Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Two perfectly conducting plates at $y = 0$ and $y = b$ , infinite in $x$ and $z$ , are filled with a lossless dielectric ( $ε = ε_{r} ε_{0}$ , $μ = μ_{r} μ_{0}$ ), and the wave travels along $z$ (Figure 7.2.1). Two facts simplify the analysis at once.

Nothing varies along $x$ : $\partial / \partial x = 0$ .
The boundary conditions on the plates are those of a perfect conductor: the tangential $E$ vanishes, that is $E_{x} = E_{z} = 0$ at $y = 0$ and $y = b$ , and the normal $H$ vanishes, $H_{y} = 0$ there.

With $\partial / \partial x = 0$ the four formulas, Eq. (7.1.11) in Section 7.1, give immediately

TE (E_{z} = 0) : H_{x} = 0, E_{y} = 0, TM (H_{z} = 0) : H_{y} = 0, E_{x} = 0, (7.2.1)

so that a TE mode has only $E_{x}$ , $H_{y}$ , $H_{z}$ , and a TM mode only $E_{y}$ , $E_{z}$ , $H_{x}$ .

(a) The guide, extending without limit in x and z. — (a) The guide, extending without limit in $x$ and $z$ .

(b) Side view, the plane x= const. — (a) The guide, extending without limit in $x$ and $z$ .

Why study it

The parallel-plate guide is the model of every two-plate structure: the stripline, the power and ground planes of a printed board, the EMC test cell (Application 7.2.1). Its modes are the simplest example of the general theory: sines and cosines in $y$ only.

7.2.1 The parallel-plate guide