|
TO‘LQINLIKLAR VA BO‘SHIK REZONATORLAR / 10
|
| bet | 4/7 | | Sana | 24.12.2023 | | Hajmi | 83,74 Kb. | | #127986 |
Bog'liq 456 TO460 TO‘LQINLIKLAR VA BO‘SHIK REZONATORLAR / 10
10-4-misol Parallel-plastinkali to'lqin o'tkazgichda tarqalayotgan TM x to'lqinining maydon eritmasi ikkita o'tkazuvchi plastinka o'rtasida oldinga va orqaga qiyshiq sakrab turadigan ikkita tekis to'lqinning superpozitsiyasi sifatida talqin qilinishi mumkinligini ko'rsating.
Solution: This can be seen readily by writing the phasor expression of E?{y) from Eq. (10-54a) for n = 1 and with the factor e~jß: restored. We have
From Chapter 8 we recognize that the first term on the right side of Eq. (10-63) represents a plane wave propagating obliquely in the +z and —y directions with phase constants fl and n/b respectively. Similarly, the second term represents, a plane wave propagating obliquely in the -t-z and + y directions with the same phase constants /i and Tt/b as those of the first plane wave. Thus, a propagating TM t wave in a parallel-plate waveguide can be regarded as the superposition of two plane waves, as depicted in Fig. 10-7.
In Subsection 8-6.2 on reflection of a parallelly polarized plane wave incident obliquely at a conducting boundary plane, we obtained an expression for the longitudinal component of the total E( field that is the sum of the longitudinal components of the incident E,- and the reflected Er. To adapt the coordinate designations of Fig. 8-10 to those of Fig. 10-5, x and z must be changed to z and —y respectively. We rewrite Ex of Eq. (8 -86a) as
Comparing the exponents of the terms in this equation with those in Eq. (10-63), we obtain two equations:
(10-64b)
(10-64a)
Fig. 10-7 Propagating wave in parallel-plate waveguide as superposition of two plane waves.
io-3 / parallel-plate waveguide 461
(The field amplitudes Involved in these equations are of no importance in the present consideration.) Solution of Eqs. (10-64a) and (10-64b) gives
which is the same as Eq. (10-58), and
where X = is the wavelength in the unbounded dielectric medium.
We observe that â solution_ol Eq. (10—65) for 0,- exists only when X/2b < 1. At /Jib = 1, or f = ujX = 1/2/) which is the cutoff frequency in Eq. (10-56) for it = 1, cos 0j = 1, and 0j — 0. This corresponds to the case when the waves bounce back and forth in the y direction, normal to the parallel plates, and there is no propagation in the r direction (/? = sin 0,- = 0). Propagation of TM, mode is possible only when X < Âf = 2b or / > Jc. Both cos 0i and sin 0i can be expressed in terms of cutoff frequency /;. From Eqs. (10-65) and (10~64a) we have
and
Equation (10—66b) is in agreement with Eqs. (10-34) and (10-36).
10-3.2 TE Waves between Parallel Plates
For transverse electric waves,
which is a simplified version of
E, = 0, we solve the following equation for H?(y), Eq. (10-41) with no x-dependence.
We note that H.(y, :) ~ H!(y)e The boundary conditions to be satisfied by H?(y) are obtained from Eq. (Id—42c). Since Ex must vanish at the surfaces of the conducting plates, we require ‘
Thcrcforc the proper solution of Eq. (10-67) is of the form
( 10—68a)
|
| |
