Page 392 - Maxwell House

372 Chapter 7

matrix transpose. To avoid any troubles with the impedance definitions, we are going to build
the following circuit matrix description on the concept of measurable power. According to the
equations (3.83) and (3.86) in Chapter 3 the density of active power carried by the harmonic
th ± 1 ± ± ∗
incident or reflected wave through the port of n line is equal to = ∙ ( ) =

2
1 ± 2 2
| | � [W/m ]. Recall that is the wave impedance of the dominant mode. Note that
2
the field subscript tan is implied but omitted while the superscripts in and ref are replaced by
sign + and -, respectively, to shorten the notations. Then the measurable incident or reflected
th
power [W] passing the n line cross section is
2
± 2
±
±
±
= ∫ = 1 ∫ | | = � � / [W] (7.2)

0
Here � � [V/m] is the E-field RMS (Root Mean Square) magnitude of incident and reflected
±
0
propagating mode, respectively, is an area of line cross-section, while is the constant

determined by the E-field dominant mode structure of 1V/m RMS intensity. Evidently, this
constant is the given and unique parameter for any mode in feed line. From (7.2) follows that
each constant is measured in [Ohms/m ] thereby the product can be interpreted as the
2

line impedance = . Therefore, we could define the incident and reflected wave

magnitudes from (7.2) directly through the average carried power density / as
±

| | = � = | |/� / , | | = � = | |/� / �√W� (7.3)
−
+
−
+

0
0
Finally, the module of reflection coefficient | | = | | | | becomes the number that is
⁄

independent on line design and its dominant mode field structure. If so,
= , = 1,2, … , (7.4)

− +
±
Here = | | ( − ) and is, respectively, the phase of the incident or reflected wave

measured at the reference plane . Evidently, the balance of active power in a passive (not

containing RF sources inside V) network takes the form
2
−
+
2
∑ − ∑ = ∑ (| | − | | ) = (7.5)
=1 =1 =1
The expression (7.5) is an undoubted observation that the total power leaving any passive
network should not surpass the total power delivered by external sources. If so, the possible
difference may be explained by the internal power absorption . The matter that the
magnitudes in (7.3) have the unusual unit dimension √W is irrelevant as long as we do not break
the energy conservation law (7.5) and “minor punishment” for such universal description.
Now we are ready to describe the network formally in Figure 7.3.1 assuming its linearity.
Loosely speaking, the latter means that some alteration of the signal in any part or network
causes proportional signal variations through the whole network. The set of linear equations
reflects this proportionality
= + + ⋯ +
12 2
11 1
1
1
2 � (7.6)
= + + ⋯ +
2 21 1 22 2
…

= + + ⋯ +

2 2
1 1