEED LINES 377

Pay attention to the asymmetrical definition of input and output column vectors that slightly
simplifies the matrix element definition (see footnote below). Evidently, the expression (7.16)
means that

= ( ⁄
= + � ���������� � 1 12 ) + ( − ⁄ ) 2 (7.17)
1
12 2
22
11 2
22
21 12
1
11
= + = (1 ) − ( ⁄ )
⁄
21 2
2
22 2
22
21
22
1
2
1
Looking back at the expression (7.6), we can come to conclusion that the components of T- and
S-matrix are connected as
⁄
11 = ⁄ ; 12 = − ⁄ ; 21 = 1 ; 22 = − ⁄ (7.18)
21 12
22
22
12
22
11
21
22
Therefore,
⁄
11 = 12 − ⁄ ; 12 = ⁄ ; 21 = − ⁄ ; 22 = 1 (7.19)
22
11 22
21
21
11
21
21
Notice that the elements of T-matrix might reach enormous values (as | | → 0 for high
21
reflective or extremely lossy circuit) in contrast to S-matrix. Clearly, such nuisance might create
some difficulties for numerical analysis based on T-matrix. Taking into account (7.15), the
reciprocal and lossless 2-port network can be described as
− 11 − 2 11
Symmetrical, i.e. 22 = and = � 11 �
11
�1−| 11 | 2 − 11 1
− 11 2 11
Asymmetrical, i.e. 22 = − and = � 11 � (7.20)
11
�1−| 11 | 2 11 1
Now assume that only two networks A and B are connected as Figure 7.3.3b demonstrates.
Then
2 2
1
� � = � 11 12 � � � and � 1 � = � 11 12 � � � (7.21)

1 21 22 2 1 21 22 2

Meanwhile, as soon as two networks are connected by the line of the same impedance, 2 =

and 2 = 1 4 , i.e. ( , ) = ( , ) , and thus
1
2
1
1
2
2
1
� � = � 11 12 � � 11 12 � � � (7.22)

1 21 22 21 22 2
Repeating this procedure, we obtain for T-matrix of the cascade network in Figure 7.3.3a
= ⋅ ⋅ … (7.23)
3

1
2
Therefore, the scattering transfer matrix of the composite network is just the matrix product
of the scattering transfer matrices of the networks in cascade. We shall see later (the expression
(7.33)) that the straightforward S-matrix transformation for the cascade network is possible but
not so trivial. Nevertheless, it can be done for any practically possible combination of networks
while the T-matrix technique is not so powerful. Notably, the connected in cascade networks
must have the equal number of input and output ports. Besides, the physical interpretation of
T-matrix elements is a little bit vague. According to (7.19) only coefficient has the clear
22

4 Now it became clear why we use the asymmetrical definition of input and output vectors