Editing Wave impedance

{{Short description|Constant related to electromagnetic wave propagation in a medium}}
The '''wave impedance''' of an [[electromagnetic wave]] is the [[ratio]] of the  transverse components of the [[electric field|electric]] and [[magnetic field]]s (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ([[Transverse mode|TEM]]) [[plane wave]] traveling through a homogeneous [[medium (optics)|medium]], the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the [[impedance of free space]]. The symbol ''Z''  is used to represent it and it is expressed in units of [[ohm]]s. The symbol ''η'' ([[eta]]) may be used instead of ''Z'' for wave impedance to avoid confusion with [[electrical impedance]].

== Definition ==
[[File:Impedance mismatch due to absorption.gif|300px|thumb|right|alt=Impedance mismatch leads to reflections.|To avoid reflections, the impedance of two media must match. On the other hand, even if the real part of the refractive index is the same, but one has a large absorption coefficient, the impedance mismatch will make the interface highly reflective.]]
The wave impedance is given by
: <math>Z = {E_0^-(x) \over H_0^-(x)}</math>
where <math>E_0^-(x)</math> is the electric field and <math>H_0^-(x)</math> is the magnetic field, in [[phasor]] representation. The impedance is, in general, a [[complex number]].

In terms of the parameters of an electromagnetic wave and the medium it travels through, the wave impedance is given by
: <math>Z = \sqrt {j \omega \mu \over \sigma + j \omega \varepsilon} </math>
where ''μ'' is the [[magnetic permeability]], ''ε'' is the (real) [[permittivity|electric permittivity]] and ''σ'' is the [[electrical conductivity]] of the material the wave is travelling through (corresponding to the imaginary component of the permittivity multiplied by omega). In the equation, ''j'' is the [[imaginary unit]], and ''ω'' is the [[angular frequency]] of the wave. Just as for [[electrical impedance]], the impedance is a function of frequency. In the case of an ideal [[dielectric]] (where the conductivity is zero), the equation reduces to the real number
: <math>Z = \sqrt {\mu \over \varepsilon }.</math>

== In free space ==
{{main|Impedance of free space}}

In [[free space]] the wave impedance of plane waves is: 
: <math>Z_0 = \sqrt{\frac{\mu_0} {\varepsilon_0}}</math>
(where ''ε''<sub>0</sub> is the [[permittivity constant]] in free space and ''μ''<sub>0</sub> is the [[permeability constant]] in free space). Now, since
: <math>c_0 = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = 299\,792\,458\text{ m/s}</math> (by definition of the [[metre]]),
: <math>Z_0 = \mu_0 c_0 = \frac{1}{\varepsilon_0 c_0}</math>.

The currently accepted value of <math>Z_0</math> is {{physconst|Z0|after=.}}

== In an unbounded dielectric ==

In an [[isotropic]], [[homogeneous]] [[dielectric]] with negligible magnetic properties, i.e. <math>\mu = \mu_0 </math> and <math>\varepsilon = \varepsilon_r \varepsilon_0</math>. So, the value of wave impedance in a perfect dielectric is
: <math>Z = \sqrt {\mu \over \varepsilon} = \sqrt {\mu_0 \over \varepsilon_0 \varepsilon_r} = {Z_0 \over \sqrt{\varepsilon_r}} \approx {377 \over \sqrt {\varepsilon_r} }\,\Omega</math>,
where <math>\varepsilon_r</math> is the relative [[dielectric constant]].

== 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.

== See also ==
* [[Characteristic impedance]]
* [[Impedance (disambiguation)]]
* [[Impedance of free space]]

== References ==
{{reflist}}
{{FS1037C MS188}}

[[Category:Wave mechanics]]
[[Category:Electromagnetic radiation]]