Page 336 - Maxwell House

316 Chapter 6

We have discussed the subject of characteristic impedance in Sections 3.4.1 of Chapter 3 (see
(3.85) and (3.86)). We also noted that this parameter does not always a fixed number and can
be the functions of the transverse spatial coordinates. The get around this non-uniqueness we
introduced the normalized impedance as = (see 3.89). It works almost fine until
⁄

our analysis is restricted by a single line containing a single mode. But what if, for example,
WR is connected to coaxial cable or any other line with different characteristic impedance or
mode. How to build the equivalent circuit of such transition if there is no unique path to define
WR impedance? Before answering this questions look at the WR impedance definitions.
According to (6.8) the characteristic impedance associated with the dominant mode in WR can
be defined one of three ways as = = 2 = ⁄⁄⁄ 2 2 2. Before commenting the

possible lack of uniqueness let us mark them as
∗
∗
⁄
= ⁄ ; = 2 ; = 2 (6.36)
⁄

Since obviously = � only two of these definitions are independent. Let us start

from assuming the propagation of TE10 mode where the E-field lines begin and end on

bottom and top wide wall, correspondingly. If so, taking (, ) from (6.32) and integrating

it along the electric force line we obtain

= ∫ = sin (/) (−) (6.37)
0
0
As expected (recall the discussion around (6.9)), this voltage associated with propagating wave
is not constant and can be any value between zero and peak (−) at x = a/2. The
0
common engineering agreement is to put into (6.36) the peak voltage. Then according to (6.37)
∗ 2 0 2 2 0 Λ (6.38)
⁄
= 2 =� � �1 − ( 2) = � �

Next step is to calculate impedance. The current in (6.36) running over top wall is the

longitudinal by meaning (see Figure 6.1.2a) and can be obtained from (6.32) and Table 2.2 as
/ (−) / (−)
2
= ∫ = − ∫ = 2 0 = 2 0 �1 − ( 2) (6.39)
⁄

0
0

0
0
Then
2
= 2 = �� ∗ 2 2 0 Λ � =
� (6.40)
2 0 Λ
= � = � � =

Note that all three definitions converge to � = � = � = � 2 0 Λ � if we formally

replace in (6.38) and (6.39) the value and with � = √ and ̃ = √. Figure 6.5.6b

depicts the renormalized WR impedance � over that is measured in the range of several
⁄
hundred Ohms.
6.6.5 Waveguide Circular
As we have found in the previous section the dominant mode in WC is TE11 (see later) which
is the equivalent of TE10 mode in WR. To keep the single mode propagation in WC, its radius
r must be chosen as 2 > > . It means that the operation wavelength must exceed the
1