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

Take a strip of width $w$ of the guide, and neglect the fringing at its edges ( $w ≫ b$ , Figure 7.2.5).

The voltage of the top plate relative to the bottom one is $V = - \int_{0}^{b} E_{y} d y = η A_{0} b e^{- j k z}$ .
The surface current on the top plate, with the normal $a_{n} = - a_{y}$ pointing into the gap, is $J_{s} = a_{n} \times H = (- a_{y}) \times A_{0} a_{x} e^{- j k z} = A_{0} e^{- j k z} a_{z}$ (the boundary condition of Section 4.2.3), so the current on the strip is $I = A_{0} w e^{- j k z}$ . The bottom plate carries $- I$ .

The ratio $V / I$ does not depend on $z$ : it is the characteristic impedance of the line. The inductance and capacitance per unit length are those of the parallel-plate geometry of Chapters 3 and 4.

The parallel-plate transmission line

Z_{0} = \frac{V}{I} = \frac{η b}{w}, L^{'} = \frac{μ b}{w}, C^{'} = \frac{εw}{b}, P = \frac{1}{2} Re {V I^{*}} = \frac{1}{2} η ∣ A_{0} ∣^{2} b w . (7.2.29)

In plain words

The TEM mode is exactly the wave on a two-conductor transmission line. Check: $1/ L^{'} C^{'} = 1/ μ ε = v$ and $L^{'} / C^{'} = η b / w = Z_{0}$ . The power $\frac{1}{2} η ∣ A_{0} ∣^{2} b w$ is the same as the integral of $\frac{1}{2} ∣ E_{y} ∣∣ H_{x} ∣$ over the cross-section $b w$ .

Cross-section of a parallel-plate line of width w, fringing neglected. The TEM field is the uniform electrostatic field of the plate capacitor. The top plate is at the voltage V and carries the current I along z, the bottom plate the return current. — **Figure 7.2.5.** Cross-section of a parallel-plate line of width $w$ , fringing neglected. The TEM field is the uniform electrostatic field of the plate capacitor. The top plate is at the voltage $V$ and carries the current $I$ along $z$ , the bottom plate the return current.

The bridge to circuits

$V$ and $I$ are well defined for the TEM mode because its transverse $E$ is a static-like field (Section 7.1.3): the integral of $E$ from one plate to the other does not depend on the path in the cross-section. TM and TE modes have $E_{z}$ or $H_{z}$ , so the voltage between the plates depends on the path, and they have no unique voltage.

Quick check

A plane pair on a printed board has $b = 0.2 mm$ of FR-4 ( $ε_{r} = 4.4$ ) and is $w = 1 mm$ wide. Find $Z_{0}$ , $L^{'}$ and $C^{'}$ , neglecting fringing.

$η = 377/ 4.4 = 179.7 Ω$ , $Z_{0} = 179.7 \times 0.2/1 = 35.9 Ω$ , $L^{'} = μ_{0} b / w = 251 nH/m$ and $C^{'} = εw / b = 195 pF/m$ . Check: $L^{'} / C^{'} = 35.9 Ω$ .

Application 7.2.1 · Parallel plates as TEM lines

Striplines and printed-board plane pairs. A stripline, a strip buried between two ground planes, and a power and ground plane pair of a printed board are parallel-plate guides. Their wave is TEM: no cutoff, no dispersion, the speed $v = c / ε_{r}$ , and an impedance set by the geometry. Higher modes need $f > f_{c 1} = v /2 b$ , far above any signal: $b = 0.2 mm$ of FR-4 ( $ε_{r} = 4.4$ ) gives $f_{c 1} = 357 GHz$ .

Parallel-plate TEM line

Z_{0} t_{d} \approx \frac{η b}{w} (wide plates), f_{c 1} = \frac{v}{2 b}, = \frac{ε _{r}}{c} per metre . (7.2.30)

For FR-4, $t_{d} = 7.0 ns/m$ , that is $70 ps/cm$ : the delay budget of every high-speed bus is counted in these units.

EMC test cells. A large plate, the septum, held over a ground plane and tapered at both ends to a coaxial source and a matched load, creates a known uniform field $E = V / b$ for immunity tests of car electronics and other equipment (automotive striplines, TEM and GTEM cells).

The TEM field is uniform only while TM $_{1}$ and TE $_{1}$ cannot propagate: $b = 0.15 m$ in air gives $f_{c 1} = 1 GHz$ , and cells are used well below it, at a few hundred MHz. Bigger cells for bigger devices work only to lower frequencies.
The field per watt follows from $P = \frac{1}{2} E^{2} b w / η$ (Worked example 7.2.1).

(b) EMC stripline (TEM cell), side view. — (a) Stripline cross-section.

7.2.5 The TEM mode as a transmission line