Question 10.1: A parallel-plate waveguide as shown in Fig. 10.3 has an air ......
A parallel-plate waveguide as shown in Fig. 10.3 has an air gap of d[m].
(a) Show that the TE modes expressed by Eq. (10-4b) are mutually orthogonal
such that \int_{y=0}^{y= d}{\pmb{E}_{m} \pmb{\cdot }\pmb{E}^{*}_{n} dy }= 0 for m≠ n.
\boxed{\pmb{E} = – j 2 E_{o} \pmb{a}_{x} \sin (k_{y} y) e^{- j k_{z}z}} (10-4b)
(b) Find the orthonormal set in TE modes satisfying
\int_{y=0}^{y= d}{\pmb{E}_{m} \pmb{\cdot }\pmb{E}^{*}_{m} dy = 1}.
(a) Upon using Eq. (10-4b), we write
(b) Assuming m = n in part (a) we have
\int_{y=0}^{y= d}{\pmb{E}_{m} \pmb{\cdot }\pmb{E}^{*}_{m} dy} = 2 d E^{2}_{o}Dividing Eq. (10-4b) by \sqrt{2 d E^{2}_{o}} , the orthonormal set in TE modes is
\boxed{\pmb{E}_{m} = – j \sqrt{\frac{2}{d} } \pmb{a}_{x} \sin \left\lgroup\frac{m \pi }{d} y \right\rgroup e^{- j k_{z}z}} (m = 1,2,3,…) (10-19)
If an electromagnetic field is propagating in a parallel-plate waveguide, which is known to have no longitudinal component of E, then its electric field can be expanded in terms of the orthonormal set given in Eq. (10-19).