Editing Wave impedance (section)

== In a waveguide ==

For any [[waveguide (electromagnetism)|waveguide]] in the form of a hollow metal tube, (such as rectangular guide, circular guide, or double-ridge guide), the wave impedance of a travelling wave is dependent on the frequency <math>f</math>, but is the same throughout the guide. For transverse electric ([[Transverse mode|TE]]) modes of propagation the wave impedance is:<ref name="Microwave Engineering">{{Cite book|last=Pozar |first=David M. |author-link=David M. Pozar|title=Microwave engineering|date=2012|publisher=Wiley|isbn=978-0-470-63155-3|edition=4th |location=Hoboken, NJ|oclc=714728044|pages=100–101}}</ref>
: <math>Z = \frac{Z_{0}}{\sqrt{1 - \left( \frac{f_{c}}{f}\right)^{2}}} \qquad \mbox{(TE modes)},</math>
where ''f''<sub>''c''</sub> is the cut-off frequency of the mode, and for transverse magnetic ([[Transverse mode|TM]]) modes of propagation the wave impedance is:<ref name="Microwave Engineering"/>
: <math>Z = Z_{0} \sqrt{1 - \left( \frac{f_{c}}{f}\right)^{2}} \qquad \mbox{(TM modes)}</math>

Above the cut-off ({{nowrap|''f'' > ''f''<sub>''c''</sub>}}), the impedance is real (resistive) and the wave carries energy. Below cut-off the impedance is imaginary (reactive) and the wave is [[evanescent wave|evanescent]]. These expressions neglect the effect of resistive loss in the walls of the waveguide. For a waveguide entirely filled with a homogeneous dielectric medium, similar expressions apply, but with the wave impedance of the medium replacing ''Z''<sub>0</sub>. The presence of the dielectric also modifies the cut-off frequency ''f''<sub>''c''</sub>.

For a waveguide or transmission line containing more than one type of dielectric medium (such as [[microstrip]]), the wave impedance will in general vary over the cross-section of the line.