Section 7.1 · Further exercises · 1 of 5
Exercise 7.1.1: TE or TM? Reading the mode from its wave impedance
Five further exercises on the general analysis of uniform guides, graded from easy to harder. None of them needs the shape of the cross-section: (or a measurement) is all that is known.
TE or TM? Reading the mode from its wave impedance
#An air-filled hollow metal guide of unknown cross-section carries a single mode at . Two small probes find that the transverse electric and magnetic fields are in phase, with . (a) Is the mode TE or TM? (b) Find its cutoff frequency, , and . (c) The guide has a second mode with the same cutoff but of the other family. What would the probes read for it? (d) At what frequency would the first mode give ? (e) What do the probes read for the first mode at , and what is the phase between and then?
Above cutoff and , so a single real number decides the family and gives . Everything else follows from alone, whatever the shape of the guide.
: the mode is TE. In phase means is real, so the mode is above cutoff.
, hence and . With ,
The TM mode with the same cutoff has the same : . In general , here .
needs , that is .
At the mode is evanescent. and
The probes read , with leading by .
. Below cutoff
the same value by a second route.
The wave impedance is the one quantity that tells TE from TM without knowing the cross-section. Below cutoff the two families differ in the sign of the reactance: TE inductive, TM capacitive. A quadrature reading therefore says "evanescent", and its sign says which family.
cf. wave impedances