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

What happens to the TM series as $n \to 0$ ? $E_{z} \propto sin (0) = 0$ vanishes, but the transverse fields need not. To see this, rewrite Eqs. (7.2.21)–(7.2.22) with a new amplitude

A_{n} = \frac{j ω ε}{h _{n}} C_{2 n}, (7.2.24)

so that $j ω ε C_{2 n} nπ / (b h_{n}^{2}) = (nπ / b h_{n}) A_{n}$ and $j τ_{n} C_{2 n} nπ / (b h_{n}^{2}) = (nπ / b h_{n}) (τ_{n} / ω ε) A_{n}$ :

E_{y} (y, z) = - n = 1 \sum \infty (\frac{nπ}{b h _{n}}) \frac{τ _{n}}{ω ε} A_{n} \times cos (\frac{nπ}{b} y) e^{- j τ_{n} z}, (7.2.25)

H_{x} (y, z) = n = 1 \sum \infty (\frac{nπ}{b h _{n}}) A_{n} \times cos (\frac{nπ}{b} y) e^{- j τ_{n} z} . (7.2.26)

$E_{z} \propto C_{2 n}$ vanishes as $n \to 0$ , but $H_{x}$ written with $A_{n}$ has the finite factor $nπ / (b h_{n}) = 1$ , and the amplitude $A_{n}$ survives the limit. The TM $_{0}$ mode ( $n = 0$ , $h_{0} = 0$ ) is the TEM mode:

E_{z} (y, z) = 0, H_{y} (y, z) = 0, E_{x} (y, z) = 0, τ_{0} = k, (7.2.27)

TEM mode of the parallel-plate guide

E_{y} (z) H_{x} (z) - \frac{E _{y}}{H _{x}} = - η A_{0} e^{- j k z}, = A_{0} e^{- j k z}, = \frac{k}{ω ε} = η . (7.2.28)

The TEM mode does not have any cutoff frequency. It propagates at every frequency.

In plain words

The TEM mode is a plane wave between the plates. Its fields are uniform across the gap, and $E \times H$ points along $+ a_{z}$ since $(- a_{y}) \times a_{x} = a_{z}$ . It is the uniform plane wave of Section 6.1: the plates sit exactly where its $E$ is normal and its $H$ is tangential, so they disturb nothing.

The TEM mode, side view, real fields at t=0: E_y=-\eta A_0\cos(\omega t-kz) and H_x=A_0\cos(\omega t-kz) with A_0 real. The electric field runs straight from plate to plate, uniform across the gap. Where \vec E points along -y, \vec H points along +x (\otimes), and (-\vec a_y)\times\vec a_x=\vec a_z. The two plates carry equal and opposite surface charges. — **Figure 7.2.4.** The TEM mode, side view, real fields at $t = 0$ : $E_{y} = - η A_{0} cos (ω t - k z)$ and $H_{x} = A_{0} cos (ω t - k z)$ with $A_{0}$ real. The electric field runs straight from plate to plate, uniform across the gap. Where $E$ points along $- y$ , $H$ points along $+ x$ ( $\otimes$ ), and $(- a_{y}) \times a_{x} = a_{z}$ . The two plates carry equal and opposite surface charges.

Common mistake · Losing

τ_{n}

or counting it twice

Rewriting Eq. (7.2.21) with $A_{n}$ and writing the factor of $E_{y}$ as $(nπ τ_{n} / b h_{n}) τ_{n} / ω ε$ . Substituting $C_{2 n} = h_{n} A_{n} / (j ω ε)$ into $- j nπ τ_{n} C_{2 n} / (b h_{n}^{2})$ gives $(nπ / b h_{n}) (τ_{n} / ω ε) A_{n}$ with a single $τ_{n}$ , as Eq. (7.2.25) shows. The check is the wave impedance: $- E_{y} / H_{x}$ must equal $τ_{n} / ω ε = Z_{T M}$ , which has the dimension of ohms only with one $τ_{n}$ .

7.2.4 The TEM mode