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

Table 7.2.1. The modes of the parallel-plate guide.

	TEM	TM $_{n}$	TE $_{n}$
fields	$E_{y}, H_{x}$	$E_{y}, E_{z}, H_{x}$	$E_{x}, H_{y}, H_{z}$
$n$	$0$	$1, 2, \dots$	$1, 2, \dots$
$f_{co, n}$	$0$	$n v /2 b$	$n v /2 b$
$τ_{n}$	$k$	$k^{2} - (nπ / b)^{2}$	$k^{2} - (nπ / b)^{2}$
$Z$	$η$	$τ_{n} / ω ε$	$ω μ / τ_{n}$

The number of propagating modes at the frequency $f$ is the TEM wave plus a TM $_{n}$ and TE $_{n}$ pair for every $n < f / f_{co, 1} = 2 b f / v$ . Figure 7.2.10 shows it as a chart: each bar starts at the cutoff of its mode. At the blue line, $f = 2.5 f_{co, 1}$ , five modes propagate: TEM, TE $_{1}$ , TM $_{1}$ , TE $_{2}$ and TM $_{2}$ . Single-mode (TEM only) operation needs $f < f_{co, 1}$ , that is $b < λ /2$ .

Mode chart of the parallel-plate guide. Each bar begins at the cutoff frequency of its mode pair. The vertical line at f=2.5f_{co,1} meets five modes. — **Figure 7.2.10.** Mode chart of the parallel-plate guide. Each bar begins at the cutoff frequency of its mode pair. The vertical line at $f = 2.5 f_{co, 1}$ meets five modes.

Quick check

How many modes propagate in an air-filled guide with $b = 2 cm$ at $20 GHz$ ?

$f_{co, 1} = c /2 b = 7.5 GHz$ and $f / f_{co, 1} = 2.67$ , so $n = 1$ and $n = 2$ propagate: TEM, TE $_{1}$ , TM $_{1}$ , TE $_{2}$ and TM $_{2}$ , five modes.

7.2.8 Parallel-plate modes at a glance