Page 398 - Maxwell House

378 Chapter 7

meaning of transfer function and thus can be measured directly by the Network Analyzer. All
others should be recalculated after the S-matrix data become available.
7.3.5 Z- and Y-matrix

We know very well that it is practically impossible to use traditional volt- and ampere-meters
at relatively high frequencies above several hundreds of megahertz. Furthermore, as we have
mentioned in Chapter 1 even at lower frequencies behind any voltage or current measure lays
the measurement of a small portion of power. If so, do we really need more matrices besides S
and T? So-called impedance matrices Z and Y proved their usefulness making applicable well-
familiar classical Kirchhoff's laws (KCL) [3]. The first obvious step is to define some equivalent
voltages and currents through the power carried by EM waves. To do so look back at the
definitions in (7.3) and rewrite one of them as
| |� = �2 = | |� [V] (7.24)
+
+

0

The term on the right-hand side of this equation is measured in Volts (� mesured in [m])

th
and thus can be interpreted as the RMS magnitude of incident voltage | | = | |� in n
+
+
0

± 2
±
line. If so, = | | ⁄ as the classical circuit theory requires. Defining the same manner

the current magnitude, we obtain
±
± ±
= � = � ∙ � [V] ⎫ (7.25)

± ⎬
± ±
= � / = �1/ ∙ � − [A] ⎭

Writing (7.25), we took into consideration that the incident and reflected waves travel toward
each other. Consequently, the products = ( ) = | | and = ( ) = −| |
2
+ ∗
+
+
− ∗
−
2
−

have opposite signs as it is required by Poynting’s theorem. We have already discussed this
issue in Chapter 3 and 4. Since both the incident and reflected waves run simultaneously in the
same line the resulting voltage and current is the superposition
−
+
= + = � ( + ) � (7.26)

+
−
= + = �1/ ( − )

These equities allow linking the voltages and currents at various ports through the so-called
mutual impedances
−1

= = � � − � ∙ ( + ) (7.27)

Therefore, we can introduce the impedance matrix as = where elements of Z-matrix are
described by (7.27) and establish through (7.27) connection with S-matrix as
−1
= ( − ) ( + )

� (7.28)
= ( − )( + ) −1

Here is the impedance matrix normalized to the impedances of lines connected to network

ports. For simplification, all of them are assumed equal. Likewise, the admittance matrix =

−1
−1 = ( + ) ( − ). It is worth to note, there are some difficulties impeding numerical