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MORE COMPLICATED ELEMENTS OF FEED LINES 399

1. Following the even- and odd-mode principle, we can represent = 1 = 05 + 0.5
1
and = 0 = 0.5 + (−0.5) and consider two excitation cases illustrated in Figure 8.2.8c.
3
2. The first case corresponds to port1 and port3 being excited equally and in phase. Such
even-mode as the whole structure cannot exist behind WRs and propagate over WR of
double width. To verify this consider the distribution of a longitudinal component of
magnetic field. If so, this even-mode should transfer itself into one or multiple modes of
double width WR. Therefore, the break in metal wall plays the role of a discontinuity
creating the reflecting waves in port1 and port3. The matching rod shown in Figure 8.2.8b
creates the other reflected waves. Evidently, they go back to these ports and could diminish
the mismatch and improve the isolation simultaneously. Remind that in a forward hybrid
port3 must be isolated.
3. We can interpret the transformation of the even-mode as the process of radiation EM waves
from port1 and port3 into the double width WR. Due to the full symmetry, the central
radiation peak of such aperture antenna must be in the plane of symmetry. If so, all the
excited TE-modes must have E-field maximum in this plane, i.e. carry the odd first
subscript like TE10, TE30, and so on. Looking back at the expression (6.21) in Chapter 6,
we see that TE30-mode propagation can be prevented if < 2 < 3/2 that we are going
to assume. According to (6.21) the propagation constant of remaining TE10-mode in
10
2
⁄
the WR of double width, i.e. 2, is equal to 10 = �1 − ( 4) .
4. The situation with the odd-mode excitation is much easier. Looking back at Figure 8.2.8c,
we see that this mode has the E-field structure of TE20-mode. Therefore, it follows to the
WR of double width without any transformation and reflection. According to (6.21) the
propagation constant of TE20-mode in the WR of double width is equal to 20 =
20
�1 − ( 2) .
2
⁄
5. Both modes propagate over the slot getting the different phase shift and . Here L
10
20
is the slot length. Reaching the end of slot, the double width WR TE10-mode breaks up into
two TE10-modes of equal magnitude and phase following to port2 and port4. Meanwhile,
TE20-mode breaks up too into two TE10-modes of equal magnitude following to port2 and
port4, but it induces these ports in opposite phase. Therefore, the total magnitude in port3
is = 0.5( − 10 + − 20 ) = − cos[( 10 − )/2] while in port4 is =
3
4
20
0.5( − 10 − − 20 ) = − − sin[( 10 − )/2]. Here = ( 10 + )/2.
20
20
6. Our final task is design 3-dB hybrid meaning that it should be | | = | |
3
4
or cos[( 10 − )/2] = sin[( 10 − )/2] and thus tan[( 10 − )/2] = 1. It
20
20
20
turns out that ( 10 − )/2 = /4 or the equity 2 = /( 10 − ) defines the slot
20
20
length at some central frequency.
7. Working out the equations above, we can evaluate the phase shift between fields in port2
and port4: / = − = −/2 at the same central frequency. Therefore, the short-slot
4
2
WR hybrid can be classified as a quadrature 3-dB coupler.
8. The reciprocal principle tells us that the same excitation of port2 and port4 in magnitude
and (−90°) phase shift means that the doubled power goes into port1 leaving port3
isolated. Switch this phase to +90° and all power appears in port3 with port1 isolated.
Such phase-manipulation technology lets develop electronically controlled and ultrafast
switches for high and extremely high power.
9. The simple CST model was developed based on these equations. The most impressive
result is shown in Figure 8.2.8d. The plot demonstrates the active part of Poynting’s vector,
i.e. the direction and density of active power flow at the central frequency. The EM power