Page 442 - Engineering Electromagnetics, 8th Edition

424 ENGINEERING ELECTROMAGNETICS

in in

Figure 12.5 A three-interface problem in which input
impedance η in,a is transformed back to the front interface
to form input impedance η in,b .

The procedure in this section for evaluating wave reflection has involved calcu-
lating an effective impedance at the first interface, η in , which is expressed in terms of
the impedances that lie beyond the front surface. This process of impedance trans-
formation is more apparent when we consider problems involving more than two
interfaces.
Forexample, consider the three-interface situation shown in Figure 12.5, where
awaveis incident from the left in region 1. We wish to determine the fraction of the
incident power that is reflected and back-propagates in region 1 and the fraction of the
incident power that is transmitted into region 4. To do this, we need to find the input
impedance at the front surface (the interface between regions 1 and 2). We start by
transforming the impedance of region 4 to form the input impedance at the boundary
between regions 2 and 3. This is shown as η in,b in Figure 12.5. Using (36), we have

η 4 cos β 3 l b + jη 3 sin β 3 l b
η in,b = η 3 (47)
η 3 cos β 3 l b + jη 4 sin β 3 l b
We have now effectively reduced the situation to a two-interface problem in which
η in,b is the impedance of all that lies beyond the second interface. The input impedance
at the front interface, η in,a ,isnow found by transforming η in,b as follows:

η in,b cos β 2 l a + jη 2 sin β 2 l a
η in,a = η 2 (48)
η 2 cos β 2 l a + jη in,b sin β 2 l a
2
The reflected power fraction is now | | , where
η in,a − η 1
=
η in,a + η 1