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

Recipe · Fields of any uniform guide

Solve $\nabla_{T}^{2} E_{z} + h^{2} E_{z} = 0$ (TM) or $\nabla_{T}^{2} H_{z} + h^{2} H_{z} = 0$ (TE) over the cross-section.
Apply the boundary conditions on the walls: they fix the allowed values of $h$ , the modes. On a perfect conductor, TM: $E_{z} = 0$ on the wall. TE: $\partial H_{z} / \partial n = 0$ on the wall (it makes the tangential transverse $E$ vanish, by Eq. (7.1.11)).
Find the transverse fields from the four formulas (7.1.11).
Compute $f_{c} = h / (2 π μ ε)$ , then everything else from Table 7.1.1.

Table 7.1.1. Universal properties of a TM or TE mode with cutoff frequency $f_{c}$ . Only $f_{c}$ depends on the shape of the guide. TEM: $τ = k$ , $Z = η$ , $v_{p} = v_{g} = v$ .

	TM	TE
$τ$	$k 1 - (f_{c} / f)^{2}$	$k 1 - (f_{c} / f)^{2}$
$λ_{g}$	$λ / 1 - (f_{c} / f)^{2}$	$λ / 1 - (f_{c} / f)^{2}$
$v_{p}$	$v / 1 - (f_{c} / f)^{2}$	$v / 1 - (f_{c} / f)^{2}$
$v_{g}$	$v 1 - (f_{c} / f)^{2}$	$v 1 - (f_{c} / f)^{2}$
$Z$	$η 1 - (f_{c} / f)^{2}$	$η / 1 - (f_{c} / f)^{2}$
below $f_{c}$	$α = h 1 - (f / f_{c})^{2}$	$α = h 1 - (f / f_{c})^{2}$

Application 7.1.1 · Pulses in a dispersive waveguide

A pulse with carrier $f_{0}$ and bandwidth $B$ travels at the group velocity. Each spectral component needs its own time $t_{g} = L / v_{g}$ , the group delay, to cross a guide of length $L$ , and the slower, lower frequencies arrive last. The spread is roughly the difference of $t_{g}$ across the band. Differentiating $t_{g} = L / (v s)$ with $s = 1 - (f_{c} / f)^{2}$ and $d s / df = (f_{c}^{2} / f^{3}) / s$ ,

Group delay and pulse spreading

t_{g} (f) Δ t = \frac{L}{v 1 - ( f _{c} / f ) ^{2}}, \approx \frac{d t _{g}}{df} B = \frac{L}{v} \frac{f _{c}^{2} / f ^{3}}{( 1 - ( f _{c} / f ) ^{2} ) ^{3/2}} B . (7.1.24)

(a) A short pulse before and after the guide.

(b) Group delay against frequency. — (a) A short pulse before and after the guide.

Typical numbers. A radar feed of $L = 30 m$ of WR-90 guide (TE $_{10}$ mode, $f_{c} = 6.562 GHz$ , Section 7.3) at $9.4 GHz$ has $t_{g} = 139.7 ns$ , against $100 ns$ in free space, and $d t_{g} / df = 14.1 ns/GHz$ (Worked example 7.1.1).

A $1 μ s$ radar pulse, $B \approx 1 MHz$ : $Δ t \approx 14 ps$ , harmless.
A $2 ns$ pulse, $B \approx 500 MHz$ : $Δ t \approx 7 ns$ , the pulse is destroyed.

The remedies are to stay well above cutoff (the standard band is $1.25$ to $1.9 f_{c}$ ), to keep wideband runs short, or to equalise the dispersion in the receiver.

7.1.8 Summary of the general analysis