Wave impedance

The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic (TEM) plane wave traveling through a homogeneous 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 ohms. The symbol η (eta) may be used instead of Z for wave impedance to avoid confusion with electrical impedance.

Definition

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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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z={E_{0}^{-}(x) \over H_{0}^{-}(x)}}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle E_{0}^{-}(x)} is the electric field and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H_{0}^{-}(x)} 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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z={\sqrt {j\omega \mu \over \sigma +j\omega \varepsilon }}}

where μ is the magnetic permeability, ε is the (real) 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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z={\sqrt {\mu \over \varepsilon }}.}

In free space

Main article: Impedance of free space

In free space the wave impedance of plane waves is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}}

(where ε₀ is the permittivity constant in free space and μ₀ is the permeability constant in free space). Now, since

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=299\,792\,458{\text{ m/s}}} (by definition of the metre),

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{0}=\mu _{0}c={\frac {1}{\varepsilon _{0}c}}} .

The currently accepted value of $Z_{0}$ is Template:Physconst

In an unbounded dielectric

In an isotropic, homogeneous dielectric with negligible magnetic properties, i.e. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mu =\mu _{0}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \varepsilon =\varepsilon _{r}\varepsilon _{0}} . So, the value of wave impedance in a perfect dielectric is

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 } ,

where $\varepsilon _{r}$ is the relative dielectric constant.

In a waveguide

For any 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 $f$, but is the same throughout the guide. For transverse electric (TE) modes of propagation the wave impedance is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z={\frac {Z_{0}}{\sqrt {1-\left({\frac {f_{c}}{f}}\right)^{2}}}}\qquad {\mbox{(TE modes)}},}

where f_c is the cut-off frequency of the mode, and for transverse magnetic (TM) modes of propagation the wave impedance is:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z=Z_{0}{\sqrt {1-\left({\frac {f_{c}}{f}}\right)^{2}}}\qquad {\mbox{(TM modes)}}}

Above the cut-off (f > f_c), the impedance is real (resistive) and the wave carries energy. Below cut-off the impedance is imaginary (reactive) and the wave is 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 Z₀. The presence of the dielectric also modifies the cut-off frequency f_c.

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

 This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C". (in support of MIL-STD-188)